diff --git a/changelog b/changelog index 160ae63..c4acf60 100644 --- a/changelog +++ b/changelog @@ -1,3 +1,4 @@ +20081122 tpd src/axiom-website put website under change control 20081122 tpd zips/gcl-2.6.8pre3.unixport.makefile.patch added 20081122 tpd zips/gcl-2.6.8pre3.unixport.init_gcl.lsp.in.patch added 20081122 tpd zips/gcl-2.6.8pre3.cmpnew.gcl_cmpflet.lsp.patch added diff --git a/src/axiom-website/CATS/index.html b/src/axiom-website/CATS/index.html new file mode 100644 index 0000000..b74f1c8 --- /dev/null +++ b/src/axiom-website/CATS/index.html @@ -0,0 +1,128 @@ + + + + (CATS) Computer Algebra Test Suite + + + +This work is part of the Computer Algebra Test Suite (CATS). +These files show the results that Axiom computes given the +set of integrals listed in +
+  Spiegel, Murray R. 
+  Mathematical Handbook of Formulas and Tables
+  Schaum's Outline Series McGraw-Hill 1968 
+
+

+Each integral is computed by Axiom and compared against the +published result. +

+Each Axiom result is differenced from the published result +and reduced to a constant (usually 0). +

+ + + Schaums 14.59-14.83  + source + pdf
+ Schaums 14.84-14.104  + source + pdf
+ Schaums 14.104-14.112  + source + pdf
+ Schaums 14.113-.119  + source + pdf
+ Schaums 14.120-14.124  + source + pdf
+ Schaums 14.125-14.143  + source + pdf
+ Schaums 14.144-14.162  + source + pdf
+ Schaums 14.163-14.181  + source + pdf
+ Schaums 14.182-14.209  + source + pdf
+ Schaums 14.210-14.236  + source + pdf
+ Schaums 14.237-14.264  + source + pdf
+ Schaums 14.265-14.279  + source + pdf
+ Schaums 14.280-14.298  + source + pdf
+ Schaums 14.299-14.310  + source + pdf
+ Schaums 14.311-14.324  + source + pdf
+ Schaums 14.325-14.338  + source + pdf
+ Schaums 14.339-14.368  + source + pdf
+ Schaums 14.369-14.398  + source + pdf
+ Schaums 14.399-14.428  + source + pdf
+ Schaums 14.429-14.439  + source + pdf
+ Schaums 14.440-14.450  + source + pdf
+ Schaums 14.451-14.460  + source + pdf
+ Schaums 14.461-14.470  + source + pdf
+ Schaums 14.471-14.508  + source + pdf
+ Schaums 14.509-14.524  + source + pdf
+ Schaums 14.525-14.539  + source + pdf
+ Schaums 14.540-14.561  + source + pdf
+ Schaums 14.562-14.589  + source + pdf
+ Schaums 14.590-14.603  + source + pdf
+ Schaums 14.604-14.614  + source + pdf
+ Schaums 14.615-14.625  + source + pdf
+ Schaums 14.626-14.635  + source + pdf
+ Schaums 14.636-14.645  + source + pdf
+ Schaums 14.646-14.677  + source + pdf
+ + \ No newline at end of file diff --git a/src/axiom-website/CATS/schaum1.input.pamphlet b/src/axiom-website/CATS/schaum1.input.pamphlet new file mode 100644 index 0000000..615d907 --- /dev/null +++ b/src/axiom-website/CATS/schaum1.input.pamphlet @@ -0,0 +1,1430 @@ +\documentclass{article} +\usepackage{axiom} +\begin{document} +\title{\$SPAD/input schaum1.input} +\author{Timothy Daly} +\maketitle +\eject +\tableofcontents +\eject +\section{\cite{1}:14.59~~~~~$\displaystyle +\int{\frac{dx}{ax+b}}$} +$$\int{\frac{1}{ax+b}}= +\frac{1}{a}~\ln(ax+b) +$$ +<<*>>= +)spool schaum1.output +)set message test on +)set message auto off +)clear all + +--S 1 +aa:=integrate(1/(a*x+b),x) +--R +--R log(a x + b) +--R (1) ------------ +--R a +--R Type: Union(Expression Integer,...) +--E 1 + +--S 2 +bb:=1/a*log(a*x+b) +--R +--R log(a x + b) +--R (2) ------------ +--R a +--R Type: Expression Integer +--E + +--S 3 14:59 Schaums and Axiom agree +cc:=bb-aa +--R +--R (3) 0 +--R Type: Expression Integer +--E +@ +\section{\cite{1}:14.60~~~~~$\displaystyle +\int{\frac{x~dx}{ax+b}}$} +$$\int{\frac{x}{ax+b}}= +\frac{x}{a}-\frac{b}{a^2}~\ln(ax+b) +$$ +<<*>>= +)clear all + +--S 4 +aa:=integrate(x/(a*x+b),x) +--R +--R +--R - b log(a x + b) + a x +--R (1) ---------------------- +--R 2 +--R a +--R Type: Union(Expression Integer,...) +--E + +--S 5 +bb:=x/a-b/a^2*log(a*x+b) +--R +--R - b log(a x + b) + a x +--R (2) ---------------------- +--R 2 +--R a +--R Type: Expression Integer +--E + +--S 6 14:60 Schaums and Axiom agree +cc:=bb-aa +--R +--R (3) 0 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.61~~~~~$\displaystyle +\int{\frac{x^2~dx}{ax+b}}$} +$$\int{\frac{x^2}{ax+b}}= +\frac{(ax+b)^2}{2a^3}-\frac{2b(ax+b)}{a^3}+\frac{b^2}{a^3}~\ln(ax+b) +$$ +<<*>>= +)clear all + +--S 7 +aa:=integrate(x^2/(a*x+b),x) +--R +--R 2 2 2 +--R 2b log(a x + b) + a x - 2a b x +--R (1) ------------------------------- +--R 3 +--R 2a +--R Type: Union(Expression Integer,...) +--E + +--S 8 +bb:=(a*x+b)^2/(2*a^3)-(2*b*(a*x+b))/a^3+b^2/a^3*log(a*x+b) +--R +--R 2 2 2 2 +--R 2b log(a x + b) + a x - 2a b x - 3b +--R (2) ------------------------------------- +--R 3 +--R 2a +--R Type: Expression Integer +--E + +--S 9 +cc:=bb-aa +--R +--R 2 +--R 3b +--R (3) - --- +--R 3 +--R 2a +--R Type: Expression Integer +--E +@ +This factor is constant with respect to $x$ as shown by taking the +derivative. It is a constant of integration. +<<*>>= +--S 10 14:61 Schaums and Axiom differ by a constant +differentiate(cc,x) +--R +--R (4) 0 +--R Type: Expression Integer +--E +@ +\section{\cite{1}:14.62~~~~~$\displaystyle +\int{\frac{x^3~dx}{ax+b}}$} +$$\int{\frac{x^3}{ax+b}}= +\frac{(ax+b)^3}{3a^4}-\frac{3b(ax+b)^2}{2a^4}+ +\frac{3b^2(ax+b)}{a^4}-\frac{b^3}{a^4}~\ln(ax+b) +$$ +<<*>>= +)clear all + +--S 11 +aa:=integrate(x^3/(a*x+b),x) +--R +--R 3 3 3 2 2 2 +--R - 6b log(a x + b) + 2a x - 3a b x + 6a b x +--R (1) -------------------------------------------- +--R 4 +--R 6a +--R Type: Union(Expression Integer,...) +--E +@ +and the book expression is: +<<*>>= +--S 12 +bb:=(a*x+b)^3/(3*a^4)-(3*b*(a*x+b)^2)/(2*a^4)+(3*b^2*(a*x+b))/a^4-(b^3/a^4)*log(a*x+b) +--R +--R 3 3 3 2 2 2 3 +--R - 6b log(a x + b) + 2a x - 3a b x + 6a b x + 11b +--R (2) --------------------------------------------------- +--R 4 +--R 6a +--R Type: Expression Integer +--E +@ + +The difference is a constant with respect to x: +<<*>>= +--S 13 +cc:=aa-bb +--R +--R 3 +--R 11b +--R (3) - ---- +--R 4 +--R 6a +--R Type: Expression Integer +--E +@ + +If we differentiate each expression we see that this is the integration +constant. +<<*>>= +--S 14 14:62 Schaums and Axiom differ by a constant +dd:=D(cc,x) +--R +--R (4) 0 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.63~~~~~$\displaystyle +\int{\frac{dx}{x~(ax+b)}}$} +$$\int{\frac{1}{x~(ax+b)}}= +\frac{1}{b}~\ln\left(\frac{x}{ax+b}\right) +$$ +<<*>>= +)clear all + +--S 15 +aa:=integrate(1/(x*(a*x+b)),x) +--R +--R - log(a x + b) + log(x) +--R (1) ----------------------- +--R b +--R Type: Union(Expression Integer,...) +--E + +--S 16 +bb:=1/b*log(x/(a*x+b)) +--R +--R x +--R log(-------) +--R a x + b +--R (2) ------------ +--R b +--R Type: Expression Integer +--E + +--S 17 +cc:=aa-bb +--R +--R x +--R - log(a x + b) + log(x) - log(-------) +--R a x + b +--R (3) -------------------------------------- +--R b +--R Type: Expression Integer +--E +@ +but we know that $$\log(a)-\log(b)=\log(\frac{a}{b})$$ + +We can express this fact as a rule: +<<*>>= +--S 18 +logdiv:=rule(log(a)-log(b) == log(a/b)) +--R +--R a +--I (4) - log(b) + log(a) + %I == log(-) + %I +--R b +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E +@ +and use this rule to rewrite the logs into divisions: +<<*>>= +--S 19 14:63 Schaums and Axiom agree +dd:=logdiv cc +--R +--R (5) 0 +--R Type: Expression Integer +--E +@ +so we can see the equivalence directly. + +\section{\cite{1}:14.64~~~~~$\displaystyle +\int{\frac{dx}{x^2~(ax+b)}}$} +$$\int{\frac{1}{x^2~(ax+b)}}= +-\frac{1}{bx}+\frac{a}{b^2}~\ln\left(\frac{ax+b}{x}\right) +$$ +<<*>>= +)clear all + +--S 20 +aa:=integrate(1/(x^2*(a*x+b)),x) +--R +--R a x log(a x + b) - a x log(x) - b +--R (1) --------------------------------- +--R 2 +--R b x +--R Type: Union(Expression Integer,...) +--E +@ + +The original form given in the book expands to: +<<*>>= +--S 21 +bb:=-1/(b*x)+a/b^2*log((a*x+b)/x) +--R +--R a x + b +--R a x log(-------) - b +--R x +--R (2) -------------------- +--R 2 +--R b x +--R Type: Expression Integer +--E + +--S 22 +cc:=aa-bb +--R +--R a x + b +--R a log(a x + b) - a log(x) - a log(-------) +--R x +--R (3) ------------------------------------------ +--R 2 +--R b +--R Type: Expression Integer +--E +@ + +We can define the following rule to expand log forms: +<<*>>= +--S 23 +divlog:=rule(log(a/b) == log(a) - log(b)) +--R +--R a +--R (4) log(-) == - log(b) + log(a) +--R b +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E +@ +and apply it to the difference +<<*>>= +--S 24 14:64 Schaums and Axiom agree +divlog cc +--R +--R (5) 0 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.65~~~~~$\displaystyle +\int{\frac{dx}{x^3~(ax+b)}}$} +$$\int{\frac{1}{x^3~(ax+b)}}= +\frac{2ax-b}{2b^2x^2}+\frac{a^2}{b^3}~\ln\left(\frac{x}{ax+b}\right) +$$ +<<*>>= +)clear all +--S 25 +aa:=integrate(1/(x^3*(a*x+b)),x) +--R +--R 2 2 2 2 2 +--R - 2a x log(a x + b) + 2a x log(x) + 2a b x - b +--R (1) ----------------------------------------------- +--R 3 2 +--R 2b x +--R Type: Union(Expression Integer,...) +--E + +--S 26 +bb:=(2*a*x-b)/(2*b^2*x^2)+a^2/b^3*log(x/(a*x+b)) +--R +--R 2 2 x 2 +--R 2a x log(-------) + 2a b x - b +--R a x + b +--R (2) ------------------------------- +--R 3 2 +--R 2b x +--R Type: Expression Integer +--E + +--S 27 +cc:=aa-bb +--R +--R 2 2 2 x +--R - a log(a x + b) + a log(x) - a log(-------) +--R a x + b +--R (3) -------------------------------------------- +--R 3 +--R b +--R Type: Expression Integer +--E + +--S 28 +divlog:=rule(log(a/b) == log(a) - log(b)) +--R +--R a +--R (4) log(-) == - log(b) + log(a) +--R b +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 29 14:65 Schaums and Axiom agree +dd:=divlog cc +--R +--R (5) 0 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.66~~~~~$\displaystyle +\int{\frac{dx}{(ax+b)^2}}$} +$$\int{\frac{1}{(ax+b)^2}}= +\frac{-1}{a~(ax+b)} +$$ +<<*>>= +)clear all + +--S 30 +aa:=integrate(1/(a*x+b)^2,x) +--R +--R 1 +--R (1) - --------- +--R 2 +--R a x + a b +--R Type: Union(Expression Integer,...) +--E + +--S 31 +bb:=-1/(a*(a*x+b)) +--R +--R 1 +--R (2) - --------- +--R 2 +--R a x + a b +--R Type: Fraction Polynomial Integer +--E + +--S 32 14:66 Schaums and Axiom agree +cc:=aa-bb +--R +--R (3) 0 +--R Type: Expression Integer +--E + +@ + +\section{\cite{1}:14.67~~~~~$\displaystyle +\int{\frac{x~dx}{(ax+b)^2}}$} +$$\int{\frac{x}{(ax+b)^2}}= +\frac{b}{a^2~(ax+b)}+\frac{1}{a^2}~\ln(ax+b) +$$ +<<*>>= +)clear all + +--S 33 +aa:=integrate(x/(a*x+b)^2,x) +--R +--R (a x + b)log(a x + b) + b +--R (1) ------------------------- +--R 3 2 +--R a x + a b +--R Type: Union(Expression Integer,...) +--E + +--S 34 +bb:=b/(a^2*(a*x+b))+1/a^2*log(a*x+b) +--R +--R (a x + b)log(a x + b) + b +--R (2) ------------------------- +--R 3 2 +--R a x + a b +--R Type: Expression Integer +--E + +--S 35 14:67 Schaums and Axiom agree +cc:=aa-bb +--R +--R (3) 0 +--R Type: Expression Integer +--E + +@ + +\section{\cite{1}:14.68~~~~~$\displaystyle +\int{\frac{x^2~dx}{(ax+b)^2}}$} +$$\int{\frac{x^2}{(ax+b)^2}}= +\frac{ax+b}{a^3}-\frac{b^2}{a^3~(ax+b)} +-\frac{2b}{a^3}~\ln(ax+b) +$$ +<<*>>= +)clear all + +--S 36 +aa:=integrate(x^2/(a*x+b)^2,x) +--R +--R 2 2 2 2 +--R (- 2a b x - 2b )log(a x + b) + a x + a b x - b +--R (1) ------------------------------------------------ +--R 4 3 +--R a x + a b +--R Type: Union(Expression Integer,...) +--E +@ +and the book expression expands into +<<*>>= +--S 37 +bb:=(a*x+b)/a^3-b^2/(a^3*(a*x+b))-((2*b)/a^3)*log(a*x+b) +--R +--R 2 2 2 +--R (- 2a b x - 2b )log(a x + b) + a x + 2a b x +--R (2) -------------------------------------------- +--R 4 3 +--R a x + a b +--R Type: Expression Integer +--E +@ + +These two expressions differ by the constant +<<*>>= +--S 38 +cc:=aa-bb +--R +--R b +--R (3) - -- +--R 3 +--R a +--R Type: Expression Integer +--E +@ + +That this expression is constant can be shown by differentiation: +<<*>>= +--S 39 14:68 Schaums and Axiom differ by a constant +D(cc,x) +--R +--R (4) 0 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.69~~~~~$\displaystyle +\int{\frac{x^3~dx}{(ax+b)^2}}$} +$$\int{\frac{x^3}{(ax+b)^2}}= +\frac{(ax+b)^2}{2a^4}-\frac{3b(ax+b)}{a^4}+\frac{b^3}{a^4(ax+b)} ++\frac{3b^2}{a^4}~\ln(ax+b) +$$ +<<*>>= +)clear all + +--S 40 +aa:=integrate(x^3/(a*x+b)^2,x) +--R +--R 2 3 3 3 2 2 2 3 +--R (6a b x + 6b )log(a x + b) + a x - 3a b x - 4a b x + 2b +--R (1) ---------------------------------------------------------- +--R 5 4 +--R 2a x + 2a b +--R Type: Union(Expression Integer,...) +--E + +--S 41 +bb:=(a*x+b)^2/(2*a^4)-(3*b*(a*x+b))/a^4+b^3/(a^4*(a*x+b))+(3*b^2/a^4)*log(a*x+b) +--R +--R 2 3 3 3 2 2 2 3 +--R (6a b x + 6b )log(a x + b) + a x - 3a b x - 9a b x - 3b +--R (2) ---------------------------------------------------------- +--R 5 4 +--R 2a x + 2a b +--R Type: Expression Integer +--E + +--S 42 +cc:=aa-bb +--R +--R 2 +--R 5b +--R (3) --- +--R 4 +--R 2a +--R Type: Expression Integer +--E + +--S 43 14:69 Schaums and Axiom differ by a constant +dd:=D(cc,x) +--R +--R (4) 0 +--R Type: Expression Integer +--E +@ +\section{\cite{1}:14.70~~~~~$\displaystyle +\int{\frac{dx}{x~(ax+b)^2}}$} +$$\int{\frac{1}{x~(ax+b)^2}}= +\frac{1}{b~(ax+b)}+\frac{1}{b^2}~\ln\left(\frac{x}{ax+b}\right) +$$ +<<*>>= +)clear all + +--S 44 +aa:=integrate(1/(x*(a*x+b)^2),x) +--R +--R (- a x - b)log(a x + b) + (a x + b)log(x) + b +--R (1) --------------------------------------------- +--R 2 3 +--R a b x + b +--R Type: Union(Expression Integer,...) +--E +@ +and the book says: +<<*>>= +--S 45 +bb:=(1/(b*(a*x+b))+(1/b^2)*log(x/(a*x+b))) +--R +--R x +--R (a x + b)log(-------) + b +--R a x + b +--R (2) ------------------------- +--R 2 3 +--R a b x + b +--R Type: Expression Integer +--E + +--S 46 +cc:=aa-bb +--R +--R x +--R - log(a x + b) + log(x) - log(-------) +--R a x + b +--R (3) -------------------------------------- +--R 2 +--R b +--R Type: Expression Integer +--E +@ +So we look at the divlog rule again: +<<*>>= +--S 47 +divlog:=rule(log(a/b) == log(a) - log(b)) +--R +--R a +--R (4) log(-) == - log(b) + log(a) +--R b +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E +@ + +we apply it: +<<*>>= +--S 48 14:70 Schaums and Axiom agree +dd:=divlog cc +--R +--R (5) 0 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.71~~~~~$\displaystyle +\int{\frac{dx}{x^2~(ax+b)^2}}$} +$$\int{\frac{1}{x^2~(ax+b)^2}}= +\frac{-a}{b^2~(ax+b)}-\frac{1}{b^2~x}+ +\frac{2a}{b^3}~\ln\left(\frac{ax+b}{x}\right) +$$ +<<*>>= +)clear all + +--S 49 +aa:=integrate(1/(x^2*(a*x+b)^2),x) +--R +--R 2 2 2 2 2 +--R (2a x + 2a b x)log(a x + b) + (- 2a x - 2a b x)log(x) - 2a b x - b +--R (1) --------------------------------------------------------------------- +--R 3 2 4 +--R a b x + b x +--R Type: Union(Expression Integer,...) +--E +@ +and the book says: +<<*>>= +--S 50 +bb:=(-a/(b^2*(a*x+b)))-(1/(b^2*x))+((2*a)/b^3)*log((a*x+b)/x) +--R +--R 2 2 a x + b 2 +--R (2a x + 2a b x)log(-------) - 2a b x - b +--R x +--R (2) ------------------------------------------ +--R 3 2 4 +--R a b x + b x +--R Type: Expression Integer +--E + +--S 51 +cc:=aa-bb +--R +--R a x + b +--R 2a log(a x + b) - 2a log(x) - 2a log(-------) +--R x +--R (3) --------------------------------------------- +--R 3 +--R b +--R Type: Expression Integer +--E +@ +which calls for our divlog rule: +<<*>>= +--S 52 +divlog:=rule(log(a/b) == log(a) - log(b)) +--R +--R a +--R (4) log(-) == - log(b) + log(a) +--R b +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E +@ +which we use to transform the result: +<<*>>= +--S 53 14:71 Schaums and Axiom agree +dd:=divlog cc +--R +--R (5) 0 +--R Type: Expression Integer +--E +@ +\section{\cite{1}:14.72~~~~~$\displaystyle +\int{\frac{dx}{x^3~(ax+b)^2}}$} +$$\int{\frac{1}{x^3~(ax+b)^2}}= +-\frac{(ax+b)^2}{2b^4x^2}+\frac{3a(ax+b)}{b^4x}- +\frac{a^3x}{b^4(ax+b)}-\frac{3a^2}{b^4}~\ln\left(\frac{ax+b}{x}\right) +$$ +<<*>>= +)clear all + +--S 54 +aa:=integrate(1/(x^3*(a*x+b)^2),x) +--R +--R (1) +--R 3 3 2 2 3 3 2 2 2 2 +--R (- 6a x - 6a b x )log(a x + b) + (6a x + 6a b x )log(x) + 6a b x +--R + +--R 2 3 +--R 3a b x - b +--R / +--R 4 3 5 2 +--R 2a b x + 2b x +--R Type: Union(Expression Integer,...) +--E + +--S 55 +bb:=-(a*x+b)^2/(2*b^4*x^2)+(3*a*(a*x+b))/(b^4*x)-(a^3*x)/(b^4*(a*x+b))-((3*a^2)/b^4)*log((a*x+b)/x) +--R +--R 3 3 2 2 a x + b 3 3 2 2 2 3 +--R (- 6a x - 6a b x )log(-------) + 3a x + 9a b x + 3a b x - b +--R x +--R (2) --------------------------------------------------------------- +--R 4 3 5 2 +--R 2a b x + 2b x +--R Type: Expression Integer +--E + +--S 56 +cc:=aa-bb +--R +--R 2 2 2 a x + b 2 +--R - 6a log(a x + b) + 6a log(x) + 6a log(-------) - 3a +--R x +--R (3) ----------------------------------------------------- +--R 4 +--R 2b +--R Type: Expression Integer +--E + +--S 57 +divlog:=rule(log(a/b) == log(a) - log(b)) +--R +--R a +--R (4) log(-) == - log(b) + log(a) +--R b +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 58 +dd:=divlog cc +--R +--R 2 +--R 3a +--R (5) - --- +--R 4 +--R 2b +--R Type: Expression Integer +--E + +--S 59 14:72 Schaums and Axiom differ by a constant +ee:=D(dd,x) +--R +--R (6) 0 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.73~~~~~$\displaystyle +\int{\frac{dx}{(ax+b)^3}}$} +$$\int{\frac{1}{(ax+b)^3}}= +\frac{-1}{2a(ax+b)^2} +$$ +<<*>>= +)clear all + +--S 60 +aa:=integrate(1/(a*x+b)^3,x) +--R +--R 1 +--R (1) - ---------------------- +--R 3 2 2 2 +--R 2a x + 4a b x + 2a b +--R Type: Union(Expression Integer,...) +--E + +--S 61 +bb:=-1/(2*(a*x+b)^2) +--R +--R 1 +--R (2) - -------------------- +--R 2 2 2 +--R 2a x + 4a b x + 2b +--R Type: Fraction Polynomial Integer +--E + +--S 62 +cc:=aa-bb +--R +--R a - 1 +--R (3) ---------------------- +--R 3 2 2 2 +--R 2a x + 4a b x + 2a b +--R Type: Expression Integer +--E + +--S 63 +dd:=aa/bb +--R +--R 1 +--R (4) - +--R a +--R Type: Expression Integer +--E + +--S 64 14:73 Schaums and Axiom differ by a constant +ee:=D(dd,x) +--R +--R (5) 0 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.74~~~~~$\displaystyle +\int{\frac{x~dx}{(ax+b)^3}}$} +$$\int{\frac{x}{(ax+b)^3}}= +\frac{-1}{a^2(ax+b)}+\frac{b}{2a^2(ax+b)^2} +$$ +<<*>>= +)clear all + +--S 65 +aa:=integrate(x/(a*x+b)^3,x) +--R +--R - 2a x - b +--R (1) ---------------------- +--R 4 2 3 2 2 +--R 2a x + 4a b x + 2a b +--R Type: Union(Expression Integer,...) +--E + +--S 66 +bb:=-1/(a^2*(a*x+b))+b/(2*a^2*(a*x+b)^2) +--R +--R - 2a x - b +--R (2) ---------------------- +--R 4 2 3 2 2 +--R 2a x + 4a b x + 2a b +--R Type: Fraction Polynomial Integer +--E + +--S 67 14:74 Schaums and Axiom agree +cc:=aa-bb +--R +--R (3) 0 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.75~~~~~$\displaystyle +\int{\frac{x^2~dx}{(ax+b)^3}}$} +$$\int{\frac{x^2}{(ax+b)^3}}= +\frac{2b}{a^3(ax+b)}-\frac{b^2}{2a^3(ax+b)^2}+ +\frac{1}{a^3}~\ln(ax+b) +$$ +<<*>>= +)clear all + +--S 68 +aa:=integrate(x^2/(a*x+b)^3,x) +--R +--R 2 2 2 2 +--R (2a x + 4a b x + 2b )log(a x + b) + 4a b x + 3b +--R (1) ------------------------------------------------- +--R 5 2 4 3 2 +--R 2a x + 4a b x + 2a b +--R Type: Union(Expression Integer,...) +--E + +--S 69 +bb:=(2*b)/(a^3*(a*x+b))-(b^2)/(2*a^3*(a*x+b)^2)+1/a^3*log(a*x+b) +--R +--R 2 2 2 2 +--R (2a x + 4a b x + 2b )log(a x + b) + 4a b x + 3b +--R (2) ------------------------------------------------- +--R 5 2 4 3 2 +--R 2a x + 4a b x + 2a b +--R Type: Expression Integer +--E + +--S 70 14:75 Schaums and Axiom agree +cc:=aa-bb +--R +--R (3) 0 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.76~~~~~$\displaystyle +\int{\frac{x^3~dx}{(ax+b)^3}}$} +$$\int{\frac{x^3}{(ax+b)^3}}= +\frac{x}{a^3}-\frac{3b^2}{a^4(ax+b)}+\frac{b^3}{2a^4(ax+b)^2}- +\frac{3b}{a^4}~\ln(ax+b) +$$ +<<*>>= +)clear all +--S 71 +aa:=integrate(x^3/(a*x+b)^3,x) +--R +--R (1) +--R 2 2 2 3 3 3 2 2 2 3 +--R (- 6a b x - 12a b x - 6b )log(a x + b) + 2a x + 4a b x - 4a b x - 5b +--R ------------------------------------------------------------------------ +--R 6 2 5 4 2 +--R 2a x + 4a b x + 2a b +--R Type: Union(Expression Integer,...) +--E + +--S 72 +bb:=(x/a^3)-(3*b^2)/(a^4*(a*x+b))+b^3/(2*a^4*(a*x+b)^2)-(3*b)/a^4*log(a*x+b) +--R +--R (2) +--R 2 2 2 3 3 3 2 2 2 3 +--R (- 6a b x - 12a b x - 6b )log(a x + b) + 2a x + 4a b x - 4a b x - 5b +--R ------------------------------------------------------------------------ +--R 6 2 5 4 2 +--R 2a x + 4a b x + 2a b +--R Type: Expression Integer +--E + +--S 73 14:76 Schaums and Axiom agree +cc:=aa-bb +--R +--R (3) 0 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.77~~~~~$\displaystyle +\int{\frac{dx}{x(ax+b)^3}}$} +$$\int{\frac{1}{x(ax+b)^3}}= +\frac{3}{2b(ax+b)^2}+\frac{2ax}{2b^2(ax+b)^2}- +\frac{1}{b^3}*\ln\left(\frac{ax+b}{x}\right) +$$ + +<<*>>= +)clear all + +--S 74 +aa:=integrate(1/(x*(a*x+b)^3),x) +--R +--R (1) +--R 2 2 2 2 2 2 +--R (- 2a x - 4a b x - 2b )log(a x + b) + (2a x + 4a b x + 2b )log(x) +--R + +--R 2 +--R 2a b x + 3b +--R / +--R 2 3 2 4 5 +--R 2a b x + 4a b x + 2b +--R Type: Union(Expression Integer,...) +--E + +--S 75 +bb:=(a^2*x^2)/(2*b^3*(a*x+b)^2)-(2*a*x)/(b^3*(a*x+b))-(1/b^3)*log((a*x+b)/x) +--R +--R 2 2 2 a x + b 2 2 +--R (- 2a x - 4a b x - 2b )log(-------) - 3a x - 4a b x +--R x +--R (2) ----------------------------------------------------- +--R 2 3 2 4 5 +--R 2a b x + 4a b x + 2b +--R Type: Expression Integer +--E + +--S 76 +cc:=aa-bb +--R +--R a x + b +--R - 2log(a x + b) + 2log(x) + 2log(-------) + 3 +--R x +--R (3) --------------------------------------------- +--R 3 +--R 2b +--R Type: Expression Integer +--E + +--S 77 +divlog:=rule(log(a/b) == log(a) - log(b)) +--R +--R a +--R (4) log(-) == - log(b) + log(a) +--R b +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 78 +dd:=divlog cc +--R +--R 3 +--R (5) --- +--R 3 +--R 2b +--R Type: Expression Integer +--E + +--S 79 14:77 Schaums and Axiom differ by a constant +ee:=D(dd,x) +--R +--R (6) 0 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.78~~~~~$\displaystyle +\int{\frac{dx}{x^2(ax+b)^3}}$} +$$\int{\frac{1}{x^2(ax+b)^3}}= +\frac{-a}{2b^2(ax+b)^2}-\frac{2a}{b^3(ax+b)}- +\frac{1}{b^3x}+\frac{3a}{b^4}~\ln\left(\frac{ax+b}{x}\right) +$$ +<<*>>= +)clear all + +--S 80 +aa:=integrate(1/(x^2*(a*x+b)^3),x) +--R +--R (1) +--R 3 3 2 2 2 +--R (6a x + 12a b x + 6a b x)log(a x + b) +--R + +--R 3 3 2 2 2 2 2 2 3 +--R (- 6a x - 12a b x - 6a b x)log(x) - 6a b x - 9a b x - 2b +--R / +--R 2 4 3 5 2 6 +--R 2a b x + 4a b x + 2b x +--R Type: Union(Expression Integer,...) +--E + +--S 81 +bb:=-a/(2*b^2*(a*x+b)^2)-(2*a)/(b^3*(a*x+b))-1/(b^3*x)+((3*a)/b^4)*log((a*x+b)/x) +--R +--R 3 3 2 2 2 a x + b 2 2 2 3 +--R (6a x + 12a b x + 6a b x)log(-------) - 6a b x - 9a b x - 2b +--R x +--R (2) ---------------------------------------------------------------- +--R 2 4 3 5 2 6 +--R 2a b x + 4a b x + 2b x +--R Type: Expression Integer +--E + +--S 82 +cc:=aa-bb +--R +--R a x + b +--R 3a log(a x + b) - 3a log(x) - 3a log(-------) +--R x +--R (3) --------------------------------------------- +--R 4 +--R b +--R Type: Expression Integer +--E + +--S 83 +divlog:=rule(log(a/b) == log(a) - log(b)) +--R +--R a +--R (4) log(-) == - log(b) + log(a) +--R b +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 84 14:78 Schaums and Axiom agree +dd:=divlog cc +--R +--R (5) 0 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.79~~~~~$\displaystyle +\int{\frac{dx}{x^3(ax+b)^3}}$} +$$\int{\frac{1}{x^3(ax+b)^3}}= +-\frac{1}{2bx^2(ax+b)^2}+ +\frac{2a}{b^2x(ax+b)^2}+ +\frac{9a^2}{b^3(ax+b)^2}+ +\frac{6a^3x}{b^4(ax+b)^2}- +\frac{6a^2}{b^5}~\ln\left(\frac{ax+b}{x}\right)$$ + +<<*>>= +)clear all + +--S 85 +aa:=integrate(1/(x^3*(a*x+b)^3),x) +--R +--R (1) +--R 4 4 3 3 2 2 2 +--R (- 12a x - 24a b x - 12a b x )log(a x + b) +--R + +--R 4 4 3 3 2 2 2 3 3 2 2 2 3 4 +--R (12a x + 24a b x + 12a b x )log(x) + 12a b x + 18a b x + 4a b x - b +--R / +--R 2 5 4 6 3 7 2 +--R 2a b x + 4a b x + 2b x +--R Type: Union(Expression Integer,...) +--E + +--S 86 +bb:=-1/(2*b*x^2*(a*x+b)^2)_ + +(2*a)/(b^2*x*(a*x+b)^2)_ + +(9*a^2)/(b^3*(a*x+b)^2)_ + +(6*a^3*x)/(b^4*(a*x+b)^2)_ + +(-6*a^2)/b^5*log((a*x+b)/x) +--R +--R (2) +--R 4 4 3 3 2 2 2 a x + b 3 3 2 2 2 +--R (- 12a x - 24a b x - 12a b x )log(-------) + 12a b x + 18a b x +--R x +--R + +--R 3 4 +--R 4a b x - b +--R / +--R 2 5 4 6 3 7 2 +--R 2a b x + 4a b x + 2b x +--R Type: Expression Integer +--E + +--S 87 +cc:=aa-bb +--R +--R 2 2 2 a x + b +--R - 6a log(a x + b) + 6a log(x) + 6a log(-------) +--R x +--R (3) ----------------------------------------------- +--R 5 +--R b +--R Type: Expression Integer +--E + +--S 88 +divlog:=rule(log(a/b) == log(a) - log(b)) +--R +--R a +--R (4) log(-) == - log(b) + log(a) +--R b +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 89 14:79 Schaums and Axiom agree +dd:=divlog cc +--R +--R (5) 0 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.80~~~~~$\displaystyle +\int{(ax+b)^n~dx}$} +$$\int{(ax+b)^n}= +\frac{(ax+b)^{n+1}}{(n+1)a}{\rm\ provided\ }n \ne -1 +$$ +<<*>>= +)clear all +--S 90 +aa:=integrate((a*x+b)^n,x) +--R +--R n log(a x + b) +--R (a x + b)%e +--R (1) ------------------------- +--R a n + a +--R Type: Union(Expression Integer,...) +--E + +--S 91 +bb:=(a*x+b)^(n+1)/((n+1)*a) +--R +--R n + 1 +--R (a x + b) +--R (2) -------------- +--R a n + a +--R Type: Expression Integer +--E + +--S 92 +cc:=aa-bb +--R +--R n log(a x + b) n + 1 +--R (a x + b)%e - (a x + b) +--R (3) ------------------------------------------ +--R a n + a +--R Type: Expression Integer +--E +@ +This messy formula can be simplified using the explog rule: +<<*>>= +--S 93 +explog:=rule(%e^(n*log(x)) == x^n) +--R +--R n log(x) n +--R (4) %e == x +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 94 +dd:=explog cc +--R +--R n + 1 n +--R - (a x + b) + (a x + b)(a x + b) +--R (5) -------------------------------------- +--R a n + a +--R Type: Expression Integer +--E + +--S 95 14:80 Schaums and Axiom agree +ee:=complexNormalize dd +--R +--R (6) 0 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.81~~~~~$\displaystyle +\int{x(ax+b)^n~dx}$} +$$\int{x(ax+b)^n}= +\frac{(ax+b)^{n+2}}{(n+2)a^2}-\frac{b(ax+b)^{n+1}}{(n+1)a^2} +{\rm\ provided\ }n \ne -1,-2 +$$ +<<*>>= +)clear all +--S 96 +aa:=integrate(x*(a*x+b)^n,x) +--R +--R 2 2 2 2 n log(a x + b) +--R ((a n + a )x + a b n x - b )%e +--R (1) --------------------------------------------- +--R 2 2 2 2 +--R a n + 3a n + 2a +--R Type: Union(Expression Integer,...) +--E + +--S 97 +bb:=((a*x+b)^(n+2))/((n+2)*a^2)-(b*(a*x+b)^(n+1))/((n+1)*a^2) +--R +--R n + 2 n + 1 +--R (n + 1)(a x + b) + (- b n - 2b)(a x + b) +--R (2) -------------------------------------------------- +--R 2 2 2 2 +--R a n + 3a n + 2a +--R Type: Expression Integer +--E + +--S 98 +cc:=aa-bb +--R +--R (3) +--R 2 2 2 2 n log(a x + b) n + 2 +--R ((a n + a )x + a b n x - b )%e + (- n - 1)(a x + b) +--R + +--R n + 1 +--R (b n + 2b)(a x + b) +--R / +--R 2 2 2 2 +--R a n + 3a n + 2a +--R Type: Expression Integer +--E + +--S 99 +explog:=rule(%e^(n*log(x)) == x^n) +--R +--R n log(x) n +--R (4) %e == x +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 100 +dd:=explog cc +--R +--R (5) +--R n + 2 n + 1 +--R (- n - 1)(a x + b) + (b n + 2b)(a x + b) +--R + +--R 2 2 2 2 n +--R ((a n + a )x + a b n x - b )(a x + b) +--R / +--R 2 2 2 2 +--R a n + 3a n + 2a +--R Type: Expression Integer +--E + +--S 101 +ee:=complexNormalize dd +--R +--R (6) 0 +--R Type: Expression Integer +--E +@ +\section{\cite{1}:14.82~~~~~$\displaystyle +\int{x^2(ax+b)^n~dx}$} +$$\int{x^2(ax+b)^n}= +\frac{(ax+b)^{n+2}}{(n+3)a^3}- +\frac{2b(ax+b)^{n+2}}{(n+2)a^3}+ +\frac{b^2(ax+b)^{n+1}}{(n+1)a^3} +{\rm\ provided\ }n \ne -1,-2,-3 +$$ + +<<*>>= +)clear all +--S 102 +aa:=integrate(x^2*(a*x+b)^n,x) +--R +--R (1) +--R 3 2 3 3 3 2 2 2 2 2 3 n log(a x + b) +--R ((a n + 3a n + 2a )x + (a b n + a b n)x - 2a b n x + 2b )%e +--R ----------------------------------------------------------------------------- +--R 3 3 3 2 3 3 +--R a n + 6a n + 11a n + 6a +--R Type: Union(Expression Integer,...) +--E + +--S 103 +bb:=(a*x+b)^(n+3)/((n+3)*a^3)-(2*b*(a*x+b)^(n+2))/((n+2)*a^3)+(b^2*(a*x+b)^(n+1))/((n+1)*a^3) +--R +--R (2) +--R 2 n + 3 2 n + 2 +--R (n + 3n + 2)(a x + b) + (- 2b n - 8b n - 6b)(a x + b) +--R + +--R 2 2 2 2 n + 1 +--R (b n + 5b n + 6b )(a x + b) +--R / +--R 3 3 3 2 3 3 +--R a n + 6a n + 11a n + 6a +--R Type: Expression Integer +--E + +--S 104 +cc:=aa-bb +--R +--R (3) +--R 3 2 3 3 3 2 2 2 2 2 3 +--R ((a n + 3a n + 2a )x + (a b n + a b n)x - 2a b n x + 2b ) +--R * +--R n log(a x + b) +--R %e +--R + +--R 2 n + 3 2 n + 2 +--R (- n - 3n - 2)(a x + b) + (2b n + 8b n + 6b)(a x + b) +--R + +--R 2 2 2 2 n + 1 +--R (- b n - 5b n - 6b )(a x + b) +--R / +--R 3 3 3 2 3 3 +--R a n + 6a n + 11a n + 6a +--R Type: Expression Integer +--E + +--S 105 +explog:=rule(%e^(n*log(x)) == x^n) +--R +--R n log(x) n +--R (4) %e == x +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 106 +dd:=explog cc +--R +--R (5) +--R 2 n + 3 2 n + 2 +--R (- n - 3n - 2)(a x + b) + (2b n + 8b n + 6b)(a x + b) +--R + +--R 2 2 2 2 n + 1 +--R (- b n - 5b n - 6b )(a x + b) +--R + +--R 3 2 3 3 3 2 2 2 2 2 3 n +--R ((a n + 3a n + 2a )x + (a b n + a b n)x - 2a b n x + 2b )(a x + b) +--R / +--R 3 3 3 2 3 3 +--R a n + 6a n + 11a n + 6a +--R Type: Expression Integer +--E + +--S 107 14:82 Schaums and Axiom agree +ee:=complexNormalize dd +--R +--R (6) 0 +--R Type: Expression Integer +--E +@ +\section{\cite{1}:14.83~~~~~$\displaystyle +\int{x^m(ax+b)^n}~dx$} +$$\int{x^m(ax+b)^n} +\left\{ +\begin{array}{l} +\displaystyle +\frac{x^{m+1}(ax+b)^n}{m+n+1} ++\frac{nb}{m+n+1}\int{x^m(ax+b)^{n-1}}\\ +\\ +\displaystyle +\frac{x^{m+1}(ax+b)^{n+1}}{(m+n+1)a} +-\frac{mb}{(m+n+1)a}\int{x^{m-1}(ax+b)^n}\\ +\\ +\displaystyle +\frac{-x^{m+1}(ax+b)^{n+1}}{(n+1)b} ++\frac{m+n+2}{(n+1)b}\int{x^m(ax+b)^{n+1}}\\ +\end{array} +\right. +$$ + +<<*>>= +--S 108 14:83 Axiom cannot do this integration +aa:=integrate(x^m*(a*x+b)^n,x) +--R +--R x +--R ++ m n +--I (1) | %U (b + %U a) d%U +--R ++ +--R Type: Union(Expression Integer,...) +--E + +)spool +)lisp (bye) +@ + +\eject +\begin{thebibliography}{99} +\bibitem{1} Spiegel, Murray R. +{\sl Mathematical Handbook of Formulas and Tables}\\ +Schaum's Outline Series McGraw-Hill 1968 pp60-61 +\end{thebibliography} +\end{document} diff --git a/src/axiom-website/CATS/schaum1.input.pdf b/src/axiom-website/CATS/schaum1.input.pdf new file mode 100644 index 0000000..1c316b5 --- /dev/null +++ b/src/axiom-website/CATS/schaum1.input.pdf @@ -0,0 +1,3029 @@ +%PDF-1.2 +7 0 obj +<< +/Type/Encoding 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+\section{\cite{1}:14.210~~~~~$\displaystyle\int{\frac{dx}{\sqrt{x^2-a^2}}}$} +$$\int{\frac{1}{\sqrt{x^2-a^2}}}=\ln\left(x+\sqrt{x^2-a^2}\right)$$ +<<*>>= +)spool schaum10.output +)set message test on +)set message auto off +)clear all + +--S 1 +aa:=integrate(1/(sqrt(x^2-a^2)),x) +--R +--R +--R +-------+ +--R | 2 2 +--R (1) - log(\|x - a - x) +--R Type: Union(Expression Integer,...) +--E + +--S 2 +bb:=log(x+sqrt(x^2-a^2)) +--R +--R +-------+ +--R | 2 2 +--R (2) log(\|x - a + x) +--R Type: Expression Integer +--E + +--S 3 +cc:=aa-bb +--R +--R +-------+ +-------+ +--R | 2 2 | 2 2 +--R (3) - log(\|x - a + x) - log(\|x - a - x) +--R Type: Expression Integer +--E + +--S 4 +logmul1:=rule(c*log(a)+c*log(b) == c*log(a*b)) +--R +--I (4) c log(b) + c log(a) + %I == c log(a b) + %I +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 5 14:210 Schaums and Axiom differ by a constant +dd:=logmul1 cc +--R +--R 2 +--R (5) - log(- a ) +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.xxx~~~~~$\displaystyle\int{\frac{x~dx}{\sqrt{x^2-a^2}}}$} +$$\int{\frac{x}{\sqrt{x^2-a^2}}}=\sqrt{x^2-a^2}$$ +<<*>>= +)clear all + +--S 6 +aa:=integrate(x/(sqrt(x^2-a^2)),x) +--R +--R +--R +-------+ +--R | 2 2 2 2 +--R - x\|x - a + x - a +--R (1) ----------------------- +--R +-------+ +--R | 2 2 +--R \|x - a - x +--R Type: Union(Expression Integer,...) +--E + +--S 7 +bb:=sqrt(x^2-a^2) +--R +--R +-------+ +--R | 2 2 +--R (2) \|x - a +--R Type: Expression Integer +--E + +--S 8 14:xxx Schaums and Axiom agree +cc:=aa-bb +--R +--R (3) 0 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.211~~~~~$\displaystyle +\int{\frac{x^2~dx}{\sqrt{x^2-a^2}}}$} +$$\int{\frac{x^2}{\sqrt{x^2-a^2}}}= +\frac{x\sqrt{x^2-a^2}}{2}+\frac{a^2}{2}\ln\left(x+\sqrt{x^2-a^2}\right) +$$ +<<*>>= +)clear all + +--S 9 +aa:=integrate(x^2/sqrt(x^2-a^2),x) +--R +--R +--R (1) +--R +-------+ +-------+ +--R 2 | 2 2 2 2 4 | 2 2 +--R (- 2a x\|x - a + 2a x - a )log(\|x - a - x) +--R + +--R +-------+ +--R 3 2 | 2 2 4 2 2 +--R (- 2x + a x)\|x - a + 2x - 2a x +--R / +--R +-------+ +--R | 2 2 2 2 +--R 4x\|x - a - 4x + 2a +--R Type: Union(Expression Integer,...) +--E + +--S 10 +bb:=(x*sqrt(x^2-a^2))/2+a^2/2*log(x+sqrt(x^2-a^2)) +--R +--R +-------+ +-------+ +--R 2 | 2 2 | 2 2 +--R a log(\|x - a + x) + x\|x - a +--R (2) ----------------------------------- +--R 2 +--R Type: Expression Integer +--E + +--S 11 +cc:=aa-bb +--R +--R +-------+ +-------+ +--R 2 | 2 2 2 | 2 2 +--R - a log(\|x - a + x) - a log(\|x - a - x) +--R (3) ----------------------------------------------- +--R 2 +--R Type: Expression Integer +--E + +--S 12 14:211 Schaums and Axiom differ by a constant +dd:=complexNormalize cc +--R +--R 2 2 +--R a log(- a ) +--R (4) - ----------- +--R 2 +--R Type: Expression Integer +--E + +@ + +\section{\cite{1}:14.212~~~~~$\displaystyle +\int{\frac{x^3~dx}{\sqrt{x^2-a^2}}}$} +$$\int{\frac{x^3}{\sqrt{x^2-a^2}}}= +\frac{(x^2-a^2)^{3/2}}{3}+a^2\sqrt{x^2-a^2} +$$ +<<*>>= +)clear all + +--S 13 +aa:=integrate(x^3/sqrt(x^2-a^2),x) +--R +--R +--R +-------+ +--R 5 2 3 4 | 2 2 6 2 4 4 2 6 +--R (- 4x - 5a x + 6a x)\|x - a + 4x + 3a x - 9a x + 2a +--R (1) ------------------------------------------------------------ +--R +-------+ +--R 2 2 | 2 2 3 2 +--R (12x - 3a )\|x - a - 12x + 9a x +--R Type: Union(Expression Integer,...) +--E + +--S 14 +bb:=(x^2-a^2)^(3/2)/3+a^2*sqrt(x^2-a^2) +--R +--R +-------+ +--R 2 2 | 2 2 +--R (x + 2a )\|x - a +--R (2) -------------------- +--R 3 +--R Type: Expression Integer +--E + +--S 15 14:212 Schaums and Axiom agree +cc:=aa-bb +--R +--R (3) 0 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.213~~~~~$\displaystyle\int{\frac{dx}{x\sqrt{x^2-a^2}}}$} +$$\int{\frac{1}{x\sqrt{x^2-a^2}}}= +\frac{1}{a}\sec^{-1}\left|\frac{x}{a}\right| +$$ +<<*>>= +)clear all + +--S 16 +aa:=integrate(1/(x*sqrt(x^2-a^2)),x) +--R +--R +--R +-------+ +--R | 2 2 +--R \|x - a - x +--R 2atan(--------------) +--R a +--R (1) --------------------- +--R a +--R Type: Union(Expression Integer,...) +--E + +--S 17 +bb:=1/a*asec(x/a) +--R +--R x +--R asec(-) +--R a +--R (2) ------- +--R a +--R Type: Expression Integer +--E + +--S 18 +cc:=aa-bb +--R +--R +-------+ +--R | 2 2 +--R \|x - a - x x +--R 2atan(--------------) - asec(-) +--R a a +--R (3) ------------------------------- +--R a +--R Type: Expression Integer +--E + +--S 19 +asecrule:=rule(asec(x) == 1/2*%pi+%i*log(sqrt(1-1/x^2)+%i/x)) +--R +--R +------+ +--R | 2 +--R |x - 1 +--R x |------ + %i +--R | 2 +--R \| x +--R 2%i log(---------------) + %pi +--R x +--R (4) asec(x) == ------------------------------ +--R 2 +--R Type: RewriteRule(Integer,Complex Integer,Expression Complex Integer) +--E + +--S 20 +dd:=asecrule cc +--R +--R +-------+ +--R | 2 2 +--R |x - a +--R x |------- + %i a +-------+ +--R | 2 | 2 2 +--R \| x \|x - a - x +--R - 2%i log(------------------) + 4atan(--------------) - %pi +--R x a +--R (5) ----------------------------------------------------------- +--R 2a +--R Type: Expression Complex Integer +--E + +--S 21 +atanrule:=rule(atan(x) == -%i/2*log((1+%i*x)/(1-%i*x))) +--R +--R - x + %i +--R %i log(--------) +--R x + %i +--R (6) atan(x) == - ---------------- +--R 2 +--R Type: RewriteRule(Integer,Complex Integer,Expression Complex Integer) +--E + +--S 22 +ee:=atanrule dd +--R +--R +-------+ +--R | 2 2 +--R |x - a +--R x |------- + %i a +-------+ +--R | 2 | 2 2 +--R \| x - \|x - a + x + %i a +--R - 2%i log(------------------) - 2%i log(-----------------------) - %pi +--R x +-------+ +--R | 2 2 +--R \|x - a - x + %i a +--R (7) ---------------------------------------------------------------------- +--R 2a +--R Type: Expression Complex Integer +--E + +--S 23 +ff:=expandLog ee +--R +--R (8) +--R +-------+ +-------+ +--R | 2 2 | 2 2 +--R 2%i log(\|x - a - x + %i a) - 2%i log(\|x - a - x - %i a) +--R + +--R +-------+ +--R | 2 2 +--R |x - a +--R - 2%i log(x |------- + %i a) + 2%i log(x) - 2%i log(- 1) - %pi +--R | 2 +--R \| x +--R / +--R 2a +--R Type: Expression Complex Integer +--E + +--S 24 +gg:=rootSimp ff +--R +--R (9) +--R +-------+ +-------+ +--R | 2 2 | 2 2 +--R - 2%i log(\|x - a + %i a) + 2%i log(\|x - a - x + %i a) +--R + +--R +-------+ +--R | 2 2 +--R - 2%i log(\|x - a - x - %i a) + 2%i log(x) - 2%i log(- 1) - %pi +--R / +--R 2a +--R Type: Expression Complex Integer +--E + +--S 25 14:213 Schaums and Axiom differ by a constant +hh:=complexNormalize gg +--R +--R %pi +--R (10) - --- +--R 2a +--R Type: Expression Complex Integer +--E +@ + +\section{\cite{1}:14.214~~~~~$\displaystyle +\int{\frac{dx}{x^2\sqrt{x^2-a^2}}}$} +$$\int{\frac{1}{x^2\sqrt{x^2-a^2}}}= +\frac{\sqrt{x^2-a^2}}{a^2x} +$$ +<<*>>= +)clear all + +--S 26 +aa:=integrate(1/(x^2*sqrt(x^2-a^2)),x) +--R +--R +--R 1 +--R (1) - ---------------- +--R +-------+ +--R | 2 2 2 +--R x\|x - a - x +--R Type: Union(Expression Integer,...) +--E + +--S 27 +bb:=sqrt(x^2-a^2)/(a^2*x) +--R +--R +-------+ +--R | 2 2 +--R \|x - a +--R (2) ---------- +--R 2 +--R a x +--R Type: Expression Integer +--E + +--S 28 14:214 Schaums and Axiom differ by a constant +cc:=aa-bb +--R +--R 1 +--R (3) -- +--R 2 +--R a +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.215~~~~~$\displaystyle\int{\frac{dx}{x^3\sqrt{x^2-a^2}}}$} +$$\int{\frac{1}{x^3\sqrt{x^2-a^2}}}= +-\frac{\sqrt{x^2-a^2}}{2a^2x^2}+\frac{1}{2a^3} +\sec^{-1}\left|\frac{x}{a}\right| +$$ +<<*>>= +)clear all + +--S 29 +aa:=integrate(1/(x^3*sqrt(x^2-a^2)),x) +--R +--R +--R (1) +--R +-------+ +--R +-------+ | 2 2 +--R 3 | 2 2 4 2 2 \|x - a - x +--R (4x \|x - a - 4x + 2a x )atan(--------------) +--R a +--R + +--R +-------+ +--R 2 3 | 2 2 3 3 +--R (- 2a x + a )\|x - a + 2a x - 2a x +--R / +--R +-------+ +--R 3 3 | 2 2 3 4 5 2 +--R 4a x \|x - a - 4a x + 2a x +--R Type: Union(Expression Integer,...) +--E + +--S 30 +bb:=sqrt(x^2-a^2)/(2*a^2*x^2)+1/(2*a^3)*asec(x/a) +--R +--R +-------+ +--R | 2 2 2 x +--R a\|x - a + x asec(-) +--R a +--R (2) ----------------------- +--R 3 2 +--R 2a x +--R Type: Expression Integer +--E + +--S 31 +cc:=aa-bb +--R +--R +--R +-------+ +--R | 2 2 +--R \|x - a - x x +--R 2atan(--------------) - asec(-) +--R a a +--R (3) ------------------------------- +--R 3 +--R 2a +--R Type: Expression Integer +--E + +--S 32 +atanrule:=rule(atan(x) == -%i/2*log((1+%i*x)/(1-%i*x))) +--R +--R - x + %i +--R %i log(--------) +--R x + %i +--R (4) atan(x) == - ---------------- +--R 2 +--R Type: RewriteRule(Integer,Complex Integer,Expression Complex Integer) +--E + +--S 33 +dd:=atanrule cc +--R +--R +-------+ +--R | 2 2 +--R - \|x - a + x + %i a x +--R - %i log(-----------------------) - asec(-) +--R +-------+ a +--R | 2 2 +--R \|x - a - x + %i a +--R (5) ------------------------------------------- +--R 3 +--R 2a +--R Type: Expression Complex Integer +--E + +--S 34 +asecrule:=rule(asec(x) == 1/2*%pi+%i*log(sqrt(1-1/x^2)+%i/x)) +--R +--R +------+ +--R | 2 +--R |x - 1 +--R x |------ + %i +--R | 2 +--R \| x +--R 2%i log(---------------) + %pi +--R x +--R (6) asec(x) == ------------------------------ +--R 2 +--R Type: RewriteRule(Integer,Complex Integer,Expression Complex Integer) +--E + +--S 35 +ee:=asecrule dd +--R +--R +-------+ +--R | 2 2 +--R |x - a +--R x |------- + %i a +-------+ +--R | 2 | 2 2 +--R \| x - \|x - a + x + %i a +--R - 2%i log(------------------) - 2%i log(-----------------------) - %pi +--R x +-------+ +--R | 2 2 +--R \|x - a - x + %i a +--R (7) ---------------------------------------------------------------------- +--R 3 +--R 4a +--R Type: Expression Complex Integer +--E + +--S 36 +ff:=expandLog ee +--R +--R (8) +--R +-------+ +-------+ +--R | 2 2 | 2 2 +--R 2%i log(\|x - a - x + %i a) - 2%i log(\|x - a - x - %i a) +--R + +--R +-------+ +--R | 2 2 +--R |x - a +--R - 2%i log(x |------- + %i a) + 2%i log(x) - 2%i log(- 1) - %pi +--R | 2 +--R \| x +--R / +--R 3 +--R 4a +--R Type: Expression Complex Integer +--E + +--S 37 +gg:=rootSimp ff +--R +--R (9) +--R +-------+ +-------+ +--R | 2 2 | 2 2 +--R - 2%i log(\|x - a + %i a) + 2%i log(\|x - a - x + %i a) +--R + +--R +-------+ +--R | 2 2 +--R - 2%i log(\|x - a - x - %i a) + 2%i log(x) - 2%i log(- 1) - %pi +--R / +--R 3 +--R 4a +--R Type: Expression Complex Integer +--E + +--S 38 14:215 Schaums and Axiom differ by a constant +hh:=complexNormalize gg +--R +--R %pi +--R (10) - --- +--R 3 +--R 4a +--R Type: Expression Complex Integer +--E + +@ + +\section{\cite{1}:14.216~~~~~$\displaystyle\int{\sqrt{x^2-a^2}}~dx$} +$$\int{\sqrt{x^2-a^2}}= +\frac{x\sqrt{x^2-a^2}}{2}-\frac{a^2}{2}\ln\left(x+\sqrt{x^2-a^2}\right) +$$ +<<*>>= +)clear all + +--S 39 +aa:=integrate(sqrt(x^2-a^2),x) +--R +--R +--R (1) +--R +-------+ +-------+ +--R 2 | 2 2 2 2 4 | 2 2 +--R (2a x\|x - a - 2a x + a )log(\|x - a - x) +--R + +--R +-------+ +--R 3 2 | 2 2 4 2 2 +--R (- 2x + a x)\|x - a + 2x - 2a x +--R / +--R +-------+ +--R | 2 2 2 2 +--R 4x\|x - a - 4x + 2a +--R Type: Union(Expression Integer,...) +--E + +--S 40 +bb:=(x*sqrt(x^2-a^2))/2-a^2/2*log(x+sqrt(x^2-a^2)) +--R +--R +-------+ +-------+ +--R 2 | 2 2 | 2 2 +--R - a log(\|x - a + x) + x\|x - a +--R (2) ------------------------------------- +--R 2 +--R Type: Expression Integer +--E + +--S 41 +cc:=aa-bb +--R +--R +-------+ +-------+ +--R 2 | 2 2 2 | 2 2 +--R a log(\|x - a + x) + a log(\|x - a - x) +--R (3) --------------------------------------------- +--R 2 +--R Type: Expression Integer +--E + +--S 42 14:216 Schaums and Axiom differ by a constant +dd:=complexNormalize cc +--R +--R 2 2 +--R a log(- a ) +--R (4) ----------- +--R 2 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.217~~~~~$\displaystyle\int{x\sqrt{x^2-a^2}}~dx$} +$$\int{x\sqrt{x^2-a^2}}= +\frac{(x^2-a^2)^{3/2}}{3} +$$ +<<*>>= +)clear all + +--S 43 +aa:=integrate(x*sqrt(x^2-a^2),x) +--R +--R +--R +-------+ +--R 5 2 3 4 | 2 2 6 2 4 4 2 6 +--R (- 4x + 7a x - 3a x)\|x - a + 4x - 9a x + 6a x - a +--R (1) ----------------------------------------------------------- +--R +-------+ +--R 2 2 | 2 2 3 2 +--R (12x - 3a )\|x - a - 12x + 9a x +--R Type: Union(Expression Integer,...) +--E + +--S 44 +bb:=(x^2-a^2)^(3/2)/3 +--R +--R +-------+ +--R 2 2 | 2 2 +--R (x - a )\|x - a +--R (2) ------------------- +--R 3 +--R Type: Expression Integer +--E + +--S 45 14:217 Schaums and Axiom agree +cc:=aa-bb +--R +--R (3) 0 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.218~~~~~$\displaystyle +\int{x^2\sqrt{x^2-a^2}}~dx$} +$$\int{x^2\sqrt{x^2-a^2}}= +\frac{x(x^2-a^2)^{3/2}}{4}+\frac{a^2x\sqrt{x^2-a^2}}{8}- +\frac{a^4}{8}\ln\left(x+\sqrt{x^2-a^2}\right) +$$ +<<*>>= +)clear all + +--S 46 +aa:=integrate(x^2*sqrt(x^2-a^2),x) +--R +--R +--R (1) +--R +-------+ +-------+ +--R 4 3 6 | 2 2 4 4 6 2 8 | 2 2 +--R ((8a x - 4a x)\|x - a - 8a x + 8a x - a )log(\|x - a - x) +--R + +--R +-------+ +--R 7 2 5 4 3 6 | 2 2 8 2 6 4 4 6 2 +--R (- 16x + 24a x - 10a x + a x)\|x - a + 16x - 32a x + 20a x - 4a x +--R / +--R +-------+ +--R 3 2 | 2 2 4 2 2 4 +--R (64x - 32a x)\|x - a - 64x + 64a x - 8a +--R Type: Union(Expression Integer,...) +--E + +--S 47 +bb:=(x*(x^2-a^2)^(3/2))/4+(a^2*x*sqrt(x^2-a^2))/8-a^4/8*log(x+sqrt(x^2-a^2)) +--R +--R +-------+ +-------+ +--R 4 | 2 2 3 2 | 2 2 +--R - a log(\|x - a + x) + (2x - a x)\|x - a +--R (2) ----------------------------------------------- +--R 8 +--R Type: Expression Integer +--E + +--S 48 +cc:=aa-bb +--R +--R +-------+ +-------+ +--R 4 | 2 2 4 | 2 2 +--R a log(\|x - a + x) + a log(\|x - a - x) +--R (3) --------------------------------------------- +--R 8 +--R Type: Expression Integer +--E + +--S 49 14:218 Schaums and Axiom differ by a constant +dd:=complexNormalize cc +--R +--R 4 2 +--R a log(- a ) +--R (4) ----------- +--R 8 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.219~~~~~$\displaystyle +\int{x^3\sqrt{x^2-a^2}}~dx$} +$$\int{x^3\sqrt{x^2-a^2}}= +\frac{(x^2-a^2)^{5/2}}{5}+\frac{a^2(x^2-a^2)^{3/2}}{3} +$$ +<<*>>= +)clear all + +--S 50 +aa:=integrate(x^3*sqrt(x^2-a^2),x) +--R +--R +--R (1) +--R +-------+ +--R 9 2 7 4 5 6 3 8 | 2 2 10 2 8 +--R (- 48x + 76a x - 3a x - 35a x + 10a x)\|x - a + 48x - 100a x +--R + +--R 4 6 6 4 8 2 10 +--R 35a x + 40a x - 25a x + 2a +--R / +--R +-------+ +--R 4 2 2 4 | 2 2 5 2 3 4 +--R (240x - 180a x + 15a )\|x - a - 240x + 300a x - 75a x +--R Type: Union(Expression Integer,...) +--E + +--S 51 +bb:=(x^2-a^2)^(5/2)/5+(a^2*(x^2-a^2)^(3/2))/3 +--R +--R +-------+ +--R 4 2 2 4 | 2 2 +--R (3x - a x - 2a )\|x - a +--R (2) ---------------------------- +--R 15 +--R Type: Expression Integer +--E + +--S 52 14:219 Schaums and Axiom agree +cc:=aa-bb +--R +--R (3) 0 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.220~~~~~$\displaystyle +\int{\frac{\sqrt{x^2-a^2}}{x}}~dx$} +$$\int{\frac{\sqrt{x^2-a^2}}{x}}= +\sqrt{x^2-a^2}-a\sec^{-1}\left|\frac{x}{a}\right| +$$ +<<*>>= +)clear all + +--S 53 +aa:=integrate(sqrt(x^2-a^2)/x,x) +--R +--R +--R +-------+ +--R +-------+ | 2 2 +-------+ +--R | 2 2 \|x - a - x | 2 2 2 2 +--R (- 2a\|x - a + 2a x)atan(--------------) - x\|x - a + x - a +--R a +--R (1) ------------------------------------------------------------------- +--R +-------+ +--R | 2 2 +--R \|x - a - x +--R Type: Union(Expression Integer,...) +--E + +--S 54 +bb:=sqrt(x^2-a^2)-a*asec(x/a) +--R +--R +-------+ +--R | 2 2 x +--R (2) \|x - a - a asec(-) +--R a +--R Type: Expression Integer +--E + +--S 55 +cc:=aa-bb +--R +--R +-------+ +--R | 2 2 +--R \|x - a - x x +--R (3) - 2a atan(--------------) + a asec(-) +--R a a +--R Type: Expression Integer +--E + +--S 56 +atanrule:=rule(atan(x) == -%i/2*log((1+%i*x)/(1-%i*x))) +--R +--R - x + %i +--R %i log(--------) +--R x + %i +--R (4) atan(x) == - ---------------- +--R 2 +--R Type: RewriteRule(Integer,Complex Integer,Expression Complex Integer) +--E + +--S 57 +dd:=atanrule cc +--R +--R +-------+ +--R | 2 2 +--R - \|x - a + x + %i a x +--R (5) %i a log(-----------------------) + a asec(-) +--R +-------+ a +--R | 2 2 +--R \|x - a - x + %i a +--R Type: Expression Complex Integer +--E + +--S 58 +asecrule:=rule(asec(x) == 1/2*%pi+%i*log(sqrt(1-1/x^2)+%i/x)) +--R +--R +------+ +--R | 2 +--R |x - 1 +--R x |------ + %i +--R | 2 +--R \| x +--R 2%i log(---------------) + %pi +--R x +--R (6) asec(x) == ------------------------------ +--R 2 +--R Type: RewriteRule(Integer,Complex Integer,Expression Complex Integer) +--E + +--S 59 +ee:=asecrule dd +--R +--R (7) +--R +-------+ +--R | 2 2 +--R |x - a +--R x |------- + %i a +-------+ +--R | 2 | 2 2 +--R \| x - \|x - a + x + %i a +--R 2%i a log(------------------) + 2%i a log(-----------------------) + a %pi +--R x +-------+ +--R | 2 2 +--R \|x - a - x + %i a +--R -------------------------------------------------------------------------- +--R 2 +--R Type: Expression Complex Integer +--E + +--S 60 +ff:=expandLog ee +--R +--R (8) +--R +-------+ +-------+ +--R | 2 2 | 2 2 +--R - 2%i a log(\|x - a - x + %i a) + 2%i a log(\|x - a - x - %i a) +--R + +--R +-------+ +--R | 2 2 +--R |x - a +--R 2%i a log(x |------- + %i a) - 2%i a log(x) + 2%i a log(- 1) + a %pi +--R | 2 +--R \| x +--R / +--R 2 +--R Type: Expression Complex Integer +--E + +--S 61 +gg:=rootSimp ff +--R +--R (9) +--R +-------+ +-------+ +--R | 2 2 | 2 2 +--R 2%i a log(\|x - a + %i a) - 2%i a log(\|x - a - x + %i a) +--R + +--R +-------+ +--R | 2 2 +--R 2%i a log(\|x - a - x - %i a) - 2%i a log(x) + 2%i a log(- 1) + a %pi +--R / +--R 2 +--R Type: Expression Complex Integer +--E + +--S 62 14:220 Schaums and Axiom differ by a constant +hh:=complexNormalize gg +--R +--R a %pi +--R (10) ----- +--R 2 +--R Type: Expression Complex Integer +--E +@ + +\section{\cite{1}:14.221~~~~~$\displaystyle +\int{\frac{\sqrt{x^2-a^2}}{x^2}}~dx$} +$$\int{\frac{\sqrt{x^2-a^2}}{x^2}}= +-\frac{\sqrt{x^2-a^2}}{x}+\ln\left(x+\sqrt{x^2-a^2}\right) +$$ +<<*>>= +)clear all + +--S 63 +aa:=integrate(sqrt(x^2-a^2)/x^2,x) +--R +--R +--R +-------+ +-------+ +--R | 2 2 2 | 2 2 2 +--R (- x\|x - a + x )log(\|x - a - x) + a +--R (1) -------------------------------------------- +--R +-------+ +--R | 2 2 2 +--R x\|x - a - x +--R Type: Union(Expression Integer,...) +--E + +--S 64 +bb:=-sqrt(x^2-a^2)/x+log(x+sqrt(x^2-a^2)) +--R +--R +-------+ +-------+ +--R | 2 2 | 2 2 +--R x log(\|x - a + x) - \|x - a +--R (2) ---------------------------------- +--R x +--R Type: Expression Integer +--E + +--S 65 +cc:=aa-bb +--R +--R +-------+ +-------+ +--R | 2 2 | 2 2 +--R (3) - log(\|x - a + x) - log(\|x - a - x) - 1 +--R Type: Expression Integer +--E + +--S 66 14:221 Schaums and Axiom differ by a constant +dd:=complexNormalize cc +--R +--R 2 +--R (4) - log(- a ) - 1 +--R Type: Expression Integer +--E + +@ + +\section{\cite{1}:14.222~~~~~$\displaystyle +\int{\frac{\sqrt{x^2-a^2}}{x^3}}~dx$} +$$\int{\frac{\sqrt{x^2-a^2}}{x^3}}= +-\frac{\sqrt{x^2-a^2}}{2x^2}+\frac{1}{2a} +\sec^{-1}\left|\frac{x}{a}\right| +$$ +<<*>>= +)clear all + +--S 67 +aa:=integrate(sqrt(x^2-a^2)/x^3,x) +--R +--R +--R (1) +--R +-------+ +--R +-------+ | 2 2 +--R 3 | 2 2 4 2 2 \|x - a - x +--R (4x \|x - a - 4x + 2a x )atan(--------------) +--R a +--R + +--R +-------+ +--R 2 3 | 2 2 3 3 +--R (2a x - a )\|x - a - 2a x + 2a x +--R / +--R +-------+ +--R 3 | 2 2 4 3 2 +--R 4a x \|x - a - 4a x + 2a x +--R Type: Union(Expression Integer,...) +--E + +--S 68 +bb:=-sqrt(x^2-a^2)/(2*x^2)+1/(2*a)*asec(x/a) +--R +--R +-------+ +--R | 2 2 2 x +--R - a\|x - a + x asec(-) +--R a +--R (2) ------------------------- +--R 2 +--R 2a x +--R Type: Expression Integer +--E + +--S 69 +cc:=aa-bb +--R +--R +-------+ +--R | 2 2 +--R \|x - a - x x +--R 2atan(--------------) - asec(-) +--R a a +--R (3) ------------------------------- +--R 2a +--R Type: Expression Integer +--E + +--S 70 +asecrule:=rule(asec(x) == 1/2*%pi+%i*log(sqrt(1-1/x^2)+%i/x)) +--R +--R +------+ +--R | 2 +--R |x - 1 +--R x |------ + %i +--R | 2 +--R \| x +--R 2%i log(---------------) + %pi +--R x +--R (4) asec(x) == ------------------------------ +--R 2 +--R Type: RewriteRule(Integer,Complex Integer,Expression Complex Integer) +--E + +--S 71 +dd:=asecrule cc +--R +--R +-------+ +--R | 2 2 +--R |x - a +--R x |------- + %i a +-------+ +--R | 2 | 2 2 +--R \| x \|x - a - x +--R - 2%i log(------------------) + 4atan(--------------) - %pi +--R x a +--R (5) ----------------------------------------------------------- +--R 4a +--R Type: Expression Complex Integer +--E + +--S 72 +atanrule:=rule(atan(x) == -%i/2*log((1+%i*x)/(1-%i*x))) +--R +--R - x + %i +--R %i log(--------) +--R x + %i +--R (6) atan(x) == - ---------------- +--R 2 +--R Type: RewriteRule(Integer,Complex Integer,Expression Complex Integer) +--E + +--S 73 +ee:=atanrule dd +--R +--R +-------+ +--R | 2 2 +--R |x - a +--R x |------- + %i a +-------+ +--R | 2 | 2 2 +--R \| x - \|x - a + x + %i a +--R - 2%i log(------------------) - 2%i log(-----------------------) - %pi +--R x +-------+ +--R | 2 2 +--R \|x - a - x + %i a +--R (7) ---------------------------------------------------------------------- +--R 4a +--R Type: Expression Complex Integer +--E + +--S 74 +ff:=expandLog ee +--R +--R (8) +--R +-------+ +-------+ +--R | 2 2 | 2 2 +--R 2%i log(\|x - a - x + %i a) - 2%i log(\|x - a - x - %i a) +--R + +--R +-------+ +--R | 2 2 +--R |x - a +--R - 2%i log(x |------- + %i a) + 2%i log(x) - 2%i log(- 1) - %pi +--R | 2 +--R \| x +--R / +--R 4a +--R Type: Expression Complex Integer +--E + +--S 75 +gg:=rootSimp ff +--R +--R (9) +--R +-------+ +-------+ +--R | 2 2 | 2 2 +--R - 2%i log(\|x - a + %i a) + 2%i log(\|x - a - x + %i a) +--R + +--R +-------+ +--R | 2 2 +--R - 2%i log(\|x - a - x - %i a) + 2%i log(x) - 2%i log(- 1) - %pi +--R / +--R 4a +--R Type: Expression Complex Integer +--E + +--S 76 14:222 Schaums and Axiom differ by a constant +hh:=complexNormalize gg +--R +--R %pi +--R (10) - --- +--R 4a +--R Type: Expression Complex Integer +--E +@ + +\section{\cite{1}:14.223~~~~~$\displaystyle\int{\frac{dx}{(x^2-a^2)^{3/2}}}$} +$$\int{\frac{1}{(x^2-a^2)^{3/2}}}= +-\frac{x}{a^2\sqrt{x^2-a^2}} +$$ +<<*>>= +)clear all + +--S 77 +aa:=integrate(1/(x^2-a^2)^(3/2),x) +--R +--R +--R 1 +--R (1) - --------------------- +--R +-------+ +--R | 2 2 2 2 +--R x\|x - a - x + a +--R Type: Union(Expression Integer,...) +--E + +--S 78 +bb:=-x/(a^2*sqrt(x^2-a^2)) +--R +--R x +--R (2) - ------------ +--R +-------+ +--R 2 | 2 2 +--R a \|x - a +--R Type: Expression Integer +--E + +--S 79 14:223 Schaums and Axiom differ by a constant +cc:=aa-bb +--R +--R 1 +--R (3) - -- +--R 2 +--R a +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.224~~~~~$\displaystyle +\int{\frac{x~dx}{(x^2-a^2)^{3/2}}}$} +$$\int{\frac{x}{(x^2-a^2)^{3/2}}}= +\frac{-1}{\sqrt{x^2-a^2}} +$$ +<<*>>= +)clear all + +--S 80 +aa:=integrate(x/(x^2-a^2)^(3/2),x) +--R +--R +--R +-------+ +--R | 2 2 +--R \|x - a - x +--R (1) --------------------- +--R +-------+ +--R | 2 2 2 2 +--R x\|x - a - x + a +--R Type: Union(Expression Integer,...) +--E + +--S 81 +bb:=-1/sqrt(x^2-a^2) +--R +--R 1 +--R (2) - ---------- +--R +-------+ +--R | 2 2 +--R \|x - a +--R Type: Expression Integer +--E + +--S 82 14:224 Schaums and Axiom agree +cc:=aa-bb +--R +--R (3) 0 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.225~~~~~$\displaystyle +\int{\frac{x^2dx}{(x^2-a^2)^{3/2}}}$} +$$\int{\frac{x^2}{(x^2-a^2)^{3/2}}}= +\frac{-x}{\sqrt{x^2-a^2}}+\ln\left(x+\sqrt{x^2-a^2}\right) +$$ +<<*>>= +)clear all + +--S 83 +aa:=integrate(x^2/(x^2-a^2)^(3/2),x) +--R +--R +--R +-------+ +-------+ +--R | 2 2 2 2 | 2 2 2 +--R (- x\|x - a + x - a )log(\|x - a - x) - a +--R (1) ------------------------------------------------- +--R +-------+ +--R | 2 2 2 2 +--R x\|x - a - x + a +--R Type: Union(Expression Integer,...) +--E + +--S 84 +bb:=-x/sqrt(x^2-a^2)+log(x+sqrt(x^2-a^2)) +--R +--R +-------+ +-------+ +--R | 2 2 | 2 2 +--R \|x - a log(\|x - a + x) - x +--R (2) --------------------------------- +--R +-------+ +--R | 2 2 +--R \|x - a +--R Type: Expression Integer +--E + +--S 85 +cc:=aa-bb +--R +--R +-------+ +-------+ +--R | 2 2 | 2 2 +--R (3) - log(\|x - a + x) - log(\|x - a - x) - 1 +--R Type: Expression Integer +--E + +--S 86 14:225 Schaums and Axiom differ by a constant +dd:=complexNormalize cc +--R +--R 2 +--R (4) - log(- a ) - 1 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.226~~~~~$\displaystyle +\int{\frac{x^3dx}{(x^2-a^2)^{3/2}}}$} +$$\int{\frac{x^3}{(x^2-a^2)^{3/2}}}= +\sqrt{x^2-a^2}-\frac{a^2}{\sqrt{x^2-a^2}} +$$ +<<*>>= +)clear all + +--S 87 +aa:=integrate(x^3/(x^2-a^2)^(3/2),x) +--R +--R +--R +-------+ +--R 3 2 | 2 2 4 2 2 4 +--R (- 2x + 4a x)\|x - a + 2x - 5a x + 2a +--R (1) -------------------------------------------- +--R +-------+ +--R 2 2 | 2 2 3 2 +--R (2x - a )\|x - a - 2x + 2a x +--R Type: Union(Expression Integer,...) +--E + +--S 88 +bb:=sqrt(x^2-a^2)-a^2/sqrt(x^2-a^2) +--R +--R 2 2 +--R x - 2a +--R (2) ---------- +--R +-------+ +--R | 2 2 +--R \|x - a +--R Type: Expression Integer +--E + +--S 89 14:226 Schaums and Axiom agree +cc:=aa-bb +--R +--R (3) 0 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.227~~~~~$\displaystyle +\int{\frac{dx}{x(x^2-a^2)^{3/2}}}$} +$$\int{\frac{1}{x(x^2-a^2)^{3/2}}}= +\frac{-1}{a^2\sqrt{x^2-a^2}}- +\frac{1}{a^3}\sec^{-1}\left|\frac{x}{a}\right| +$$ +<<*>>= +)clear all + +--S 90 +aa:=integrate(1/(x*(x^2-a^2)^(3/2)),x) +--R +--R +--R +-------+ +--R +-------+ | 2 2 +-------+ +--R | 2 2 2 2 \|x - a - x | 2 2 +--R (- 2x\|x - a + 2x - 2a )atan(--------------) + a\|x - a - a x +--R a +--R (1) -------------------------------------------------------------------- +--R +-------+ +--R 3 | 2 2 3 2 5 +--R a x\|x - a - a x + a +--R Type: Union(Expression Integer,...) +--E + +--S 91 +bb:=-1/(a^2*sqrt(x^2-a^2))-1/a^3*asec(x/a) +--R +--R +-------+ +--R x | 2 2 +--R - asec(-)\|x - a - a +--R a +--R (2) ----------------------- +--R +-------+ +--R 3 | 2 2 +--R a \|x - a +--R Type: Expression Integer +--E + +--S 92 +cc:=aa-bb +--R +--R +-------+ +--R | 2 2 +--R \|x - a - x x +--R - 2atan(--------------) + asec(-) +--R a a +--R (3) --------------------------------- +--R 3 +--R a +--R Type: Expression Integer +--E + +--S 93 +atanrule:=rule(atan(x) == -%i/2*log((1+%i*x)/(1-%i*x))) +--R +--R - x + %i +--R %i log(--------) +--R x + %i +--R (4) atan(x) == - ---------------- +--R 2 +--R Type: RewriteRule(Integer,Complex Integer,Expression Complex Integer) +--E + +--S 94 +dd:=atanrule cc +--R +--R +-------+ +--R | 2 2 +--R - \|x - a + x + %i a x +--R %i log(-----------------------) + asec(-) +--R +-------+ a +--R | 2 2 +--R \|x - a - x + %i a +--R (5) ----------------------------------------- +--R 3 +--R a +--R Type: Expression Complex Integer +--E + +--S 95 +asecrule:=rule(asec(x) == 1/2*%pi+%i*log(sqrt(1-1/x^2)+%i/x)) +--R +--R +------+ +--R | 2 +--R |x - 1 +--R x |------ + %i +--R | 2 +--R \| x +--R 2%i log(---------------) + %pi +--R x +--R (6) asec(x) == ------------------------------ +--R 2 +--R Type: RewriteRule(Integer,Complex Integer,Expression Complex Integer) +--E + +--S 96 +ee:=asecrule dd +--R +--R +-------+ +--R | 2 2 +--R |x - a +--R x |------- + %i a +-------+ +--R | 2 | 2 2 +--R \| x - \|x - a + x + %i a +--R 2%i log(------------------) + 2%i log(-----------------------) + %pi +--R x +-------+ +--R | 2 2 +--R \|x - a - x + %i a +--R (7) -------------------------------------------------------------------- +--R 3 +--R 2a +--R Type: Expression Complex Integer +--E + +--S 97 +ff:=expandLog ee +--R +--R (8) +--R +-------+ +-------+ +--R | 2 2 | 2 2 +--R - 2%i log(\|x - a - x + %i a) + 2%i log(\|x - a - x - %i a) +--R + +--R +-------+ +--R | 2 2 +--R |x - a +--R 2%i log(x |------- + %i a) - 2%i log(x) + 2%i log(- 1) + %pi +--R | 2 +--R \| x +--R / +--R 3 +--R 2a +--R Type: Expression Complex Integer +--E + +--S 98 +gg:=rootSimp ff +--R +--R (9) +--R +-------+ +-------+ +--R | 2 2 | 2 2 +--R 2%i log(\|x - a + %i a) - 2%i log(\|x - a - x + %i a) +--R + +--R +-------+ +--R | 2 2 +--R 2%i log(\|x - a - x - %i a) - 2%i log(x) + 2%i log(- 1) + %pi +--R / +--R 3 +--R 2a +--R Type: Expression Complex Integer +--E + +--S 99 14:227 Schaums and Axiom differ by a constant +hh:=complexNormalize gg +--R +--R %pi +--R (10) --- +--R 3 +--R 2a +--R Type: Expression Complex Integer +--E + +@ + +\section{\cite{1}:14.228~~~~~$\displaystyle +\int{\frac{dx}{x^2(x^2-a^2)^{3/2}}}$} +$$\int{\frac{1}{x^2(x^2-a^2)^{3/2}}}= +-\frac{\sqrt{x^2-a^2}}{a^4x}-\frac{x}{a^4\sqrt{x^2-a^2}} +$$ +<<*>>= +)clear all + +--S 100 +aa:=integrate(1/(x^2*(x^2-a^2)^(3/2)),x) +--R +--R +--R 1 +--R (1) - ----------------------------------- +--R +-------+ +--R 3 2 | 2 2 4 2 2 +--R (2x - a x)\|x - a - 2x + 2a x +--R Type: Union(Expression Integer,...) +--E + +--S 101 +bb:=-sqrt(x^2-a^2)/(a^4*x)-x/(a^4*sqrt(x^2-a^2)) +--R +--R 2 2 +--R - 2x + a +--R (2) ------------- +--R +-------+ +--R 4 | 2 2 +--R a x\|x - a +--R Type: Expression Integer +--E + +--S 102 14:228 Schaums and Axiom differ by a constant +cc:=aa-bb +--R +--R 2 +--R (3) - -- +--R 4 +--R a +--R Type: Expression Integer +--E + +@ + +\section{\cite{1}:14.229~~~~~$\displaystyle +\int{\frac{dx}{x^3(x^2-a^2)^{3/2}}}$} +$$\int{\frac{1}{x^3(x^2-a^2)^{3/2}}}= +\frac{1}{2a^2x^2\sqrt{x^2-a^2}}- +\frac{3}{2a^4\sqrt{x^2-a^2}}- +\frac{3}{2a^5}\sec^{-1}\left|\frac{x}{a}\right| +$$ +<<*>>= +)clear all + +--S 103 +aa:=integrate(1/(x^3*(x^2-a^2)^(3/2)),x) +--R +--R +--R (1) +--R +-------+ +--R +-------+ | 2 2 +--R 5 2 3 | 2 2 6 2 4 4 2 \|x - a - x +--R ((- 24x + 18a x )\|x - a + 24x - 30a x + 6a x )atan(--------------) +--R a +--R + +--R +-------+ +--R 4 3 2 5 | 2 2 5 3 3 5 +--R (12a x - 7a x + a )\|x - a - 12a x + 13a x - 3a x +--R / +--R +-------+ +--R 5 5 7 3 | 2 2 5 6 7 4 9 2 +--R (8a x - 6a x )\|x - a - 8a x + 10a x - 2a x +--R Type: Union(Expression Integer,...) +--E + +--S 104 +bb:=1/(2*a^2*x^2*sqrt(x^2-a^2))-3/(2*a^4*sqrt(x^2-a^2))-3/(2*a^5)*asec(x/a) +--R +--R +-------+ +--R 2 x | 2 2 2 3 +--R - 3x asec(-)\|x - a - 3a x + a +--R a +--R (2) ----------------------------------- +--R +-------+ +--R 5 2 | 2 2 +--R 2a x \|x - a +--R Type: Expression Integer +--E + +--S 105 +cc:=aa-bb +--R +--R +-------+ +--R | 2 2 +--R \|x - a - x x +--R - 6atan(--------------) + 3asec(-) +--R a a +--R (3) ---------------------------------- +--R 5 +--R 2a +--R Type: Expression Integer +--E + +--S 106 +atanrule:=rule(atan(x) == -%i/2*log((1+%i*x)/(1-%i*x))) +--R +--R - x + %i +--R %i log(--------) +--R x + %i +--R (4) atan(x) == - ---------------- +--R 2 +--R Type: RewriteRule(Integer,Complex Integer,Expression Complex Integer) +--E + +--S 107 +dd:=atanrule cc +--R +--R +-------+ +--R | 2 2 +--R - \|x - a + x + %i a x +--R 3%i log(-----------------------) + 3asec(-) +--R +-------+ a +--R | 2 2 +--R \|x - a - x + %i a +--R (5) ------------------------------------------- +--R 5 +--R 2a +--R Type: Expression Complex Integer +--E + +--S 108 +asecrule:=rule(asec(x) == 1/2*%pi+%i*log(sqrt(1-1/x^2)+%i/x)) +--R +--R +------+ +--R | 2 +--R |x - 1 +--R x |------ + %i +--R | 2 +--R \| x +--R 2%i log(---------------) + %pi +--R x +--R (6) asec(x) == ------------------------------ +--R 2 +--R Type: RewriteRule(Integer,Complex Integer,Expression Complex Integer) +--E + +--S 109 +ee:=asecrule dd +--R +--R +-------+ +--R | 2 2 +--R |x - a +--R x |------- + %i a +-------+ +--R | 2 | 2 2 +--R \| x - \|x - a + x + %i a +--R 6%i log(------------------) + 6%i log(-----------------------) + 3%pi +--R x +-------+ +--R | 2 2 +--R \|x - a - x + %i a +--R (7) --------------------------------------------------------------------- +--R 5 +--R 4a +--R Type: Expression Complex Integer +--E + +--S 110 +ff:=expandLog ee +--R +--R (8) +--R +-------+ +-------+ +--R | 2 2 | 2 2 +--R - 6%i log(\|x - a - x + %i a) + 6%i log(\|x - a - x - %i a) +--R + +--R +-------+ +--R | 2 2 +--R |x - a +--R 6%i log(x |------- + %i a) - 6%i log(x) + 6%i log(- 1) + 3%pi +--R | 2 +--R \| x +--R / +--R 5 +--R 4a +--R Type: Expression Complex Integer +--E + +--S 111 +gg:=rootSimp ff +--R +--R (9) +--R +-------+ +-------+ +--R | 2 2 | 2 2 +--R 6%i log(\|x - a + %i a) - 6%i log(\|x - a - x + %i a) +--R + +--R +-------+ +--R | 2 2 +--R 6%i log(\|x - a - x - %i a) - 6%i log(x) + 6%i log(- 1) + 3%pi +--R / +--R 5 +--R 4a +--R Type: Expression Complex Integer +--E + +--S 112 14:229 Schaums and Axiom differ by a constant +hh:=complexNormalize gg +--R +--R 3%pi +--R (10) ---- +--R 5 +--R 4a +--R Type: Expression Complex Integer +--E +@ +\section{\cite{1}:14.230~~~~~$\displaystyle\int{(x^2-a^2)^{3/2}}~dx$} +$$\int{(x^2-a^2)^{3/2}}= +\frac{x(x^2-a^2)^{3/2}}{4}-\frac{3a^2x\sqrt{x^2-a^2}}{8}+ +\frac{3}{8}a^4\ln\left(x+\sqrt{x^2-a^2}\right) +$$ +<<*>>= +)clear all + +--S 113 +aa:=integrate((x^2-a^2)^(3/2),x) +--R +--R +--R (1) +--R +-------+ +-------+ +--R 4 3 6 | 2 2 4 4 6 2 8 | 2 2 +--R ((- 24a x + 12a x)\|x - a + 24a x - 24a x + 3a )log(\|x - a - x) +--R + +--R +-------+ +--R 7 2 5 4 3 6 | 2 2 8 2 6 4 4 +--R (- 16x + 56a x - 42a x + 5a x)\|x - a + 16x - 64a x + 68a x +--R + +--R 6 2 +--R - 20a x +--R / +--R +-------+ +--R 3 2 | 2 2 4 2 2 4 +--R (64x - 32a x)\|x - a - 64x + 64a x - 8a +--R Type: Union(Expression Integer,...) +--E + +--S 114 +bb:=(x*(x^2-a^2)^(3/2))/4-(3*a^2*x*sqrt(x^2-a^2))/8+3/8*a^4*log(x+sqrt(x^2-a^2)) +--R +--R +-------+ +-------+ +--R 4 | 2 2 3 2 | 2 2 +--R 3a log(\|x - a + x) + (2x - 5a x)\|x - a +--R (2) ----------------------------------------------- +--R 8 +--R Type: Expression Integer +--E + +--S 115 +cc:=aa-bb +--R +--R +-------+ +-------+ +--R 4 | 2 2 4 | 2 2 +--R - 3a log(\|x - a + x) - 3a log(\|x - a - x) +--R (3) ------------------------------------------------- +--R 8 +--R Type: Expression Integer +--E + +--S 116 14:230 Schaums and Axiom differ by a constant +dd:=complexNormalize cc +--R +--R 4 2 +--R 3a log(- a ) +--R (4) - ------------ +--R 8 +--R Type: Expression Integer +--E + +@ + +\section{\cite{1}:14.231~~~~~$\displaystyle\int{x(x^2-a^2)^{3/2}}~dx$} +$$\int{x(x^2-a^2)^{3/2}}=\frac{(x^2-a^2)^{5/2}}{5}$$ +<<*>>= +)clear all + +--S 117 +aa:=integrate(x*(x^2-a^2)^(3/2),x) +--R +--R +--R (1) +--R +-------+ +--R 9 2 7 4 5 6 3 8 | 2 2 10 2 8 +--R (- 16x + 52a x - 61a x + 30a x - 5a x)\|x - a + 16x - 60a x +--R + +--R 4 6 6 4 8 2 10 +--R 85a x - 55a x + 15a x - a +--R / +--R +-------+ +--R 4 2 2 4 | 2 2 5 2 3 4 +--R (80x - 60a x + 5a )\|x - a - 80x + 100a x - 25a x +--R Type: Union(Expression Integer,...) +--E + +--S 118 +bb:=(x^2-a^2)^(5/2)/5 +--R +--R +-------+ +--R 4 2 2 4 | 2 2 +--R (x - 2a x + a )\|x - a +--R (2) --------------------------- +--R 5 +--R Type: Expression Integer +--E + +--S 119 14:231 Schaums and Axiom agree +cc:=aa-bb +--R +--R (3) 0 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.232~~~~~$\displaystyle\int{x^2(x^2-a^2)^{3/2}}~dx$} +$$\int{x^2(x^2-a^2)^{3/2}}= +\frac{x(x^2-a^2)^{5/2}}{6}+\frac{a^2x(x^2-a^2)^{3/2}}{24}- +\frac{a^4x\sqrt{x^2-a^2}}{16}+ +\frac{a^6}{16}\ln\left(x+\sqrt{x^2-a^2}\right) +$$ +<<*>>= +)clear all + +--S 120 +aa:=integrate(x^2*(x^2-a^2)^(3/2),x) +--R +--R +--R (1) +--R +-------+ +--R 6 5 8 3 10 | 2 2 6 6 8 4 10 2 +--R (- 96a x + 96a x - 18a x)\|x - a + 96a x - 144a x + 54a x +--R + +--R 12 +--R - 3a +--R * +--R +-------+ +--R | 2 2 +--R log(\|x - a - x) +--R + +--R +-------+ +--R 11 2 9 4 7 6 5 8 3 10 | 2 2 +--R (- 256x + 832a x - 912a x + 404a x - 68a x + 3a x)\|x - a +--R + +--R 12 2 10 4 8 6 6 8 4 10 2 +--R 256x - 960a x + 1296a x - 772a x + 198a x - 18a x +--R / +--R +-------+ +--R 5 2 3 4 | 2 2 6 2 4 4 2 6 +--R (1536x - 1536a x + 288a x)\|x - a - 1536x + 2304a x - 864a x + 48a +--R Type: Union(Expression Integer,...) +--E + +--S 121 +bb:=(x*(x^2-a^2)^(5/2))/6+(a^2*x*(x^2-a^2)^(3/2))/24-(a^4*x*sqrt(x^2-a^2))/16+a^6/16*log(x+sqrt(x^2-a^2)) +--R +--R +-------+ +-------+ +--R 6 | 2 2 5 2 3 4 | 2 2 +--R 3a log(\|x - a + x) + (8x - 14a x + 3a x)\|x - a +--R (2) -------------------------------------------------------- +--R 48 +--R Type: Expression Integer +--E + +--S 122 +cc:=aa-bb +--R +--R +-------+ +-------+ +--R 6 | 2 2 6 | 2 2 +--R - a log(\|x - a + x) - a log(\|x - a - x) +--R (3) ----------------------------------------------- +--R 16 +--R Type: Expression Integer +--E + +--S 123 14:232 Schaums and Axiom differ by a constant +dd:=complexNormalize cc +--R +--R 6 2 +--R a log(- a ) +--R (4) - ----------- +--R 16 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.233~~~~~$\displaystyle\int{x^3(x^2-a^2)^{3/2}}~dx$} +$$\int{x^3(x^2-a^2)^{3/2}}= +\frac{(x^2-a^2)^{7/2}}{7}+\frac{a^2(x^2-a^2)^{5/2}}{5} +$$ +<<*>>= +)clear all + +--S 124 +aa:=integrate(x^3*(x^2-a^2)^(3/2),x) +--R +--R +--R (1) +--R 13 2 11 4 9 6 7 8 5 10 3 +--R - 320x + 1072a x - 1240a x + 467a x + 112a x - 105a x +--R + +--R 12 +--R 14a x +--R * +--R +-------+ +--R | 2 2 +--R \|x - a +--R + +--R 14 2 12 4 10 6 8 8 6 10 4 12 2 +--R 320x - 1232a x + 1736a x - 973a x + 21a x + 175a x - 49a x +--R + +--R 14 +--R 2a +--R / +--R +-------+ +--R 6 2 4 4 2 6 | 2 2 7 2 5 +--R (2240x - 2800a x + 840a x - 35a )\|x - a - 2240x + 3920a x +--R + +--R 4 3 6 +--R - 1960a x + 245a x +--R Type: Union(Expression Integer,...) +--E + +--S 125 +bb:=(x^2-a^2)^(7/2)/7+(a^2*(x^2-a^2)^(5/2))/5 +--R +--R +-------+ +--R 6 2 4 4 2 6 | 2 2 +--R (5x - 8a x + a x + 2a )\|x - a +--R (2) ------------------------------------ +--R 35 +--R Type: Expression Integer +--E + +--S 126 14:233 Schaums and Axiom agree +cc:=aa-bb +--R +--R (3) 0 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.234~~~~~$\displaystyle +\int{\frac{(x^2-a^2)^{3/2}}{x}}~dx$} +$$\int{\frac{(x^2-a^2)^{3/2}}{x}}= +\frac{(x^2-a^2)^{3/2}}{3}-a^2\sqrt{x^2-a^2}+ +a^3\sec^{-1}\left|\frac{x}{a}\right| +$$ +<<*>>= +)clear all + +--S 127 +aa:=integrate((x^2-a^2)^(3/2)/x,x) +--R +--R +--R (1) +--R +-------+ +--R +-------+ | 2 2 +--R 3 2 5 | 2 2 3 3 5 \|x - a - x +--R ((24a x - 6a )\|x - a - 24a x + 18a x)atan(--------------) +--R a +--R + +--R +-------+ +--R 5 2 3 4 | 2 2 6 2 4 4 2 6 +--R (- 4x + 19a x - 12a x)\|x - a + 4x - 21a x + 21a x - 4a +--R / +--R +-------+ +--R 2 2 | 2 2 3 2 +--R (12x - 3a )\|x - a - 12x + 9a x +--R Type: Union(Expression Integer,...) +--E + +--S 128 +bb:=(x^2-a^2)^(3/2)/3-a^2*sqrt(x^2-a^2)+a^3*asec(x/a) +--R +--R +-------+ +--R 2 2 | 2 2 3 x +--R (x - 4a )\|x - a + 3a asec(-) +--R a +--R (2) --------------------------------- +--R 3 +--R Type: Expression Integer +--E + +--S 129 +cc:=aa-bb +--R +--R +-------+ +--R | 2 2 +--R 3 \|x - a - x 3 x +--R (3) 2a atan(--------------) - a asec(-) +--R a a +--R Type: Expression Integer +--E + +--S 130 +asecrule:=rule(asec(x) == 1/2*%pi+%i*log(sqrt(1-1/x^2)+%i/x)) +--R +--R +------+ +--R | 2 +--R |x - 1 +--R x |------ + %i +--R | 2 +--R \| x +--R 2%i log(---------------) + %pi +--R x +--R (4) asec(x) == ------------------------------ +--R 2 +--R Type: RewriteRule(Integer,Complex Integer,Expression Complex Integer) +--E + +--S 131 +dd:=asecrule cc +--R +--R +-------+ +--R | 2 2 +--R |x - a +--R x |------- + %i a +-------+ +--R | 2 | 2 2 +--R 3 \| x 3 \|x - a - x 3 +--R - 2%i a log(------------------) + 4a atan(--------------) - a %pi +--R x a +--R (5) ----------------------------------------------------------------- +--R 2 +--R Type: Expression Complex Integer +--E + +--S 132 +atanrule:=rule(atan(x) == -%i/2*log((1+%i*x)/(1-%i*x))) +--R +--R - x + %i +--R %i log(--------) +--R x + %i +--R (6) atan(x) == - ---------------- +--R 2 +--R Type: RewriteRule(Integer,Complex Integer,Expression Complex Integer) +--E + +--S 133 +ee:=atanrule dd +--R +--R (7) +--R +-------+ +--R | 2 2 +--R |x - a +--R x |------- + %i a +-------+ +--R | 2 | 2 2 +--R 3 \| x 3 - \|x - a + x + %i a 3 +--R - 2%i a log(------------------) - 2%i a log(-----------------------) - a %pi +--R x +-------+ +--R | 2 2 +--R \|x - a - x + %i a +--R ---------------------------------------------------------------------------- +--R 2 +--R Type: Expression Complex Integer +--E + +--S 134 +ff:=expandLog ee +--R +--R (8) +--R +-------+ +-------+ +--R 3 | 2 2 3 | 2 2 +--R 2%i a log(\|x - a - x + %i a) - 2%i a log(\|x - a - x - %i a) +--R + +--R +-------+ +--R | 2 2 +--R 3 |x - a 3 3 3 +--R - 2%i a log(x |------- + %i a) + 2%i a log(x) - 2%i a log(- 1) - a %pi +--R | 2 +--R \| x +--R / +--R 2 +--R Type: Expression Complex Integer +--E + +--S 135 +gg:=rootSimp ff +--R +--R (9) +--R +-------+ +-------+ +--R 3 | 2 2 3 | 2 2 +--R - 2%i a log(\|x - a + %i a) + 2%i a log(\|x - a - x + %i a) +--R + +--R +-------+ +--R 3 | 2 2 3 3 3 +--R - 2%i a log(\|x - a - x - %i a) + 2%i a log(x) - 2%i a log(- 1) - a %pi +--R / +--R 2 +--R Type: Expression Complex Integer +--E + +--S 136 14:234 Schaums and Axiom differ by a constant +hh:=complexNormalize gg +--R +--R 3 +--R a %pi +--R (10) - ----- +--R 2 +--R Type: Expression Complex Integer +--E +@ + +\section{\cite{1}:14.235~~~~~$\displaystyle +\int{\frac{(x^2-a^2)^{3/2}}{x^2}}~dx$} +$$\int{\frac{(x^2-a^2)^{3/2}}{x^2}}= +-\frac{(x^2-a^2)^{3/2}}{x}+\frac{3x\sqrt{x^2-a^2}}{2}- +\frac{3}{2}a^2\ln\left(x+\sqrt{x^2-a^2}\right) +$$ +<<*>>= +)clear all + +--S 137 +aa:=integrate((x^2-a^2)^{3/2}/x^2,x) +--R +--R +--R (1) +--R +-------+ +-------+ +--R 2 3 4 | 2 2 2 4 4 2 | 2 2 +--R ((12a x - 3a x)\|x - a - 12a x + 9a x )log(\|x - a - x) +--R + +--R +-------+ +--R 5 2 3 4 | 2 2 6 2 4 4 2 6 +--R (- 4x + 3a x + 4a x)\|x - a + 4x - 5a x - 3a x + 2a +--R / +--R +-------+ +--R 3 2 | 2 2 4 2 2 +--R (8x - 2a x)\|x - a - 8x + 6a x +--R Type: Union(Expression Integer,...) +--E + +--S 138 +bb:=-(x^2-a^2)^(3/2)/x+3*x*sqrt(x^2-a^2)/2-3/2*a^2*log(x+sqrt(x^2-a^2)) +--R +--R +-------+ +-------+ +--R 2 | 2 2 2 2 | 2 2 +--R - 3a x log(\|x - a + x) + (x + 2a )\|x - a +--R (2) ------------------------------------------------- +--R 2x +--R Type: Expression Integer +--E + +--S 139 +cc:=aa-bb +--R +--R +-------+ +-------+ +--R 2 | 2 2 2 | 2 2 2 +--R 3a log(\|x - a + x) + 3a log(\|x - a - x) + 2a +--R (3) ----------------------------------------------------- +--R 2 +--R Type: Expression Integer +--E + +--S 140 14:235 Schaums and Axiom differ by a constant +dd:=complexNormalize cc +--R +--R 2 2 2 +--R 3a log(- a ) + 2a +--R (4) ------------------ +--R 2 +--R Type: Expression Integer +--E + +@ + +\section{\cite{1}:14.236~~~~~$\displaystyle +\int{\frac{(x^2-a^2)^{3/2}}{x^3}}~dx$} +$$\int{\frac{(x^2-a^2)^{3/2}}{x^3}}= +-\frac{(x^2-a^2)^{3/2}}{2x^2}+\frac{3}{2}\sqrt{x^2-a^2}- +\frac{3}{2}a\sec^{-1}\left|\frac{x}{a}\right| +$$ +<<*>>= +)clear all + +--S 141 +aa:=integrate((x^2-a^2)^(3/2)/x^3,x) +--R +--R +--R (1) +--R +-------+ +--R +-------+ | 2 2 +--R 4 3 2 | 2 2 5 3 3 \|x - a - x +--R ((- 24a x + 6a x )\|x - a + 24a x - 18a x )atan(--------------) +--R a +--R + +--R +-------+ +--R 5 2 3 4 | 2 2 6 2 4 4 2 6 +--R (- 8x + 2a x + 3a x)\|x - a + 8x - 6a x - 3a x + a +--R / +--R +-------+ +--R 4 2 2 | 2 2 5 2 3 +--R (8x - 2a x )\|x - a - 8x + 6a x +--R Type: Union(Expression Integer,...) +--E + +--S 142 +bb:=-(x^2-a^2)^(3/2)/(2*x^2)+(3*sqrt(x^2-a^2))/2-3/2*a*asec(x/a) +--R +--R +-------+ +--R 2 2 | 2 2 2 x +--R (2x + a )\|x - a - 3a x asec(-) +--R a +--R (2) ----------------------------------- +--R 2 +--R 2x +--R Type: Expression Integer +--E + +--S 143 +cc:=aa-bb +--R +--R +-------+ +--R | 2 2 +--R \|x - a - x x +--R - 6a atan(--------------) + 3a asec(-) +--R a a +--R (3) -------------------------------------- +--R 2 +--R Type: Expression Integer +--E + +--S 144 +atanrule:=rule(atan(x) == -%i/2*log((1+%i*x)/(1-%i*x))) +--R +--R - x + %i +--R %i log(--------) +--R x + %i +--R (4) atan(x) == - ---------------- +--R 2 +--R Type: RewriteRule(Integer,Complex Integer,Expression Complex Integer) +--E + +--S 145 +dd:=atanrule cc +--R +--R +-------+ +--R | 2 2 +--R - \|x - a + x + %i a x +--R 3%i a log(-----------------------) + 3a asec(-) +--R +-------+ a +--R | 2 2 +--R \|x - a - x + %i a +--R (5) ----------------------------------------------- +--R 2 +--R Type: Expression Complex Integer +--E + +--S 146 +asecrule:=rule(asec(x) == 1/2*%pi+%i*log(sqrt(1-1/x^2)+%i/x)) +--R +--R +------+ +--R | 2 +--R |x - 1 +--R x |------ + %i +--R | 2 +--R \| x +--R 2%i log(---------------) + %pi +--R x +--R (6) asec(x) == ------------------------------ +--R 2 +--R Type: RewriteRule(Integer,Complex Integer,Expression Complex Integer) +--E + +--S 147 +ee:=asecrule dd +--R +--R (7) +--R +-------+ +--R | 2 2 +--R |x - a +--R x |------- + %i a +-------+ +--R | 2 | 2 2 +--R \| x - \|x - a + x + %i a +--R 6%i a log(------------------) + 6%i a log(-----------------------) + 3a %pi +--R x +-------+ +--R | 2 2 +--R \|x - a - x + %i a +--R --------------------------------------------------------------------------- +--R 4 +--R Type: Expression Complex Integer +--E + +--S 148 +ff:=expandLog ee +--R +--R (8) +--R +-------+ +-------+ +--R | 2 2 | 2 2 +--R - 6%i a log(\|x - a - x + %i a) + 6%i a log(\|x - a - x - %i a) +--R + +--R +-------+ +--R | 2 2 +--R |x - a +--R 6%i a log(x |------- + %i a) - 6%i a log(x) + 6%i a log(- 1) + 3a %pi +--R | 2 +--R \| x +--R / +--R 4 +--R Type: Expression Complex Integer +--E + +--S 149 +gg:=rootSimp ff +--R +--R (9) +--R +-------+ +-------+ +--R | 2 2 | 2 2 +--R 6%i a log(\|x - a + %i a) - 6%i a log(\|x - a - x + %i a) +--R + +--R +-------+ +--R | 2 2 +--R 6%i a log(\|x - a - x - %i a) - 6%i a log(x) + 6%i a log(- 1) + 3a %pi +--R / +--R 4 +--R Type: Expression Complex Integer +--E + +--S 150 14:236 Schaums and Axiom differ by a constant +hh:=complexNormalize gg +--R +--R 3a %pi +--R (10) ------ +--R 4 +--R Type: Expression Complex Integer +--E + +)spool +)lisp (bye) +@ + +\eject +\begin{thebibliography}{99} +\bibitem{1} Spiegel, Murray R. +{\sl Mathematical Handbook of Formulas and Tables}\\ +Schaum's Outline Series McGraw-Hill 1968 pp68-69 +\end{thebibliography} +\end{document} diff --git a/src/axiom-website/CATS/schaum10.input.pdf b/src/axiom-website/CATS/schaum10.input.pdf new file mode 100644 index 0000000..8271532 --- /dev/null +++ b/src/axiom-website/CATS/schaum10.input.pdf @@ -0,0 +1,3941 @@ +%PDF-1.2 +7 0 obj +<< +/Type/Encoding 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+\section{\cite{1}:14.237~~~~~$\displaystyle\int{\frac{dx}{\sqrt{a^2-x^2}}}$} +$$\int{\frac{1}{\sqrt{a^2-x^2}}}=\ln\left(x+\sqrt{a^2-x^2}\right)$$ +<<*>>= +)spool schaum11.output +)set message test on +)set message auto off +)clear all + +--S 1 +aa:=integrate(1/(sqrt(a^2-x^2)),x) +--R +--R +--R +---------+ +--R | 2 2 +--R \|- x + a - a +--R (1) - 2atan(----------------) +--R x +--R Type: Union(Expression Integer,...) +--E + +--S 2 +bb:=asin(x/a) +--R +--R x +--R (2) asin(-) +--R a +--R Type: Expression Integer +--E + +--S 3 +cc:=aa-bb +--R +--R +---------+ +--R | 2 2 +--R \|- x + a - a x +--R (3) - 2atan(----------------) - asin(-) +--R x a +--R Type: Expression Integer +--E + +--S 4 +atanrule:=rule(atan(x) == -%i/2*log((1+%i*x)/(1-%i*x))) +--R +--R - x + %i +--R %i log(--------) +--R x + %i +--R (4) atan(x) == - ---------------- +--R 2 +--R Type: RewriteRule(Integer,Complex Integer,Expression Complex Integer) +--E + +--S 5 +dd:=atanrule cc +--R +--R +---------+ +--R | 2 2 +--R - \|- x + a + %i x + a x +--R (5) %i log(-------------------------) - asin(-) +--R +---------+ a +--R | 2 2 +--R \|- x + a + %i x - a +--R Type: Expression Complex Integer +--E + +--S 6 +asinrule:=rule(asin(x) == %i*log(-%i*x+sqrt(1-x^2))) +--R +--R +--------+ +--R | 2 +--R (6) asin(x) == %i log(\|- x + 1 - %i x) +--R Type: RewriteRule(Integer,Complex Integer,Expression Complex Integer) +--E + +--S 7 +ee:=asinrule dd +--R +--R +---------+ +--R | 2 2 +--R |- x + a +--R a |--------- - %i x +---------+ +--R | 2 | 2 2 +--R \| a - \|- x + a + %i x + a +--R (7) - %i log(--------------------) + %i log(-------------------------) +--R a +---------+ +--R | 2 2 +--R \|- x + a + %i x - a +--R Type: Expression Complex Integer +--E + +--S 8 +ff:=rootSimp ee +--R +--R +-------+ +-------+ +--R | 2 2 | 2 2 +--R %i\|x - a - %i x - \|x - a + x - %i a +--R (8) - %i log(-------------------) + %i log(-----------------------) +--R a +-------+ +--R | 2 2 +--R \|x - a + x + %i a +--R Type: Expression Complex Integer +--E + +--S 9 14:238 Schaums and Axiom agree +gg:=complexNormalize ff +--R +--R (9) 0 +--R Type: Expression Complex Integer +--E + +@ + +\section{\cite{1}:14.238~~~~~$\displaystyle\int{\frac{x~dx}{\sqrt{a^2-x^2}}}$} +$$\int{\frac{x}{\sqrt{a^2-x^2}}}=\sqrt{a^2-x^2}$$ +<<*>>= +)clear all + +--S 10 +aa:=integrate(x/(sqrt(a^2-x^2)),x) +--R +--R +--R 2 +--R x +--R (1) ---------------- +--R +---------+ +--R | 2 2 +--R \|- x + a - a +--R Type: Union(Expression Integer,...) +--E + +--S 11 +bb:=-sqrt(a^2-x^2) +--R +--R +---------+ +--R | 2 2 +--R (2) - \|- x + a +--R Type: Expression Integer +--E + +--S 12 14:238 Schaums and Axiom differ by a constant +cc:=aa-bb +--R +--R (3) - a +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.239~~~~~$\displaystyle +\int{\frac{x^2~dx}{\sqrt{a^2-x^2}}}$} +$$\int{\frac{x^2}{\sqrt{a^2-x^2}}}= +\frac{x\sqrt{a^2-x^2}}{2}+\frac{a^2}{2}\ln\left(x+\sqrt{a^2-x^2}\right) +$$ +<<*>>= +)clear all + +--S 13 +aa:=integrate(x^2/sqrt(a^2-x^2),x) +--R +--R +--R (1) +--R +---------+ +--R +---------+ | 2 2 +--R 3 | 2 2 2 2 4 \|- x + a - a +--R (- 4a \|- x + a - 2a x + 4a )atan(----------------) +--R x +--R + +--R +---------+ +--R 3 2 | 2 2 3 3 +--R (- x + 2a x)\|- x + a + 2a x - 2a x +--R / +--R +---------+ +--R | 2 2 2 2 +--R 4a\|- x + a + 2x - 4a +--R Type: Union(Expression Integer,...) +--E + +--S 14 +bb:=-(x*sqrt(a^2-x^2))/2+a^2/2*asin(x/a) +--R +--R +---------+ +--R | 2 2 2 x +--R - x\|- x + a + a asin(-) +--R a +--R (2) --------------------------- +--R 2 +--R Type: Expression Integer +--E + +--S 15 +cc:=aa-bb +--R +--R +---------+ +--R | 2 2 +--R 2 \|- x + a - a 2 x +--R - 2a atan(----------------) - a asin(-) +--R x a +--R (3) --------------------------------------- +--R 2 +--R Type: Expression Integer +--E + +--S 16 +atanrule:=rule(atan(x) == -%i/2*log((1+%i*x)/(1-%i*x))) +--R +--R - x + %i +--R %i log(--------) +--R x + %i +--R (4) atan(x) == - ---------------- +--R 2 +--R Type: RewriteRule(Integer,Complex Integer,Expression Complex Integer) +--E + +--S 17 +dd:=atanrule cc +--R +--R +---------+ +--R | 2 2 +--R 2 - \|- x + a + %i x + a 2 x +--R %i a log(-------------------------) - a asin(-) +--R +---------+ a +--R | 2 2 +--R \|- x + a + %i x - a +--R (5) ----------------------------------------------- +--R 2 +--R Type: Expression Complex Integer +--E + +--S 18 +asinrule:=rule(asin(x) == %i*log(-%i*x+sqrt(1-x^2))) +--R +--R +--------+ +--R | 2 +--R (6) asin(x) == %i log(\|- x + 1 - %i x) +--R Type: RewriteRule(Integer,Complex Integer,Expression Complex Integer) +--E + +--S 19 +ee:=asinrule dd +--R +--R +---------+ +--R | 2 2 +--R |- x + a +--R a |--------- - %i x +---------+ +--R | 2 | 2 2 +--R 2 \| a 2 - \|- x + a + %i x + a +--R - %i a log(--------------------) + %i a log(-------------------------) +--R a +---------+ +--R | 2 2 +--R \|- x + a + %i x - a +--R (7) ---------------------------------------------------------------------- +--R 2 +--R Type: Expression Complex Integer +--E + +--S 20 +ff:=expandLog ee +--R +--R (8) +--R +---------+ +--R | 2 2 +---------+ +--R 2 |- x + a 2 | 2 2 +--R - %i a log(a |--------- - %i x) - %i a log(\|- x + a + %i x - a) +--R | 2 +--R \| a +--R + +--R +---------+ +--R 2 | 2 2 2 2 +--R %i a log(\|- x + a - %i x - a) + %i a log(a) + %i a log(- 1) +--R / +--R 2 +--R Type: Expression Complex Integer +--E + +--S 21 +gg:=rootSimp ff +--R +--R (9) +--R +-------+ +-------+ +--R 2 | 2 2 2 | 2 2 +--R - %i a log(%i\|x - a + %i x - a) - %i a log(%i\|x - a - %i x) +--R + +--R +-------+ +--R 2 | 2 2 2 2 +--R %i a log(%i\|x - a - %i x - a) + %i a log(a) + %i a log(- 1) +--R / +--R 2 +--R Type: Expression Complex Integer +--E + +--S 22 14:239 Schaums and Axiom agree +hh:=complexNormalize gg +--R +--R (10) 0 +--R Type: Expression Complex Integer +--E + +@ + +\section{\cite{1}:14.240~~~~~$\displaystyle +\int{\frac{x^3~dx}{\sqrt{a^2-x^2}}}$} +$$\int{\frac{x^3}{\sqrt{a^2-x^2}}}= +\frac{(a^2-x^2)^{3/2}}{3}+a^2\sqrt{a^2-x^2} +$$ +<<*>>= +)clear all + +--S 23 +aa:=integrate(x^3/sqrt(a^2-x^2),x) +--R +--R +--R +---------+ +--R 4 | 2 2 6 2 4 +--R 3a x \|- x + a + x - 3a x +--R (1) --------------------------------------- +--R +---------+ +--R 2 2 | 2 2 2 3 +--R (3x - 12a )\|- x + a - 9a x + 12a +--R Type: Union(Expression Integer,...) +--E + +--S 24 +bb:=(a^2-x^2)^(3/2)/3-a^2*sqrt(a^2-x^2) +--R +--R +---------+ +--R 2 2 | 2 2 +--R (- x - 2a )\|- x + a +--R (2) ------------------------ +--R 3 +--R Type: Expression Integer +--E + +--S 25 14:240 Schaums and Axiom differ by a constant +cc:=aa-bb +--R +--R 3 +--R 2a +--R (3) - --- +--R 3 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.241~~~~~$\displaystyle\int{\frac{dx}{x\sqrt{a^2-x^2}}}$} +$$\int{\frac{1}{x\sqrt{a^2-x^2}}}= +\frac{1}{a}\sec^{-1}\left|\frac{x}{a}\right| +$$ +<<*>>= +)clear all + +--S 26 +aa:=integrate(1/(x*sqrt(a^2-x^2)),x) +--R +--R +--R +---------+ +--R | 2 2 +--R \|- x + a - a +--R log(----------------) +--R x +--R (1) --------------------- +--R a +--R Type: Union(Expression Integer,...) +--E + +--S 27 +bb:=-1/a*log((a+sqrt(a^2-x^2))/x) +--R +--R +---------+ +--R | 2 2 +--R \|- x + a + a +--R log(----------------) +--R x +--R (2) - --------------------- +--R a +--R Type: Expression Integer +--E + +--S 28 +cc:=aa-bb +--R +--R +---------+ +---------+ +--R | 2 2 | 2 2 +--R \|- x + a + a \|- x + a - a +--R log(----------------) + log(----------------) +--R x x +--R (3) --------------------------------------------- +--R a +--R Type: Expression Integer +--E + +--S 29 +dd:=expandLog cc +--R +--R +---------+ +---------+ +--R | 2 2 | 2 2 +--R log(\|- x + a + a) + log(\|- x + a - a) - 2log(x) +--R (4) ------------------------------------------------------- +--R a +--R Type: Expression Integer +--E + +--S 30 +ee:=complexNormalize dd +--R +--R x +--R 2log(-------) +--R +----+ +--R | 2 +--R \|- x +--R (5) - ------------- +--R a +--R Type: Expression Integer +--E + +--S 31 14:241 Schaums and Axiom differ by a constant +ff:=rootSimp ee +--R +--R +---+ +--R 2log(\|- 1 ) +--R (6) ------------ +--R a +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.242~~~~~$\displaystyle +\int{\frac{dx}{x^2\sqrt{a^2-x^2}}}$} +$$\int{\frac{1}{x^2\sqrt{a^2-x^2}}}= +\frac{\sqrt{a^2-x^2}}{a^2x} +$$ +<<*>>= +)clear all + +--S 32 +aa:=integrate(1/(x^2*sqrt(a^2-x^2)),x) +--R +--R +--R +---------+ +--R | 2 2 2 2 +--R a\|- x + a + x - a +--R (1) ----------------------- +--R +---------+ +--R 2 | 2 2 3 +--R a x\|- x + a - a x +--R Type: Union(Expression Integer,...) +--E + +--S 33 +bb:=-sqrt(a^2-x^2)/(a^2*x) +--R +--R +---------+ +--R | 2 2 +--R \|- x + a +--R (2) - ------------ +--R 2 +--R a x +--R Type: Expression Integer +--E + +--S 34 14:242 Schaums and Axiom agree +cc:=aa-bb +--R +--R (3) 0 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.243~~~~~$\displaystyle\int{\frac{dx}{x^3\sqrt{a^2-x^2}}}$} +$$\int{\frac{1}{x^3\sqrt{a^2-x^2}}}= +-\frac{\sqrt{a^2-x^2}}{2a^2x^2}+\frac{1}{2a^3} +\sec^{-1}\left|\frac{x}{a}\right| +$$ +<<*>>= +)clear all + +--S 35 +aa:=integrate(1/(x^3*sqrt(a^2-x^2)),x) +--R +--R +--R (1) +--R +---------+ +--R +---------+ | 2 2 +--R 2 | 2 2 4 2 2 \|- x + a - a +--R (2a x \|- x + a + x - 2a x )log(----------------) +--R x +--R + +--R +---------+ +--R 2 3 | 2 2 2 2 4 +--R (- a x + 2a )\|- x + a + 2a x - 2a +--R / +--R +---------+ +--R 4 2 | 2 2 3 4 5 2 +--R 4a x \|- x + a + 2a x - 4a x +--R Type: Union(Expression Integer,...) +--E + +--S 36 +bb:=-sqrt(a^2-x^2)/(2*a^2*x^2)-1/(2*a^3)*log((a+sqrt(a^2-x^2))/x) +--R +--R +---------+ +--R | 2 2 +---------+ +--R 2 \|- x + a + a | 2 2 +--R - x log(----------------) - a\|- x + a +--R x +--R (2) ----------------------------------------- +--R 3 2 +--R 2a x +--R Type: Expression Integer +--E + +--S 37 +cc:=aa-bb +--R +--R +---------+ +---------+ +--R | 2 2 | 2 2 +--R \|- x + a + a \|- x + a - a +--R log(----------------) + log(----------------) +--R x x +--R (3) --------------------------------------------- +--R 3 +--R 2a +--R Type: Expression Integer +--E + +--S 38 +dd:=expandLog cc +--R +--R +---------+ +---------+ +--R | 2 2 | 2 2 +--R log(\|- x + a + a) + log(\|- x + a - a) - 2log(x) +--R (4) ------------------------------------------------------- +--R 3 +--R 2a +--R Type: Expression Integer +--E + +--S 39 +ee:=complexNormalize dd +--R +--R x +--R log(-------) +--R +----+ +--R | 2 +--R \|- x +--R (5) - ------------ +--R 3 +--R a +--R Type: Expression Integer +--E + +--S 40 14:243 Schaums and Axiom differ by a constant +ff:=rootSimp ee +--R +--R +---+ +--R log(\|- 1 ) +--R (6) ----------- +--R 3 +--R a +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.244~~~~~$\displaystyle\int{\sqrt{a^2-x^2}}~dx$} +$$\int{\sqrt{a^2-x^2}}= +\frac{x\sqrt{a^2-x^2}}{2}-\frac{a^2}{2}\ln\left(x+\sqrt{a^2-x^2}\right) +$$ +<<*>>= +)clear all + +--S 41 +aa:=integrate(sqrt(a^2-x^2),x) +--R +--R +--R (1) +--R +---------+ +--R +---------+ | 2 2 +--R 3 | 2 2 2 2 4 \|- x + a - a +--R (- 4a \|- x + a - 2a x + 4a )atan(----------------) +--R x +--R + +--R +---------+ +--R 3 2 | 2 2 3 3 +--R (x - 2a x)\|- x + a - 2a x + 2a x +--R / +--R +---------+ +--R | 2 2 2 2 +--R 4a\|- x + a + 2x - 4a +--R Type: Union(Expression Integer,...) +--E + +--S 42 +bb:=(x*sqrt(a^2-x^2))/2+a^2/2*asin(x/a) +--R +--R +---------+ +--R | 2 2 2 x +--R x\|- x + a + a asin(-) +--R a +--R (2) ------------------------- +--R 2 +--R Type: Expression Integer +--E + +--S 43 +cc:=aa-bb +--R +--R +---------+ +--R | 2 2 +--R 2 \|- x + a - a 2 x +--R - 2a atan(----------------) - a asin(-) +--R x a +--R (3) --------------------------------------- +--R 2 +--R Type: Expression Integer +--E + +--S 44 +asinrule:=rule(asin(x) == %i*log(-%i*x+sqrt(1-x^2))) +--R +--R +--------+ +--R | 2 +--R (4) asin(x) == %i log(\|- x + 1 - %i x) +--R Type: RewriteRule(Integer,Complex Integer,Expression Complex Integer) +--E + +--S 45 +dd:=asinrule cc +--R +--R +---------+ +--R | 2 2 +--R |- x + a +--R a |--------- - %i x +---------+ +--R | 2 | 2 2 +--R 2 \| a 2 \|- x + a - a +--R - %i a log(--------------------) - 2a atan(----------------) +--R a x +--R (5) ------------------------------------------------------------ +--R 2 +--R Type: Expression Complex Integer +--E + +--S 46 +atanrule:=rule(atan(x) == -%i/2*log((1+%i*x)/(1-%i*x))) +--R +--R - x + %i +--R %i log(--------) +--R x + %i +--R (6) atan(x) == - ---------------- +--R 2 +--R Type: RewriteRule(Integer,Complex Integer,Expression Complex Integer) +--E + +--S 47 +ee:=atanrule dd +--R +--R +---------+ +--R | 2 2 +--R |- x + a +--R a |--------- - %i x +---------+ +--R | 2 | 2 2 +--R 2 \| a 2 - \|- x + a + %i x + a +--R - %i a log(--------------------) + %i a log(-------------------------) +--R a +---------+ +--R | 2 2 +--R \|- x + a + %i x - a +--R (7) ---------------------------------------------------------------------- +--R 2 +--R Type: Expression Complex Integer +--E + +--S 48 +ff:=expandLog ee +--R +--R (8) +--R +---------+ +--R | 2 2 +---------+ +--R 2 |- x + a 2 | 2 2 +--R - %i a log(a |--------- - %i x) - %i a log(\|- x + a + %i x - a) +--R | 2 +--R \| a +--R + +--R +---------+ +--R 2 | 2 2 2 2 +--R %i a log(\|- x + a - %i x - a) + %i a log(a) + %i a log(- 1) +--R / +--R 2 +--R Type: Expression Complex Integer +--E + +--S 49 +gg:=rootSimp ff +--R +--R (9) +--R +-------+ +-------+ +--R 2 | 2 2 2 | 2 2 +--R - %i a log(%i\|x - a + %i x - a) - %i a log(%i\|x - a - %i x) +--R + +--R +-------+ +--R 2 | 2 2 2 2 +--R %i a log(%i\|x - a - %i x - a) + %i a log(a) + %i a log(- 1) +--R / +--R 2 +--R Type: Expression Complex Integer +--E + +--S 50 14:244 Schaums and Axiom agree +hh:=complexNormalize gg +--R +--R (10) 0 +--R Type: Expression Complex Integer +--E +@ + +\section{\cite{1}:14.245~~~~~$\displaystyle\int{x\sqrt{a^2-x^2}}~dx$} +$$\int{x\sqrt{a^2-x^2}}= +\frac{(a^2-x^2)^{3/2}}{3} +$$ +<<*>>= +)clear all + +--S 51 +aa:=integrate(x*sqrt(a^2-x^2),x) +--R +--R +--R +---------+ +--R 4 3 2 | 2 2 6 2 4 4 2 +--R (- 3a x + 6a x )\|- x + a - x + 6a x - 6a x +--R (1) -------------------------------------------------- +--R +---------+ +--R 2 2 | 2 2 2 3 +--R (3x - 12a )\|- x + a - 9a x + 12a +--R Type: Union(Expression Integer,...) +--E + +--S 52 +bb:=-(a^2-x^2)^(3/2)/3 +--R +--R +---------+ +--R 2 2 | 2 2 +--R (x - a )\|- x + a +--R (2) --------------------- +--R 3 +--R Type: Expression Integer +--E + +--S 53 14:245 Schaums and Axiom differ by a constant +cc:=aa-bb +--R +--R 3 +--R a +--R (3) - -- +--R 3 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.246~~~~~$\displaystyle +\int{x^2\sqrt{a^2-x^2}}~dx$} +$$\int{x^2\sqrt{a^2-x^2}}= +\frac{x(a^2-x^2)^{3/2}}{4}+\frac{a^2x\sqrt{a^2-x^2}}{8}- +\frac{a^4}{8}\ln\left(x+\sqrt{a^2-x^2}\right) +$$ +<<*>>= +)clear all + +--S 54 +aa:=integrate(x^2*sqrt(a^2-x^2),x) +--R +--R +--R (1) +--R +---------+ +--R 5 2 7 | 2 2 4 4 6 2 8 +--R ((- 8a x + 16a )\|- x + a - 2a x + 16a x - 16a ) +--R * +--R +---------+ +--R | 2 2 +--R \|- x + a - a +--R atan(----------------) +--R x +--R + +--R +---------+ +--R 7 2 5 4 3 6 | 2 2 7 3 5 5 3 7 +--R (2x - 17a x + 24a x - 8a x)\|- x + a - 8a x + 28a x - 28a x + 8a x +--R / +--R +---------+ +--R 2 3 | 2 2 4 2 2 4 +--R (32a x - 64a )\|- x + a + 8x - 64a x + 64a +--R Type: Union(Expression Integer,...) +--E + +--S 55 +bb:=-(x*(a^2-x^2)^(3/2))/4+(a^2*x*sqrt(a^2-x^2))/8+a^4/8*asin(x/a) +--R +--R +---------+ +--R 3 2 | 2 2 4 x +--R (2x - a x)\|- x + a + a asin(-) +--R a +--R (2) ----------------------------------- +--R 8 +--R Type: Expression Integer +--E + +--S 56 +cc:=aa-bb +--R +--R +---------+ +--R | 2 2 +--R 4 \|- x + a - a 4 x +--R - 2a atan(----------------) - a asin(-) +--R x a +--R (3) --------------------------------------- +--R 8 +--R Type: Expression Integer +--E + +--S 57 +atanrule:=rule(atan(x) == -%i/2*log((1+%i*x)/(1-%i*x))) +--R +--R - x + %i +--R %i log(--------) +--R x + %i +--R (4) atan(x) == - ---------------- +--R 2 +--R Type: RewriteRule(Integer,Complex Integer,Expression Complex Integer) +--E + +--S 58 +dd:=atanrule cc +--R +--R +---------+ +--R | 2 2 +--R 4 - \|- x + a + %i x + a 4 x +--R %i a log(-------------------------) - a asin(-) +--R +---------+ a +--R | 2 2 +--R \|- x + a + %i x - a +--R (5) ----------------------------------------------- +--R 8 +--R Type: Expression Complex Integer +--E + +--S 59 +asinrule:=rule(asin(x) == %i*log(-%i*x+sqrt(1-x^2))) +--R +--R +--------+ +--R | 2 +--R (6) asin(x) == %i log(\|- x + 1 - %i x) +--R Type: RewriteRule(Integer,Complex Integer,Expression Complex Integer) +--E + +--S 60 +ee:=asinrule dd +--R +--R +---------+ +--R | 2 2 +--R |- x + a +--R a |--------- - %i x +---------+ +--R | 2 | 2 2 +--R 4 \| a 4 - \|- x + a + %i x + a +--R - %i a log(--------------------) + %i a log(-------------------------) +--R a +---------+ +--R | 2 2 +--R \|- x + a + %i x - a +--R (7) ---------------------------------------------------------------------- +--R 8 +--R Type: Expression Complex Integer +--E + +--S 61 +ff:=expandLog ee +--R +--R (8) +--R +---------+ +--R | 2 2 +---------+ +--R 4 |- x + a 4 | 2 2 +--R - %i a log(a |--------- - %i x) - %i a log(\|- x + a + %i x - a) +--R | 2 +--R \| a +--R + +--R +---------+ +--R 4 | 2 2 4 4 +--R %i a log(\|- x + a - %i x - a) + %i a log(a) + %i a log(- 1) +--R / +--R 8 +--R Type: Expression Complex Integer +--E + +--S 62 +gg:=rootSimp ff +--R +--R (9) +--R +-------+ +-------+ +--R 4 | 2 2 4 | 2 2 +--R - %i a log(%i\|x - a + %i x - a) - %i a log(%i\|x - a - %i x) +--R + +--R +-------+ +--R 4 | 2 2 4 4 +--R %i a log(%i\|x - a - %i x - a) + %i a log(a) + %i a log(- 1) +--R / +--R 8 +--R Type: Expression Complex Integer +--E + +--S 63 14:246 Schaums and Axiom agree +hh:=complexNormalize gg +--R +--R (10) 0 +--R Type: Expression Complex Integer +--E +@ + +\section{\cite{1}:14.247~~~~~$\displaystyle +\int{x^3\sqrt{a^2-x^2}}~dx$} +$$\int{x^3\sqrt{a^2-x^2}}= +\frac{(a^2-x^2)^{5/2}}{5}+\frac{a^2(a^2-x^2)^{3/2}}{3} +$$ +<<*>>= +)clear all + +--S 64 +aa:=integrate(x^3*sqrt(a^2-x^2),x) +--R +--R +--R (1) +--R +---------+ +--R 8 3 6 5 4 | 2 2 10 2 8 4 6 6 4 +--R (- 15a x + 65a x - 60a x )\|- x + a - 3x + 40a x - 95a x + 60a x +--R -------------------------------------------------------------------------- +--R +---------+ +--R 4 2 2 4 | 2 2 4 3 2 5 +--R (15x - 180a x + 240a )\|- x + a - 75a x + 300a x - 240a +--R Type: Union(Expression Integer,...) +--E + +--S 65 +bb:=(a^2-x^2)^(5/2)/5-(a^2*(a^2-x^2)^(3/2))/3 +--R +--R +---------+ +--R 4 2 2 4 | 2 2 +--R (3x - a x - 2a )\|- x + a +--R (2) ------------------------------ +--R 15 +--R Type: Expression Integer +--E + +--S 66 14:247 Schaums and Axiom differ by a constant +cc:=aa-bb +--R +--R 5 +--R 2a +--R (3) - --- +--R 15 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.248~~~~~$\displaystyle +\int{\frac{\sqrt{a^2-x^2}}{x}}~dx$} +$$\int{\frac{\sqrt{a^2-x^2}}{x}}= +\sqrt{a^2-x^2}-a\sec^{-1}\left|\frac{x}{a}\right| +$$ +<<*>>= +)clear all + +--S 67 +aa:=integrate(sqrt(a^2-x^2)/x,x) +--R +--R +--R +---------+ +--R +---------+ | 2 2 +--R | 2 2 2 \|- x + a - a 2 +--R (a\|- x + a - a )log(----------------) - x +--R x +--R (1) ---------------------------------------------- +--R +---------+ +--R | 2 2 +--R \|- x + a - a +--R Type: Union(Expression Integer,...) +--E + +--S 68 +bb:=sqrt(a^2-x^2)-a*log((a+sqrt(a^2-x^2))/x) +--R +--R +---------+ +--R | 2 2 +---------+ +--R \|- x + a + a | 2 2 +--R (2) - a log(----------------) + \|- x + a +--R x +--R Type: Expression Integer +--E + +--S 69 +cc:=aa-bb +--R +--R +---------+ +---------+ +--R | 2 2 | 2 2 +--R \|- x + a + a \|- x + a - a +--R (3) a log(----------------) + a log(----------------) + a +--R x x +--R Type: Expression Integer +--E + +--S 70 +dd:=expandLog cc +--R +--R +---------+ +---------+ +--R | 2 2 | 2 2 +--R (4) a log(\|- x + a + a) + a log(\|- x + a - a) - 2a log(x) + a +--R Type: Expression Integer +--E + +--S 71 +ee:=complexNormalize dd +--R +--R x +--R (5) - 2a log(-------) + a +--R +----+ +--R | 2 +--R \|- x +--R Type: Expression Integer +--E + +--S 72 14:248 Schaums and Axiom differ by a constant +ff:=rootSimp ee +--R +--R +---+ +--R (6) 2a log(\|- 1 ) + a +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.249~~~~~$\displaystyle +\int{\frac{\sqrt{a^2-x^2}}{x^2}}~dx$} +$$\int{\frac{\sqrt{a^2-x^2}}{x^2}}= +-\frac{\sqrt{a^2-x^2}}{x}+\ln\left(x+\sqrt{a^2-x^2}\right) +$$ +<<*>>= +)clear all + +--S 73 +aa:=integrate(sqrt(a^2-x^2)/x^2,x) +--R +--R +--R (1) +--R +---------+ +--R +---------+ | 2 2 +---------+ +--R | 2 2 \|- x + a - a | 2 2 2 2 +--R (2x\|- x + a - 2a x)atan(----------------) + a\|- x + a + x - a +--R x +--R ----------------------------------------------------------------------- +--R +---------+ +--R | 2 2 +--R x\|- x + a - a x +--R Type: Union(Expression Integer,...) +--E + +--S 74 +bb:=-sqrt(a^2-x^2)/x-asin(x/a) +--R +--R +---------+ +--R | 2 2 x +--R - \|- x + a - x asin(-) +--R a +--R (2) -------------------------- +--R x +--R Type: Expression Integer +--E + +--S 75 +cc:=aa-bb +--R +--R +---------+ +--R | 2 2 +--R \|- x + a - a x +--R (3) 2atan(----------------) + asin(-) +--R x a +--R Type: Expression Integer +--E + +--S 76 +asinrule:=rule(asin(x) == %i*log(-%i*x+sqrt(1-x^2))) +--R +--R +--------+ +--R | 2 +--R (4) asin(x) == %i log(\|- x + 1 - %i x) +--R Type: RewriteRule(Integer,Complex Integer,Expression Complex Integer) +--E + +--S 77 +dd:=asinrule cc +--R +--R +---------+ +--R | 2 2 +--R |- x + a +--R a |--------- - %i x +---------+ +--R | 2 | 2 2 +--R \| a \|- x + a - a +--R (5) %i log(--------------------) + 2atan(----------------) +--R a x +--R Type: Expression Complex Integer +--E + +--S 78 +atanrule:=rule(atan(x) == -%i/2*log((1+%i*x)/(1-%i*x))) +--R +--R - x + %i +--R %i log(--------) +--R x + %i +--R (6) atan(x) == - ---------------- +--R 2 +--R Type: RewriteRule(Integer,Complex Integer,Expression Complex Integer) +--E + +--S 79 +ee:=atanrule dd +--R +--R +---------+ +--R | 2 2 +--R |- x + a +--R a |--------- - %i x +---------+ +--R | 2 | 2 2 +--R \| a - \|- x + a + %i x + a +--R (7) %i log(--------------------) - %i log(-------------------------) +--R a +---------+ +--R | 2 2 +--R \|- x + a + %i x - a +--R Type: Expression Complex Integer +--E + +--S 80 +ff:=expandLog ee +--R +--R (8) +--R +---------+ +--R | 2 2 +---------+ +--R |- x + a | 2 2 +--R %i log(a |--------- - %i x) + %i log(\|- x + a + %i x - a) +--R | 2 +--R \| a +--R + +--R +---------+ +--R | 2 2 +--R - %i log(\|- x + a - %i x - a) - %i log(a) - %i log(- 1) +--R Type: Expression Complex Integer +--E + +--S 81 +gg:=rootSimp ff +--R +--R (9) +--R +-------+ +-------+ +--R | 2 2 | 2 2 +--R %i log(%i\|x - a + %i x - a) + %i log(%i\|x - a - %i x) +--R + +--R +-------+ +--R | 2 2 +--R - %i log(%i\|x - a - %i x - a) - %i log(a) - %i log(- 1) +--R Type: Expression Complex Integer +--E + +--S 82 14:249 Schaums and Axiom agree +hh:=complexNormalize gg +--R +--R (10) 0 +--R Type: Expression Complex Integer +--E +@ + +\section{\cite{1}:14.250~~~~~$\displaystyle +\int{\frac{\sqrt{a^2-x^2}}{x^3}}~dx$} +$$\int{\frac{\sqrt{a^2-x^2}}{x^3}}= +-\frac{\sqrt{a^2-x^2}}{2x^2}+\frac{1}{2a} +\sec^{-1}\left|\frac{x}{a}\right| +$$ +<<*>>= +)clear all + +--S 83 +aa:=integrate(sqrt(a^2-x^2)/x^3,x) +--R +--R +--R (1) +--R +---------+ +--R +---------+ | 2 2 +--R 2 | 2 2 4 2 2 \|- x + a - a +--R (- 2a x \|- x + a - x + 2a x )log(----------------) +--R x +--R + +--R +---------+ +--R 2 3 | 2 2 2 2 4 +--R (- a x + 2a )\|- x + a + 2a x - 2a +--R / +--R +---------+ +--R 2 2 | 2 2 4 3 2 +--R 4a x \|- x + a + 2a x - 4a x +--R Type: Union(Expression Integer,...) +--E + +--S 84 +bb:=-sqrt(a^2-x^2)/(2*x^2)+1/(2*a)*log((a+sqrt(a^2-x^2))/x) +--R +--R +---------+ +--R | 2 2 +---------+ +--R 2 \|- x + a + a | 2 2 +--R x log(----------------) - a\|- x + a +--R x +--R (2) --------------------------------------- +--R 2 +--R 2a x +--R Type: Expression Integer +--E + +--S 85 +cc:=aa-bb +--R +--R +---------+ +---------+ +--R | 2 2 | 2 2 +--R \|- x + a + a \|- x + a - a +--R - log(----------------) - log(----------------) +--R x x +--R (3) ----------------------------------------------- +--R 2a +--R Type: Expression Integer +--E + +--S 86 +dd:=expandLog cc +--R +--R +---------+ +---------+ +--R | 2 2 | 2 2 +--R - log(\|- x + a + a) - log(\|- x + a - a) + 2log(x) +--R (4) --------------------------------------------------------- +--R 2a +--R Type: Expression Integer +--E + +--S 87 +ee:=complexNormalize dd +--R +--R x +--R log(-------) +--R +----+ +--R | 2 +--R \|- x +--R (5) ------------ +--R a +--R Type: Expression Integer +--E + +--S 88 14:250 Schaums and Axiom differ by a constant +ff:=rootSimp ee +--R +--R +---+ +--R log(\|- 1 ) +--R (6) - ----------- +--R a +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.251~~~~~$\displaystyle\int{\frac{dx}{(a^2-x^2)^{3/2}}}$} +$$\int{\frac{1}{(a^2-x^2)^{3/2}}}= +-\frac{x}{a^2\sqrt{a^2-x^2}} +$$ +<<*>>= +)clear all + +--S 89 +aa:=integrate(1/(a^2-x^2)^(3/2),x) +--R +--R +--R +---------+ +--R | 2 2 +--R - x\|- x + a + a x +--R (1) -------------------------- +--R +---------+ +--R 3 | 2 2 2 2 4 +--R a \|- x + a + a x - a +--R Type: Union(Expression Integer,...) +--E + +--S 90 +bb:=x/(a^2*sqrt(a^2-x^2)) +--R +--R x +--R (2) -------------- +--R +---------+ +--R 2 | 2 2 +--R a \|- x + a +--R Type: Expression Integer +--E + +--S 91 14:251 Schaums and Axiom agree +cc:=aa-bb +--R +--R (3) 0 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.252~~~~~$\displaystyle +\int{\frac{x~dx}{(a^2-x^2)^{3/2}}}$} +$$\int{\frac{x}{(a^2-x^2)^{3/2}}}= +\frac{-1}{\sqrt{a^2-x^2}} +$$ +<<*>>= +)clear all + +--S 92 +aa:=integrate(x/(a^2-x^2)^(3/2),x) +--R +--R +--R 2 +--R x +--R (1) -------------------------- +--R +---------+ +--R 2 | 2 2 2 3 +--R a \|- x + a + a x - a +--R Type: Union(Expression Integer,...) +--E + +--S 93 +bb:=1/sqrt(a^2-x^2) +--R +--R 1 +--R (2) ------------ +--R +---------+ +--R | 2 2 +--R \|- x + a +--R Type: Expression Integer +--E + +--S 94 14:252 Schaums and Axiom differ by a constant +cc:=aa-bb +--R +--R 1 +--R (3) - +--R a +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.253~~~~~$\displaystyle +\int{\frac{x^2dx}{(a^2-x^2)^{3/2}}}$} +$$\int{\frac{x^2}{(a^2-x^2)^{3/2}}}= +\frac{-x}{\sqrt{a^2-x^2}}+\ln\left(x+\sqrt{a^2-x^2}\right) +$$ +<<*>>= +)clear all + +--S 95 +aa:=integrate(x^2/(a^2-x^2)^(3/2),x) +--R +--R +--R (1) +--R +---------+ +--R +---------+ | 2 2 +---------+ +--R | 2 2 2 2 \|- x + a - a | 2 2 +--R (2a\|- x + a + 2x - 2a )atan(----------------) - x\|- x + a + a x +--R x +--R ------------------------------------------------------------------------ +--R +---------+ +--R | 2 2 2 2 +--R a\|- x + a + x - a +--R Type: Union(Expression Integer,...) +--E + +--S 96 +bb:=x/sqrt(a^2-x^2)-asin(x/a) +--R +--R +---------+ +--R x | 2 2 +--R - asin(-)\|- x + a + x +--R a +--R (2) ------------------------- +--R +---------+ +--R | 2 2 +--R \|- x + a +--R Type: Expression Integer +--E + +--S 97 +cc:=aa-bb +--R +--R +---------+ +--R | 2 2 +--R \|- x + a - a x +--R (3) 2atan(----------------) + asin(-) +--R x a +--R Type: Expression Integer +--E + +--S 98 +atanrule:=rule(atan(x) == -%i/2*log((1+%i*x)/(1-%i*x))) +--R +--R - x + %i +--R %i log(--------) +--R x + %i +--R (4) atan(x) == - ---------------- +--R 2 +--R Type: RewriteRule(Integer,Complex Integer,Expression Complex Integer) +--E + +--S 99 +dd:=atanrule cc +--R +--R +---------+ +--R | 2 2 +--R - \|- x + a + %i x + a x +--R (5) - %i log(-------------------------) + asin(-) +--R +---------+ a +--R | 2 2 +--R \|- x + a + %i x - a +--R Type: Expression Complex Integer +--E + +--S 100 +asinrule:=rule(asin(x) == %i*log(-%i*x+sqrt(1-x^2))) +--R +--R +--------+ +--R | 2 +--R (6) asin(x) == %i log(\|- x + 1 - %i x) +--R Type: RewriteRule(Integer,Complex Integer,Expression Complex Integer) +--E + +--S 101 +ee:=asinrule dd +--R +--R +---------+ +--R | 2 2 +--R |- x + a +--R a |--------- - %i x +---------+ +--R | 2 | 2 2 +--R \| a - \|- x + a + %i x + a +--R (7) %i log(--------------------) - %i log(-------------------------) +--R a +---------+ +--R | 2 2 +--R \|- x + a + %i x - a +--R Type: Expression Complex Integer +--E + +--S 102 +ff:=expandLog ee +--R +--R (8) +--R +---------+ +--R | 2 2 +---------+ +--R |- x + a | 2 2 +--R %i log(a |--------- - %i x) + %i log(\|- x + a + %i x - a) +--R | 2 +--R \| a +--R + +--R +---------+ +--R | 2 2 +--R - %i log(\|- x + a - %i x - a) - %i log(a) - %i log(- 1) +--R Type: Expression Complex Integer +--E + +--S 103 +gg:=rootSimp ff +--R +--R (9) +--R +-------+ +-------+ +--R | 2 2 | 2 2 +--R %i log(%i\|x - a + %i x - a) + %i log(%i\|x - a - %i x) +--R + +--R +-------+ +--R | 2 2 +--R - %i log(%i\|x - a - %i x - a) - %i log(a) - %i log(- 1) +--R Type: Expression Complex Integer +--E + +--S 104 14:253 Schaums and Axiom agree +hh:=complexNormalize gg +--R +--R (10) 0 +--R Type: Expression Complex Integer +--E +@ + +\section{\cite{1}:14.254~~~~~$\displaystyle +\int{\frac{x^3dx}{(a^2-x^2)^{3/2}}}$} +$$\int{\frac{x^3}{(a^2-x^2)^{3/2}}}= +\sqrt{a^2-x^2}-\frac{a^2}{\sqrt{a^2-x^2}} +$$ +<<*>>= +)clear all + +--S 105 +aa:=integrate(x^3/(a^2-x^2)^(3/2),x) +--R +--R +--R 4 +--R x +--R (1) - ------------------------------------ +--R +---------+ +--R 2 2 | 2 2 2 3 +--R (x - 2a )\|- x + a - 2a x + 2a +--R Type: Union(Expression Integer,...) +--E + +--S 106 +bb:=sqrt(a^2-x^2)+a^2/sqrt(a^2-x^2) +--R +--R 2 2 +--R - x + 2a +--R (2) ------------ +--R +---------+ +--R | 2 2 +--R \|- x + a +--R Type: Expression Integer +--E + +--S 107 14:254 Schaums and Axiom differ by a constant +cc:=aa-bb +--R +--R (3) 2a +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.255~~~~~$\displaystyle +\int{\frac{dx}{x(a^2-x^2)^{3/2}}}$} +$$\int{\frac{1}{x(a^2-x^2)^{3/2}}}= +\frac{-1}{a^2\sqrt{a^2-x^2}}- +\frac{1}{a^3}\sec^{-1}\left|\frac{x}{a}\right| +$$ +<<*>>= +)clear all + +--S 108 +aa:=integrate(1/(x*(a^2-x^2)^(3/2)),x) +--R +--R +--R +---------+ +--R +---------+ | 2 2 +--R | 2 2 2 2 \|- x + a - a 2 +--R (a\|- x + a + x - a )log(----------------) + x +--R x +--R (1) --------------------------------------------------- +--R +---------+ +--R 4 | 2 2 3 2 5 +--R a \|- x + a + a x - a +--R Type: Union(Expression Integer,...) +--E + +--S 109 +bb:=1/(a^2*sqrt(a^2-x^2))-1/a^3*log((a+sqrt(a^2-x^2))/x) +--R +--R +---------+ +--R +---------+ | 2 2 +--R | 2 2 \|- x + a + a +--R - \|- x + a log(----------------) + a +--R x +--R (2) --------------------------------------- +--R +---------+ +--R 3 | 2 2 +--R a \|- x + a +--R Type: Expression Integer +--E + +--S 110 +cc:=aa-bb +--R +--R +---------+ +---------+ +--R | 2 2 | 2 2 +--R \|- x + a + a \|- x + a - a +--R log(----------------) + log(----------------) + 1 +--R x x +--R (3) ------------------------------------------------- +--R 3 +--R a +--R Type: Expression Integer +--E + +--S 111 +dd:=expandLog cc +--R +--R +---------+ +---------+ +--R | 2 2 | 2 2 +--R log(\|- x + a + a) + log(\|- x + a - a) - 2log(x) + 1 +--R (4) ----------------------------------------------------------- +--R 3 +--R a +--R Type: Expression Integer +--E + +--S 112 +ee:=complexNormalize dd +--R +--R x +--R - 2log(-------) + 1 +--R +----+ +--R | 2 +--R \|- x +--R (5) ------------------- +--R 3 +--R a +--R Type: Expression Integer +--E + +--S 113 14:255 Schaums and Axiom differ by a constant +ff:=rootSimp ee +--R +--R +---+ +--R 2log(\|- 1 ) + 1 +--R (6) ---------------- +--R 3 +--R a +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.256~~~~~$\displaystyle +\int{\frac{dx}{x^2(a^2-x^2)^{3/2}}}$} +$$\int{\frac{1}{x^2(a^2-x^2)^{3/2}}}= +-\frac{\sqrt{a^2-x^2}}{a^4x}-\frac{x}{a^4\sqrt{a^2-x^2}} +$$ +<<*>>= +)clear all + +--S 114 +aa:=integrate(1/(x^2*(a^2-x^2)^(3/2)),x) +--R +--R +--R +---------+ +--R 2 3 | 2 2 4 2 2 4 +--R (4a x - 2a )\|- x + a + 2x - 5a x + 2a +--R (1) --------------------------------------------- +--R +---------+ +--R 4 3 6 | 2 2 5 3 7 +--R (a x - 2a x)\|- x + a - 2a x + 2a x +--R Type: Union(Expression Integer,...) +--E + +--S 115 +bb:=-sqrt(a^2-x^2)/(a^4*x)+x/(a^4*sqrt(a^2-x^2)) +--R +--R 2 2 +--R 2x - a +--R (2) --------------- +--R +---------+ +--R 4 | 2 2 +--R a x\|- x + a +--R Type: Expression Integer +--E + +--S 116 14:256 Schaums and Axiom agree +cc:=aa-bb +--R +--R (3) 0 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.257~~~~~$\displaystyle +\int{\frac{dx}{x^3(a^2-x^2)^{3/2}}}$} +$$\int{\frac{1}{x^3(a^2-x^2)^{3/2}}}= +\frac{1}{2a^2x^2\sqrt{a^2-x^2}}- +\frac{3}{2a^4\sqrt{a^2-x^2}}- +\frac{3}{2a^5}\sec^{-1}\left|\frac{x}{a}\right| +$$ +<<*>>= +)clear all + +--S 117 +aa:=integrate(1/(x^3*(a^2-x^2)^(3/2)),x) +--R +--R +--R (1) +--R +---------+ +--R 4 3 2 | 2 2 6 2 4 4 2 +--R ((9a x - 12a x )\|- x + a + 3x - 15a x + 12a x ) +--R * +--R +---------+ +--R | 2 2 +--R \|- x + a - a +--R log(----------------) +--R x +--R + +--R +---------+ +--R 4 3 2 5 | 2 2 6 2 4 4 2 6 +--R (3a x + 5a x - 4a )\|- x + a + 2x - a x - 7a x + 4a +--R / +--R +---------+ +--R 6 4 8 2 | 2 2 5 6 7 4 9 2 +--R (6a x - 8a x )\|- x + a + 2a x - 10a x + 8a x +--R Type: Union(Expression Integer,...) +--E + +--S 118 +bb:=-1/(2*a^2*x^2*sqrt(a^2-x^2))+3/(2*a^4*sqrt(a^2-x^2))-3/(2*a^5)*log((a+sqrt(a^2-x^2))/x) +--R +--R +---------+ +--R +---------+ | 2 2 +--R 2 | 2 2 \|- x + a + a 2 3 +--R - 3x \|- x + a log(----------------) + 3a x - a +--R x +--R (2) --------------------------------------------------- +--R +---------+ +--R 5 2 | 2 2 +--R 2a x \|- x + a +--R Type: Expression Integer +--E + +--S 119 +cc:=aa-bb +--R +--R +---------+ +---------+ +--R | 2 2 | 2 2 +--R \|- x + a + a \|- x + a - a +--R 3log(----------------) + 3log(----------------) + 2 +--R x x +--R (3) --------------------------------------------------- +--R 5 +--R 2a +--R Type: Expression Integer +--E + +--S 120 +dd:=expandLog cc +--R +--R +---------+ +---------+ +--R | 2 2 | 2 2 +--R 3log(\|- x + a + a) + 3log(\|- x + a - a) - 6log(x) + 2 +--R (4) ------------------------------------------------------------- +--R 5 +--R 2a +--R Type: Expression Integer +--E + +--S 121 +ee:=complexNormalize dd +--R +--R x +--R - 3log(-------) + 1 +--R +----+ +--R | 2 +--R \|- x +--R (5) ------------------- +--R 5 +--R a +--R Type: Expression Integer +--E + +--S 122 14:257 Schaums and Axiom differ by a constant +ff:=rootSimp ee +--R +--R +---+ +--R 3log(\|- 1 ) + 1 +--R (6) ---------------- +--R 5 +--R a +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.258~~~~~$\displaystyle\int{(a^2-x^2)^{3/2}}~dx$} +$$\int{(a^2-x^2)^{3/2}}= +\frac{x(a^2-x^2)^{3/2}}{4}-\frac{3a^2x\sqrt{a^2-x^2}}{8}+ +\frac{3}{8}a^4\ln\left(x+\sqrt{a^2-x^2}\right) +$$ +<<*>>= +)clear all + +--S 123 +aa:=integrate((a^2-x^2)^(3/2),x) +--R +--R +--R (1) +--R +---------+ +--R 5 2 7 | 2 2 4 4 6 2 8 +--R ((- 24a x + 48a )\|- x + a - 6a x + 48a x - 48a ) +--R * +--R +---------+ +--R | 2 2 +--R \|- x + a - a +--R atan(----------------) +--R x +--R + +--R +---------+ +--R 7 2 5 4 3 6 | 2 2 7 3 5 5 3 +--R (- 2x + 21a x - 56a x + 40a x)\|- x + a + 8a x - 44a x + 76a x +--R + +--R 7 +--R - 40a x +--R / +--R +---------+ +--R 2 3 | 2 2 4 2 2 4 +--R (32a x - 64a )\|- x + a + 8x - 64a x + 64a +--R Type: Union(Expression Integer,...) +--E + +--S 124 +bb:=(x*(a^2-x^2)^(3/2))/4+(3*a^2*x*sqrt(a^2-x^2))/8+3/8*a^4*asin(x/a) +--R +--R +---------+ +--R 3 2 | 2 2 4 x +--R (- 2x + 5a x)\|- x + a + 3a asin(-) +--R a +--R (2) --------------------------------------- +--R 8 +--R Type: Expression Integer +--E + +--S 125 +cc:=aa-bb +--R +--R +---------+ +--R | 2 2 +--R 4 \|- x + a - a 4 x +--R - 6a atan(----------------) - 3a asin(-) +--R x a +--R (3) ---------------------------------------- +--R 8 +--R Type: Expression Integer +--E + +--S 126 +asinrule:=rule(asin(x) == %i*log(-%i*x+sqrt(1-x^2))) +--R +--R +--------+ +--R | 2 +--R (4) asin(x) == %i log(\|- x + 1 - %i x) +--R Type: RewriteRule(Integer,Complex Integer,Expression Complex Integer) +--E + +--S 127 +ee:=asinrule cc +--R +--R +---------+ +--R | 2 2 +--R |- x + a +--R a |--------- - %i x +---------+ +--R | 2 | 2 2 +--R 4 \| a 4 \|- x + a - a +--R - 3%i a log(--------------------) - 6a atan(----------------) +--R a x +--R (5) ------------------------------------------------------------- +--R 8 +--R Type: Expression Complex Integer +--E + +--S 128 +atanrule:=rule(atan(x) == -%i/2*log((1+%i*x)/(1-%i*x))) +--R +--R - x + %i +--R %i log(--------) +--R x + %i +--R (6) atan(x) == - ---------------- +--R 2 +--R Type: RewriteRule(Integer,Complex Integer,Expression Complex Integer) +--E + +--S 129 +ff:=atanrule ee +--R +--R (7) +--R +---------+ +--R | 2 2 +--R |- x + a +--R a |--------- - %i x +---------+ +--R | 2 | 2 2 +--R 4 \| a 4 - \|- x + a + %i x + a +--R - 3%i a log(--------------------) + 3%i a log(-------------------------) +--R a +---------+ +--R | 2 2 +--R \|- x + a + %i x - a +--R ------------------------------------------------------------------------ +--R 8 +--R Type: Expression Complex Integer +--E + +--S 130 +gg:=expandLog ff +--R +--R (8) +--R +---------+ +--R | 2 2 +---------+ +--R 4 |- x + a 4 | 2 2 +--R - 3%i a log(a |--------- - %i x) - 3%i a log(\|- x + a + %i x - a) +--R | 2 +--R \| a +--R + +--R +---------+ +--R 4 | 2 2 4 4 +--R 3%i a log(\|- x + a - %i x - a) + 3%i a log(a) + 3%i a log(- 1) +--R / +--R 8 +--R Type: Expression Complex Integer +--E + +--S 131 +hh:=rootSimp gg +--R +--R (9) +--R +-------+ +-------+ +--R 4 | 2 2 4 | 2 2 +--R - 3%i a log(%i\|x - a + %i x - a) - 3%i a log(%i\|x - a - %i x) +--R + +--R +-------+ +--R 4 | 2 2 4 4 +--R 3%i a log(%i\|x - a - %i x - a) + 3%i a log(a) + 3%i a log(- 1) +--R / +--R 8 +--R Type: Expression Complex Integer +--E + +--S 132 14:258 Schaums and Axiom agree +ii:=complexNormalize hh +--R +--R (10) 0 +--R Type: Expression Complex Integer +--E +@ + +\section{\cite{1}:14.259~~~~~$\displaystyle\int{x(a^2-x^2)^{3/2}}~dx$} +$$\int{x(a^2-x^2)^{3/2}}=\frac{(a^2-x^2)^{5/2}}{5}$$ +<<*>>= +)clear all + +--S 133 +aa:=integrate(x*(a^2-x^2)^(3/2),x) +--R +--R +--R (1) +--R +---------+ +--R 8 3 6 5 4 7 2 | 2 2 10 2 8 4 6 +--R (5a x - 30a x + 60a x - 40a x )\|- x + a + x - 15a x + 55a x +--R + +--R 6 4 8 2 +--R - 80a x + 40a x +--R / +--R +---------+ +--R 4 2 2 4 | 2 2 4 3 2 5 +--R (5x - 60a x + 80a )\|- x + a - 25a x + 100a x - 80a +--R Type: Union(Expression Integer,...) +--E + +--S 134 +bb:=-(a^2-x^2)^(5/2)/5 +--R +--R +---------+ +--R 4 2 2 4 | 2 2 +--R (- x + 2a x - a )\|- x + a +--R (2) ------------------------------- +--R 5 +--R Type: Expression Integer +--E + +--S 135 14:259 Schaums and Axiom differ by a constant +cc:=aa-bb +--R +--R 5 +--R a +--R (3) - -- +--R 5 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.260~~~~~$\displaystyle\int{x^2(a^2-x^2)^{3/2}}~dx$} +$$\int{x^2(a^2-x^2)^{3/2}}= +\frac{x(a^2-x^2)^{5/2}}{6}+\frac{a^2x(a^2-x^2)^{3/2}}{24}- +\frac{a^4x\sqrt{a^2-x^2}}{16}+ +\frac{a^6}{16}\ln\left(x+\sqrt{a^2-x^2}\right) +$$ +<<*>>= +)clear all + +--S 136 +aa:=integrate(x^2*(a^2-x^2)^(3/2),x) +--R +--R +--R (1) +--R +---------+ +--R 7 4 9 2 11 | 2 2 6 6 8 4 +--R (- 36a x + 192a x - 192a )\|- x + a - 6a x + 108a x +--R + +--R 10 2 12 +--R - 288a x + 192a +--R * +--R +---------+ +--R | 2 2 +--R \|- x + a - a +--R atan(----------------) +--R x +--R + +--R +---------+ +--R 11 2 9 4 7 6 5 8 3 10 | 2 2 +--R (- 8x + 158a x - 639a x + 982a x - 592a x + 96a x)\|- x + a +--R + +--R 11 3 9 5 7 7 5 9 3 11 +--R 48a x - 388a x + 1062a x - 1266a x + 640a x - 96a x +--R / +--R +---------+ +--R 4 3 2 5 | 2 2 6 2 4 4 2 +--R (288a x - 1536a x + 1536a )\|- x + a + 48x - 864a x + 2304a x +--R + +--R 6 +--R - 1536a +--R Type: Union(Expression Integer,...) +--E + +--S 137 +bb:=-(x*(a^2-x^2)^(5/2))/6+(a^2*x*(a^2-x^2)^(3/2))/24+(a^4*x*sqrt(a^2-x^2))/16+a^6/16*asin(x/a) +--R +--R +---------+ +--R 5 2 3 4 | 2 2 6 x +--R (- 8x + 14a x - 3a x)\|- x + a + 3a asin(-) +--R a +--R (2) ------------------------------------------------ +--R 48 +--R Type: Expression Integer +--E + +--S 138 +cc:=aa-bb +--R +--R +---------+ +--R | 2 2 +--R 6 \|- x + a - a 6 x +--R - 2a atan(----------------) - a asin(-) +--R x a +--R (3) --------------------------------------- +--R 16 +--R Type: Expression Integer +--E + +--S 139 +atanrule:=rule(atan(x) == -%i/2*log((1+%i*x)/(1-%i*x))) +--R +--R - x + %i +--R %i log(--------) +--R x + %i +--R (4) atan(x) == - ---------------- +--R 2 +--R Type: RewriteRule(Integer,Complex Integer,Expression Complex Integer) +--E + +--S 140 +dd:=atanrule cc +--R +--R +---------+ +--R | 2 2 +--R 6 - \|- x + a + %i x + a 6 x +--R %i a log(-------------------------) - a asin(-) +--R +---------+ a +--R | 2 2 +--R \|- x + a + %i x - a +--R (5) ----------------------------------------------- +--R 16 +--R Type: Expression Complex Integer +--E + +--S 141 +asinrule:=rule(asin(x) == %i*log(-%i*x+sqrt(1-x^2))) +--R +--R +--------+ +--R | 2 +--R (6) asin(x) == %i log(\|- x + 1 - %i x) +--R Type: RewriteRule(Integer,Complex Integer,Expression Complex Integer) +--E + +--S 142 +ee:=asinrule dd +--R +--R +---------+ +--R | 2 2 +--R |- x + a +--R a |--------- - %i x +---------+ +--R | 2 | 2 2 +--R 6 \| a 6 - \|- x + a + %i x + a +--R - %i a log(--------------------) + %i a log(-------------------------) +--R a +---------+ +--R | 2 2 +--R \|- x + a + %i x - a +--R (7) ---------------------------------------------------------------------- +--R 16 +--R Type: Expression Complex Integer +--E + +--S 143 +ff:=expandLog ee +--R +--R (8) +--R +---------+ +--R | 2 2 +---------+ +--R 6 |- x + a 6 | 2 2 +--R - %i a log(a |--------- - %i x) - %i a log(\|- x + a + %i x - a) +--R | 2 +--R \| a +--R + +--R +---------+ +--R 6 | 2 2 6 6 +--R %i a log(\|- x + a - %i x - a) + %i a log(a) + %i a log(- 1) +--R / +--R 16 +--R Type: Expression Complex Integer +--E + +--S 144 +gg:=rootSimp ff +--R +--R (9) +--R +-------+ +-------+ +--R 6 | 2 2 6 | 2 2 +--R - %i a log(%i\|x - a + %i x - a) - %i a log(%i\|x - a - %i x) +--R + +--R +-------+ +--R 6 | 2 2 6 6 +--R %i a log(%i\|x - a - %i x - a) + %i a log(a) + %i a log(- 1) +--R / +--R 16 +--R Type: Expression Complex Integer +--E + +--S 145 14:260 Schaums and Axiom agree +hh:=complexNormalize gg +--R +--R (10) 0 +--R Type: Expression Complex Integer +--E +@ + +\section{\cite{1}:14.261~~~~~$\displaystyle\int{x^3(a^2-x^2)^{3/2}}~dx$} +$$\int{x^3(a^2-x^2)^{3/2}}= +\frac{(a^2-x^2)^{7/2}}{7}+\frac{a^2(a^2-x^2)^{5/2}}{5} +$$ +<<*>>= +)clear all + +--S 146 +aa:=integrate(x^3*(a^2-x^2)^(3/2),x) +--R +--R +--R (1) +--R +---------+ +--R 12 3 10 5 8 7 6 9 4 | 2 2 14 +--R (35a x - 336a x + 1015a x - 1260a x + 560a x )\|- x + a + 5x +--R + +--R 2 12 4 10 6 8 8 6 10 4 +--R - 133a x + 721a x - 1575a x + 1540a x - 560a x +--R / +--R +---------+ +--R 6 2 4 4 2 6 | 2 2 6 3 4 +--R (35x - 840a x + 2800a x - 2240a )\|- x + a - 245a x + 1960a x +--R + +--R 5 2 7 +--R - 3920a x + 2240a +--R Type: Union(Expression Integer,...) +--E + +--S 147 +bb:=(a^2-x^2)^(7/2)/7-(a^2*(a^2-x^2)^(5/2))/5 +--R +--R +---------+ +--R 6 2 4 4 2 6 | 2 2 +--R (- 5x + 8a x - a x - 2a )\|- x + a +--R (2) ---------------------------------------- +--R 35 +--R Type: Expression Integer +--E + +--S 148 14:261 Schaums and Axiom differ by a constant +cc:=aa-bb +--R +--R 7 +--R 2a +--R (3) - --- +--R 35 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.262~~~~~$\displaystyle +\int{\frac{(a^2-x^2)^{3/2}}{x}}~dx$} +$$\int{\frac{(a^2-x^2)^{3/2}}{x}}= +\frac{(a^2-x^2)^{3/2}}{3}-a^2\sqrt{a^2-x^2}+ +a^3\sec^{-1}\left|\frac{x}{a}\right| +$$ +<<*>>= +)clear all + +--S 149 +aa:=integrate((a^2-x^2)^(3/2)/x,x) +--R +--R +--R (1) +--R +---------+ +--R +---------+ | 2 2 +--R 3 2 5 | 2 2 4 2 6 \|- x + a - a +--R ((3a x - 12a )\|- x + a - 9a x + 12a )log(----------------) +--R x +--R + +--R +---------+ +--R 4 3 2 | 2 2 6 2 4 4 2 +--R (3a x - 12a x )\|- x + a + x - 9a x + 12a x +--R / +--R +---------+ +--R 2 2 | 2 2 2 3 +--R (3x - 12a )\|- x + a - 9a x + 12a +--R Type: Union(Expression Integer,...) +--E + +--S 150 +bb:=(a^2-x^2)^(3/2)/3+a^2*sqrt(a^2-x^2)-a^3*log((a+sqrt(a^2-x^2))/x) +--R +--R +---------+ +--R | 2 2 +---------+ +--R 3 \|- x + a + a 2 2 | 2 2 +--R - 3a log(----------------) + (- x + 4a )\|- x + a +--R x +--R (2) ----------------------------------------------------- +--R 3 +--R Type: Expression Integer +--E + +--S 151 +cc:=aa-bb +--R +--R +---------+ +---------+ +--R | 2 2 | 2 2 +--R 3 \|- x + a + a 3 \|- x + a - a 3 +--R 3a log(----------------) + 3a log(----------------) + 4a +--R x x +--R (3) --------------------------------------------------------- +--R 3 +--R Type: Expression Integer +--E + +--S 152 +dd:=expandLog cc +--R +--R +---------+ +---------+ +--R 3 | 2 2 3 | 2 2 3 3 +--R 3a log(\|- x + a + a) + 3a log(\|- x + a - a) - 6a log(x) + 4a +--R (4) --------------------------------------------------------------------- +--R 3 +--R Type: Expression Integer +--E + +--S 153 +ee:=complexNormalize dd +--R +--R 3 x 3 +--R - 6a log(-------) + 4a +--R +----+ +--R | 2 +--R \|- x +--R (5) ----------------------- +--R 3 +--R Type: Expression Integer +--E + +--S 154 14:262 Schaums and Axiom differ by a constant +ff:=rootSimp ee +--R +--R 3 +---+ 3 +--R 6a log(\|- 1 ) + 4a +--R (6) -------------------- +--R 3 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.263~~~~~$\displaystyle +\int{\frac{(a^2-x^2)^{3/2}}{x^2}}~dx$} +$$\int{\frac{(a^2-x^2)^{3/2}}{x^2}}= +-\frac{(a^2-x^2)^{3/2}}{x}+\frac{3x\sqrt{a^2-x^2}}{2}- +\frac{3}{2}a^2\ln\left(x+\sqrt{a^2-x^2}\right) +$$ +<<*>>= +)clear all + +--S 155 +aa:=integrate((a^2-x^2)^{3/2}/x^2,x) +--R +--R +--R (1) +--R +---------+ +--R +---------+ | 2 2 +--R 2 3 4 | 2 2 3 3 5 \|- x + a - a +--R ((6a x - 24a x)\|- x + a - 18a x + 24a x)atan(----------------) +--R x +--R + +--R +---------+ +--R 4 3 2 5 | 2 2 6 2 4 4 2 6 +--R (3a x + 2a x - 8a )\|- x + a + x - 3a x - 6a x + 8a +--R / +--R +---------+ +--R 3 2 | 2 2 3 3 +--R (2x - 8a x)\|- x + a - 6a x + 8a x +--R Type: Union(Expression Integer,...) +--E + +--S 156 +bb:=-(a^2-x^2)^(3/2)/x-(3*x*sqrt(a^2-x^2))/2-3/2*a^2*asin(x/a) +--R +--R +---------+ +--R 2 2 | 2 2 2 x +--R (- x - 2a )\|- x + a - 3a x asin(-) +--R a +--R (2) --------------------------------------- +--R 2x +--R Type: Expression Integer +--E + +--S 157 +cc:=aa-bb +--R +--R +---------+ +--R | 2 2 +--R 2 \|- x + a - a 2 x +--R 6a atan(----------------) + 3a asin(-) +--R x a +--R (3) -------------------------------------- +--R 2 +--R Type: Expression Integer +--E + +--S 158 +asinrule:=rule(asin(x) == %i*log(-%i*x+sqrt(1-x^2))) +--R +--R +--------+ +--R | 2 +--R (4) asin(x) == %i log(\|- x + 1 - %i x) +--R Type: RewriteRule(Integer,Complex Integer,Expression Complex Integer) +--E + +--S 159 +dd:=asinrule cc +--R +--R +---------+ +--R | 2 2 +--R |- x + a +--R a |--------- - %i x +---------+ +--R | 2 | 2 2 +--R 2 \| a 2 \|- x + a - a +--R 3%i a log(--------------------) + 6a atan(----------------) +--R a x +--R (5) ----------------------------------------------------------- +--R 2 +--R Type: Expression Complex Integer +--E + +--S 160 +atanrule:=rule(atan(x) == -%i/2*log((1+%i*x)/(1-%i*x))) +--R +--R - x + %i +--R %i log(--------) +--R x + %i +--R (6) atan(x) == - ---------------- +--R 2 +--R Type: RewriteRule(Integer,Complex Integer,Expression Complex Integer) +--E + +--S 161 +ee:=atanrule dd +--R +--R +---------+ +--R | 2 2 +--R |- x + a +--R a |--------- - %i x +---------+ +--R | 2 | 2 2 +--R 2 \| a 2 - \|- x + a + %i x + a +--R 3%i a log(--------------------) - 3%i a log(-------------------------) +--R a +---------+ +--R | 2 2 +--R \|- x + a + %i x - a +--R (7) ---------------------------------------------------------------------- +--R 2 +--R Type: Expression Complex Integer +--E + +--S 162 +ff:=expandLog ee +--R +--R (8) +--R +---------+ +--R | 2 2 +---------+ +--R 2 |- x + a 2 | 2 2 +--R 3%i a log(a |--------- - %i x) + 3%i a log(\|- x + a + %i x - a) +--R | 2 +--R \| a +--R + +--R +---------+ +--R 2 | 2 2 2 2 +--R - 3%i a log(\|- x + a - %i x - a) - 3%i a log(a) - 3%i a log(- 1) +--R / +--R 2 +--R Type: Expression Complex Integer +--E + +--S 163 +gg:=rootSimp ff +--R +--R (9) +--R +-------+ +-------+ +--R 2 | 2 2 2 | 2 2 +--R 3%i a log(%i\|x - a + %i x - a) + 3%i a log(%i\|x - a - %i x) +--R + +--R +-------+ +--R 2 | 2 2 2 2 +--R - 3%i a log(%i\|x - a - %i x - a) - 3%i a log(a) - 3%i a log(- 1) +--R / +--R 2 +--R Type: Expression Complex Integer +--E + +--S 164 14:263 Schaums and Axiom agree +hh:=complexNormalize gg +--R +--R (10) 0 +--R Type: Expression Complex Integer +--E +@ + +\section{\cite{1}:14.264~~~~~$\displaystyle +\int{\frac{(a^2-x^2)^{3/2}}{x^3}}~dx$} +$$\int{\frac{(a^2-x^2)^{3/2}}{x^3}}= +-\frac{(a^2-x^2)^{3/2}}{2x^2}+\frac{3}{2}\sqrt{a^2-x^2}- +\frac{3}{2}a\sec^{-1}\left|\frac{x}{a}\right| +$$ +<<*>>= +)clear all + +--S 165 +aa:=integrate((a^2-x^2)^(3/2)/x^3,x) +--R +--R +--R (1) +--R +---------+ +--R +---------+ | 2 2 +--R 4 3 2 | 2 2 2 4 4 2 \|- x + a - a +--R ((- 3a x + 12a x )\|- x + a + 9a x - 12a x )log(----------------) +--R x +--R + +--R +---------+ +--R 4 3 2 5 | 2 2 6 2 4 4 2 6 +--R (4a x + 3a x - 4a )\|- x + a + 2x - 3a x - 5a x + 4a +--R / +--R +---------+ +--R 4 2 2 | 2 2 4 3 2 +--R (2x - 8a x )\|- x + a - 6a x + 8a x +--R Type: Union(Expression Integer,...) +--E + +--S 166 +bb:=-(a^2-x^2)^(3/2)/(2*x^2)-(3*sqrt(a^2-x^2))/2+3/2*a*log((a+sqrt(a^2-x^2))/x) +--R +--R +---------+ +--R | 2 2 +---------+ +--R 2 \|- x + a + a 2 2 | 2 2 +--R 3a x log(----------------) + (- 2x - a )\|- x + a +--R x +--R (2) ----------------------------------------------------- +--R 2 +--R 2x +--R Type: Expression Integer +--E + +--S 167 +cc:=aa-bb +--R +--R +---------+ +---------+ +--R | 2 2 | 2 2 +--R \|- x + a + a \|- x + a - a +--R - 3a log(----------------) - 3a log(----------------) - 2a +--R x x +--R (3) ---------------------------------------------------------- +--R 2 +--R Type: Expression Integer +--E + +--S 168 +dd:=expandLog cc +--R +--R +---------+ +---------+ +--R | 2 2 | 2 2 +--R - 3a log(\|- x + a + a) - 3a log(\|- x + a - a) + 6a log(x) - 2a +--R (4) ---------------------------------------------------------------------- +--R 2 +--R Type: Expression Integer +--E + +--S 169 +ee:=complexNormalize dd +--R +--R x +--R (5) 3a log(-------) - a +--R +----+ +--R | 2 +--R \|- x +--R Type: Expression Integer +--E + +--S 170 14:264 Schaums and Axiom differ by a constant +ff:=rootSimp ee +--R +--R +---+ +--R (6) - 3a log(\|- 1 ) - a +--R Type: Expression Integer +--E + +)spool +)lisp (bye) +@ + +\eject +\begin{thebibliography}{99} +\bibitem{1} Spiegel, Murray R. +{\sl Mathematical Handbook of Formulas and Tables}\\ +Schaum's Outline Series McGraw-Hill 1968 pp68-69 +\end{thebibliography} +\end{document} diff --git a/src/axiom-website/CATS/schaum11.input.pdf b/src/axiom-website/CATS/schaum11.input.pdf new file mode 100644 index 0000000..cb7ac08 --- /dev/null +++ b/src/axiom-website/CATS/schaum11.input.pdf @@ -0,0 +1,3980 @@ +%PDF-1.2 +7 0 obj +<< +/Type/Encoding 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b/src/axiom-website/CATS/schaum12.input.pamphlet @@ -0,0 +1,2870 @@ +\documentclass{article} +\usepackage{axiom} +\begin{document} +\title{\$SPAD/input schaum12.input} +\author{Timothy Daly} +\maketitle +\eject +\tableofcontents +\eject +\section{\cite{1}:14.265~~~~~$\displaystyle +\int{\frac{dx}{ax^2+bx+c}}$} +$$\int{\frac{1}{ax^2+bx+c}}= +\left\{ +\begin{array}{l} +\displaystyle\frac{2}{\sqrt{4ac-b^2}}\tan^{-1}\frac{2ax+b}{\sqrt{4ac-b^2}}\\ +\displaystyle\frac{1}{\sqrt{b^2-4ac}}\ln\left( +\frac{2ax+b-\sqrt{b^2-4ac}}{2ax+b+\sqrt{b^2-4ac}}\right) +\end{array} +\right. +$$ +<<*>>= +)spool schaum12.output +)set message test on +)set message auto off +)clear all + +--S 1 +aa:=integrate(1/(a*x^2+b*x+c),x) +--R +--R (1) +--R [ +--R log +--R +-----------+ +--R 2 2 2 | 2 2 2 +--R (2a x + 2a b x - 2a c + b )\|- 4a c + b + (8a c - 2a b )x +--R + +--R 3 +--R 4a b c - b +--R / +--R 2 +--R a x + b x + c +--R / +--R +-----------+ +--R | 2 +--R \|- 4a c + b +--R , +--R +---------+ +--R | 2 +--R (2a x + b)\|4a c - b +--R 2atan(----------------------) +--R 2 +--R 4a c - b +--R -----------------------------] +--R +---------+ +--R | 2 +--R \|4a c - b +--R Type: Union(List Expression Integer,...) +--E + +--S 2 +bb1:=2/sqrt(4*a*c-b^2)*atan((2*a*x+b)/sqrt(4*a*c-b^2)) +--R +--R +--R 2a x + b +--R 2atan(------------) +--R +---------+ +--R | 2 +--R \|4a c - b +--R (2) ------------------- +--R +---------+ +--R | 2 +--R \|4a c - b +--R Type: Expression Integer +--E + +--S 3 +bb2:=1/sqrt(b^2-4*a*c)*log((2*a*x+b-sqrt(b^2-4*a*c))/(2*a*x+b+sqrt(b^2-4*a*c))) +--R +--R +--R +-----------+ +--R | 2 +--R - \|- 4a c + b + 2a x + b +--R log(---------------------------) +--R +-----------+ +--R | 2 +--R \|- 4a c + b + 2a x + b +--R (3) -------------------------------- +--R +-----------+ +--R | 2 +--R \|- 4a c + b +--R Type: Expression Integer +--E + +--S 4 +cc1:=aa.1-bb1 +--R +--R (4) +--R +---------+ +--R | 2 +--R \|4a c - b +--R * +--R log +--R +-----------+ +--R 2 2 2 | 2 2 2 +--R (2a x + 2a b x - 2a c + b )\|- 4a c + b + (8a c - 2a b )x +--R + +--R 3 +--R 4a b c - b +--R / +--R 2 +--R a x + b x + c +--R + +--R +-----------+ +--R | 2 2a x + b +--R - 2\|- 4a c + b atan(------------) +--R +---------+ +--R | 2 +--R \|4a c - b +--R / +--R +-----------+ +---------+ +--R | 2 | 2 +--R \|- 4a c + b \|4a c - b +--R Type: Expression Integer +--E + +--S 5 +cc2:=aa.1-bb2 +--R +--R (5) +--R log +--R +-----------+ +--R 2 2 2 | 2 2 2 +--R (2a x + 2a b x - 2a c + b )\|- 4a c + b + (8a c - 2a b )x +--R + +--R 3 +--R 4a b c - b +--R / +--R 2 +--R a x + b x + c +--R + +--R +-----------+ +--R | 2 +--R - \|- 4a c + b + 2a x + b +--R - log(---------------------------) +--R +-----------+ +--R | 2 +--R \|- 4a c + b + 2a x + b +--R / +--R +-----------+ +--R | 2 +--R \|- 4a c + b +--R Type: Expression Integer +--E + +--S 6 +cc3:=aa.2-bb1 +--R +--R +---------+ +--R | 2 +--R (2a x + b)\|4a c - b 2a x + b +--R 2atan(----------------------) - 2atan(------------) +--R 2 +---------+ +--R 4a c - b | 2 +--R \|4a c - b +--R (6) --------------------------------------------------- +--R +---------+ +--R | 2 +--R \|4a c - b +--R Type: Expression Integer +--E + +--S 7 +cc4:=aa.2-bb2 +--R +--R (7) +--R +-----------+ +--R +---------+ | 2 +--R | 2 - \|- 4a c + b + 2a x + b +--R - \|4a c - b log(---------------------------) +--R +-----------+ +--R | 2 +--R \|- 4a c + b + 2a x + b +--R + +--R +---------+ +--R +-----------+ | 2 +--R | 2 (2a x + b)\|4a c - b +--R 2\|- 4a c + b atan(----------------------) +--R 2 +--R 4a c - b +--R / +--R +-----------+ +---------+ +--R | 2 | 2 +--R \|- 4a c + b \|4a c - b +--R Type: Expression Integer +--E + +--S 8 +atanrule:=rule(atan(x) == -%i/2*log((1+%i*x)/(1-%i*x))) +--R +--R - x + %i +--R %i log(--------) +--R x + %i +--R (8) atan(x) == - ---------------- +--R 2 +--R Type: RewriteRule(Integer,Complex Integer,Expression Complex Integer) +--E + +--S 9 +dd3:=atanrule cc3 +--R +--R (9) +--R +---------+ +--R | 2 +--R \|4a c - b + 2%i a x + %i b +--R %i log(-----------------------------) +--R +---------+ +--R | 2 +--R \|4a c - b - 2%i a x - %i b +--R + +--R +---------+ +--R | 2 2 +--R (- 2a x - b)\|4a c - b + 4%i a c - %i b +--R - %i log(------------------------------------------) +--R +---------+ +--R | 2 2 +--R (2a x + b)\|4a c - b + 4%i a c - %i b +--R / +--R +---------+ +--R | 2 +--R \|4a c - b +--R Type: Expression Complex Integer +--E + +--S 10 +ee3:=expandLog dd3 +--R +--R (10) +--R +---------+ +--R | 2 2 +--R %i log((2a x + b)\|4a c - b + 4%i a c - %i b ) +--R + +--R +---------+ +--R | 2 2 +--R - %i log((2a x + b)\|4a c - b - 4%i a c + %i b ) +--R + +--R +---------+ +--R | 2 +--R %i log(\|4a c - b + 2%i a x + %i b) +--R + +--R +---------+ +--R | 2 +--R - %i log(\|4a c - b - 2%i a x - %i b) - %i log(- 1) +--R / +--R +---------+ +--R | 2 +--R \|4a c - b +--R Type: Expression Complex Integer +--E + +--S 11 14:265 Schaums and Axiom agree +ff3:=complexNormalize ee3 +--R +--R (11) 0 +--R Type: Expression Complex Integer +--E +@ + +\section{\cite{1}:14.266~~~~~$\displaystyle +\int{\frac{x~dx}{ax^2+bx+c}}$} +$$\int{\frac{x}{ax^2+bx+c}}= +\frac{1}{2a}\ln(ax^2+bx+c)-\frac{b}{2a}\int{\frac{1}{ax^2+bx+c}} +$$ +<<*>>= +)clear all + +--S 12 +aa:=integrate(x/(a*x^2+b*x+c),x) +--R +--R (1) +--R [ +--R b +--R * +--R log +--R +-----------+ +--R 2 2 2 | 2 2 2 +--R (2a x + 2a b x - 2a c + b )\|- 4a c + b + (- 8a c + 2a b )x +--R + +--R 3 +--R - 4a b c + b +--R / +--R 2 +--R a x + b x + c +--R + +--R +-----------+ +--R 2 | 2 +--R log(a x + b x + c)\|- 4a c + b +--R / +--R +-----------+ +--R | 2 +--R 2a\|- 4a c + b +--R , +--R +---------+ +--R | 2 +---------+ +--R (2a x + b)\|4a c - b 2 | 2 +--R - 2b atan(----------------------) + log(a x + b x + c)\|4a c - b +--R 2 +--R 4a c - b +--R -------------------------------------------------------------------] +--R +---------+ +--R | 2 +--R 2a\|4a c - b +--R Type: Union(List Expression Integer,...) +--E + +--S 13 +t1:=integrate(1/(a*x^2+b*x+c),x) +--R +--R +--R (2) +--R [ +--R log +--R +-----------+ +--R 2 2 2 | 2 2 2 +--R (2a x + 2a b x - 2a c + b )\|- 4a c + b + (8a c - 2a b )x +--R + +--R 3 +--R 4a b c - b +--R / +--R 2 +--R a x + b x + c +--R / +--R +-----------+ +--R | 2 +--R \|- 4a c + b +--R , +--R +---------+ +--R | 2 +--R (2a x + b)\|4a c - b +--R 2atan(----------------------) +--R 2 +--R 4a c - b +--R -----------------------------] +--R +---------+ +--R | 2 +--R \|4a c - b +--R Type: Union(List Expression Integer,...) +--E + +--S 14 +bb1:=1/(2*a)*log(a*x^2+b*x+c)-b/(2*a)*t1.1 +--R +--R +--R (3) +--R - +--R b +--R * +--R log +--R +-----------+ +--R 2 2 2 | 2 2 2 +--R (2a x + 2a b x - 2a c + b )\|- 4a c + b + (8a c - 2a b )x +--R + +--R 3 +--R 4a b c - b +--R / +--R 2 +--R a x + b x + c +--R + +--R +-----------+ +--R 2 | 2 +--R log(a x + b x + c)\|- 4a c + b +--R / +--R +-----------+ +--R | 2 +--R 2a\|- 4a c + b +--R Type: Expression Integer +--E + +--S 15 +bb2:=1/(2*a)*log(a*x^2+b*x+c)-b/(2*a)*t1.2 +--R +--R +--R +---------+ +--R | 2 +---------+ +--R (2a x + b)\|4a c - b 2 | 2 +--R - 2b atan(----------------------) + log(a x + b x + c)\|4a c - b +--R 2 +--R 4a c - b +--R (4) ------------------------------------------------------------------- +--R +---------+ +--R | 2 +--R 2a\|4a c - b +--R Type: Expression Integer +--E + +--S 16 +cc1:=aa.1-bb1 +--R +--R (5) +--R b +--R * +--R log +--R +-----------+ +--R 2 2 2 | 2 2 2 +--R (2a x + 2a b x - 2a c + b )\|- 4a c + b + (8a c - 2a b )x +--R + +--R 3 +--R 4a b c - b +--R / +--R 2 +--R a x + b x + c +--R + +--R b +--R * +--R log +--R +-----------+ +--R 2 2 2 | 2 2 2 +--R (2a x + 2a b x - 2a c + b )\|- 4a c + b + (- 8a c + 2a b )x +--R + +--R 3 +--R - 4a b c + b +--R / +--R 2 +--R a x + b x + c +--R / +--R +-----------+ +--R | 2 +--R 2a\|- 4a c + b +--R Type: Expression Integer +--E + +--S 17 +cc2:=aa.2-bb1 +--R +--R (6) +--R +---------+ +--R | 2 +--R b\|4a c - b +--R * +--R log +--R +-----------+ +--R 2 2 2 | 2 2 2 +--R (2a x + 2a b x - 2a c + b )\|- 4a c + b + (8a c - 2a b )x +--R + +--R 3 +--R 4a b c - b +--R / +--R 2 +--R a x + b x + c +--R + +--R +---------+ +--R +-----------+ | 2 +--R | 2 (2a x + b)\|4a c - b +--R - 2b\|- 4a c + b atan(----------------------) +--R 2 +--R 4a c - b +--R / +--R +-----------+ +---------+ +--R | 2 | 2 +--R 2a\|- 4a c + b \|4a c - b +--R Type: Expression Integer +--E + +--S 18 +cc3:=aa.2-bb1 +--R +--R (7) +--R +---------+ +--R | 2 +--R b\|4a c - b +--R * +--R log +--R +-----------+ +--R 2 2 2 | 2 2 2 +--R (2a x + 2a b x - 2a c + b )\|- 4a c + b + (8a c - 2a b )x +--R + +--R 3 +--R 4a b c - b +--R / +--R 2 +--R a x + b x + c +--R + +--R +---------+ +--R +-----------+ | 2 +--R | 2 (2a x + b)\|4a c - b +--R - 2b\|- 4a c + b atan(----------------------) +--R 2 +--R 4a c - b +--R / +--R +-----------+ +---------+ +--R | 2 | 2 +--R 2a\|- 4a c + b \|4a c - b +--R Type: Expression Integer +--E + +--S 19 14:266 Schaums and Axiom agree +cc4:=aa.2-bb2 +--R +--R (8) 0 +--R Type: Expression Integer +--E + +@ + +\section{\cite{1}:14.267~~~~~$\displaystyle +\int{\frac{x^2dx}{ax^2+bx+c}}$} +$$\int{\frac{x^2}{ax^2+bx+c}}= +\frac{x}{a}-\frac{b}{2a^2}\ln(ax^2+bx+c)+\frac{b^2-2ac}{2a^2} +\int{\frac{1}{ax^2+bx+c}} +$$ +<<*>>= +)clear all + +--S 20 +aa:=integrate(x^2/(a*x^2+b*x+c),x) +--R +--R +--R (1) +--R [ +--R 2 +--R (2a c - b ) +--R * +--R log +--R +-----------+ +--R 2 2 2 | 2 2 2 +--R (2a x + 2a b x - 2a c + b )\|- 4a c + b + (- 8a c + 2a b )x +--R + +--R 3 +--R - 4a b c + b +--R / +--R 2 +--R a x + b x + c +--R + +--R +-----------+ +--R 2 | 2 +--R (- b log(a x + b x + c) + 2a x)\|- 4a c + b +--R / +--R +-----------+ +--R 2 | 2 +--R 2a \|- 4a c + b +--R , +--R +--R +---------+ +--R | 2 +--R 2 (2a x + b)\|4a c - b +--R (- 4a c + 2b )atan(----------------------) +--R 2 +--R 4a c - b +--R + +--R +---------+ +--R 2 | 2 +--R (- b log(a x + b x + c) + 2a x)\|4a c - b +--R / +--R +---------+ +--R 2 | 2 +--R 2a \|4a c - b +--R ] +--R Type: Union(List Expression Integer,...) +--E + +--S 21 +t1:=integrate(1/(a*x^2+b*x+c),x) +--R +--R +--R (2) +--R [ +--R log +--R +-----------+ +--R 2 2 2 | 2 2 2 +--R (2a x + 2a b x - 2a c + b )\|- 4a c + b + (8a c - 2a b )x +--R + +--R 3 +--R 4a b c - b +--R / +--R 2 +--R a x + b x + c +--R / +--R +-----------+ +--R | 2 +--R \|- 4a c + b +--R , +--R +---------+ +--R | 2 +--R (2a x + b)\|4a c - b +--R 2atan(----------------------) +--R 2 +--R 4a c - b +--R -----------------------------] +--R +---------+ +--R | 2 +--R \|4a c - b +--R Type: Union(List Expression Integer,...) +--E + +--S 22 +bb1:=x/a-b/(2*a^2)*log(a*x^2+b*x+c)+(b^2-2*a*c)/(2*a^2)*t1.1 +--R +--R +--R (3) +--R 2 +--R (- 2a c + b ) +--R * +--R log +--R +-----------+ +--R 2 2 2 | 2 2 2 +--R (2a x + 2a b x - 2a c + b )\|- 4a c + b + (8a c - 2a b )x +--R + +--R 3 +--R 4a b c - b +--R / +--R 2 +--R a x + b x + c +--R + +--R +-----------+ +--R 2 | 2 +--R (- b log(a x + b x + c) + 2a x)\|- 4a c + b +--R / +--R +-----------+ +--R 2 | 2 +--R 2a \|- 4a c + b +--R Type: Expression Integer +--E + +--S 23 +bb2:=x/a-b/(2*a^2)*log(a*x^2+b*x+c)+(b^2-2*a*c)/(2*a^2)*t1.2 +--R +--R +--R (4) +--R +---------+ +--R | 2 +--R 2 (2a x + b)\|4a c - b +--R (- 4a c + 2b )atan(----------------------) +--R 2 +--R 4a c - b +--R + +--R +---------+ +--R 2 | 2 +--R (- b log(a x + b x + c) + 2a x)\|4a c - b +--R / +--R +---------+ +--R 2 | 2 +--R 2a \|4a c - b +--R Type: Expression Integer +--E + +--S 24 +cc1:=bb1-aa.1 +--R +--R (5) +--R 2 +--R (- 2a c + b ) +--R * +--R log +--R +-----------+ +--R 2 2 2 | 2 2 2 +--R (2a x + 2a b x - 2a c + b )\|- 4a c + b + (8a c - 2a b )x +--R + +--R 3 +--R 4a b c - b +--R / +--R 2 +--R a x + b x + c +--R + +--R 2 +--R (- 2a c + b ) +--R * +--R log +--R +-----------+ +--R 2 2 2 | 2 2 2 +--R (2a x + 2a b x - 2a c + b )\|- 4a c + b + (- 8a c + 2a b )x +--R + +--R 3 +--R - 4a b c + b +--R / +--R 2 +--R a x + b x + c +--R / +--R +-----------+ +--R 2 | 2 +--R 2a \|- 4a c + b +--R Type: Expression Integer +--E + +--S 25 14:267 Schaums and Axiom differ by a constant +dd1:=complexNormalize cc1 +--R +--R 2 3 2 2 +--R (- 2a c + b )log(- 16a c + 4a b ) +--R (6) --------------------------------- +--R +-----------+ +--R 2 | 2 +--R 2a \|- 4a c + b +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.268~~~~~$\displaystyle +\int{\frac{x^m~dx}{ax^2+bx+c}}$} +$$\int{\frac{x^m}{ax^2+bx+c}}= +\frac{x^{m-1}}{(m-1)a}-\frac{c}{a}\int{\frac{x^{m-2}}{ax^2+bx+c}}- +\frac{b}{a}\int{\frac{x^{m-1}}{ax^2+bx+c}} +$$ +<<*>>= +)clear all + +--S 26 14:268 Axiom cannot compute this integral +aa:=integrate(x^m/(a*x^2+b*x+c),x) +--R +--R +--R x m +--I ++ %N +--I (1) | --------------- d%N +--R ++ 2 +--I c + %N b + %N a +--R Type: Union(Expression Integer,...) +--E + +@ + +\section{\cite{1}:14.269~~~~~$\displaystyle +\int{\frac{dx}{x(ax^2+bx+c)}}$} +$$\int{\frac{1}{x(ax^2+bx+c)}}= +\frac{1}{2c}\ln\left(\frac{x^2}{ax^2+bx+c}\right)- +\frac{b}{2c}\int{\frac{1}{ax^2+bx+c}} +$$ +<<*>>= +)clear all + +--S 27 +aa:=integrate(1/(x*(a*x^2+b*x+c)),x) +--R +--R +--R (1) +--R [ +--R b +--R * +--R log +--R +-----------+ +--R 2 2 2 | 2 2 2 +--R (2a x + 2a b x - 2a c + b )\|- 4a c + b + (- 8a c + 2a b )x +--R + +--R 3 +--R - 4a b c + b +--R / +--R 2 +--R a x + b x + c +--R + +--R +-----------+ +--R 2 | 2 +--R (- log(a x + b x + c) + 2log(x))\|- 4a c + b +--R / +--R +-----------+ +--R | 2 +--R 2c\|- 4a c + b +--R , +--R +--R +---------+ +--R | 2 +--R (2a x + b)\|4a c - b +--R - 2b atan(----------------------) +--R 2 +--R 4a c - b +--R + +--R +---------+ +--R 2 | 2 +--R (- log(a x + b x + c) + 2log(x))\|4a c - b +--R / +--R +---------+ +--R | 2 +--R 2c\|4a c - b +--R ] +--R Type: Union(List Expression Integer,...) +--E + +--S 28 +t1:=integrate(1/(a*x^2+b*x+c),x) +--R +--R +--R (2) +--R [ +--R log +--R +-----------+ +--R 2 2 2 | 2 2 2 +--R (2a x + 2a b x - 2a c + b )\|- 4a c + b + (8a c - 2a b )x +--R + +--R 3 +--R 4a b c - b +--R / +--R 2 +--R a x + b x + c +--R / +--R +-----------+ +--R | 2 +--R \|- 4a c + b +--R , +--R +---------+ +--R | 2 +--R (2a x + b)\|4a c - b +--R 2atan(----------------------) +--R 2 +--R 4a c - b +--R -----------------------------] +--R +---------+ +--R | 2 +--R \|4a c - b +--R Type: Union(List Expression Integer,...) +--E + +--S 29 +bb1:=1/(2*c)*log(x^2/(a*x^2+b*x+c))-b/(2*c)*t1.1 +--R +--R +--R (3) +--R - +--R b +--R * +--R log +--R +-----------+ +--R 2 2 2 | 2 2 2 +--R (2a x + 2a b x - 2a c + b )\|- 4a c + b + (8a c - 2a b )x +--R + +--R 3 +--R 4a b c - b +--R / +--R 2 +--R a x + b x + c +--R + +--R 2 +-----------+ +--R x | 2 +--R log(--------------)\|- 4a c + b +--R 2 +--R a x + b x + c +--R / +--R +-----------+ +--R | 2 +--R 2c\|- 4a c + b +--R Type: Expression Integer +--E + +--S 30 +bb2:=1/(2*c)*log(x^2/(a*x^2+b*x+c))-b/(2*c)*t1.2 +--R +--R +--R +---------+ +--R | 2 2 +---------+ +--R (2a x + b)\|4a c - b x | 2 +--R - 2b atan(----------------------) + log(--------------)\|4a c - b +--R 2 2 +--R 4a c - b a x + b x + c +--R (4) ------------------------------------------------------------------- +--R +---------+ +--R | 2 +--R 2c\|4a c - b +--R Type: Expression Integer +--E + +--S 31 +cc1:=bb1-aa.1 +--R +--R (5) +--R - +--R b +--R * +--R log +--R +-----------+ +--R 2 2 2 | 2 2 2 +--R (2a x + 2a b x - 2a c + b )\|- 4a c + b + (8a c - 2a b )x +--R + +--R 3 +--R 4a b c - b +--R / +--R 2 +--R a x + b x + c +--R + +--R - +--R b +--R * +--R log +--R +-----------+ +--R 2 2 2 | 2 +--R (2a x + 2a b x - 2a c + b )\|- 4a c + b +--R + +--R 2 2 3 +--R (- 8a c + 2a b )x - 4a b c + b +--R / +--R 2 +--R a x + b x + c +--R + +--R 2 +-----------+ +--R 2 x | 2 +--R (log(a x + b x + c) - 2log(x) + log(--------------))\|- 4a c + b +--R 2 +--R a x + b x + c +--R / +--R +-----------+ +--R | 2 +--R 2c\|- 4a c + b +--R Type: Expression Integer +--E + +--S 32 +dd1:=expandLog cc1 +--R +--R (6) +--R - +--R b +--R * +--R log +--R +-----------+ +--R 2 2 2 | 2 2 2 +--R (2a x + 2a b x - 2a c + b )\|- 4a c + b + (8a c - 2a b )x +--R + +--R 3 +--R 4a b c - b +--R + +--R - +--R b +--R * +--R log +--R +-----------+ +--R 2 2 2 | 2 2 2 +--R (2a x + 2a b x - 2a c + b )\|- 4a c + b + (- 8a c + 2a b )x +--R + +--R 3 +--R - 4a b c + b +--R + +--R 2 +--R 2b log(a x + b x + c) +--R / +--R +-----------+ +--R | 2 +--R 2c\|- 4a c + b +--R Type: Expression Integer +--E + +--S 33 14:269 Schaums and Axiom differ by a constant +ee1:=complexNormalize dd1 +--R +--R 3 2 2 +--R b log(- 16a c + 4a b ) +--R (7) - ---------------------- +--R +-----------+ +--R | 2 +--R 2c\|- 4a c + b +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.270~~~~~$\displaystyle +\int{\frac{dx}{x^2(ax^2+bx+c)}}$} +$$\int{\frac{1}{x^2(ax^2+bx+c)}}= +\frac{b}{2c^2}\ln\left(\frac{ax^2+bx+c}{x^2}\right)-\frac{1}{cx}+ +\frac{b^2-2ac}{2c^2}\int{\frac{1}{ax^2+bx+c}} +$$ +<<*>>= +)clear all + +--S 34 +aa:=integrate(1/(x^2*(a*x^2+b*x+c)),x) +--R +--R +--R (1) +--R [ +--R 2 +--R (2a c - b )x +--R * +--R log +--R +-----------+ +--R 2 2 2 | 2 2 2 +--R (2a x + 2a b x - 2a c + b )\|- 4a c + b + (- 8a c + 2a b )x +--R + +--R 3 +--R - 4a b c + b +--R / +--R 2 +--R a x + b x + c +--R + +--R +-----------+ +--R 2 | 2 +--R (b x log(a x + b x + c) - 2b x log(x) - 2c)\|- 4a c + b +--R / +--R +-----------+ +--R 2 | 2 +--R 2c x\|- 4a c + b +--R , +--R +--R +---------+ +--R | 2 +--R 2 (2a x + b)\|4a c - b +--R (- 4a c + 2b )x atan(----------------------) +--R 2 +--R 4a c - b +--R + +--R +---------+ +--R 2 | 2 +--R (b x log(a x + b x + c) - 2b x log(x) - 2c)\|4a c - b +--R / +--R +---------+ +--R 2 | 2 +--R 2c x\|4a c - b +--R ] +--R Type: Union(List Expression Integer,...) +--E + +--S 35 +t1:=integrate(1/(a*x^2+b*x+c),x) +--R +--R +--R (2) +--R [ +--R log +--R +-----------+ +--R 2 2 2 | 2 2 2 +--R (2a x + 2a b x - 2a c + b )\|- 4a c + b + (8a c - 2a b )x +--R + +--R 3 +--R 4a b c - b +--R / +--R 2 +--R a x + b x + c +--R / +--R +-----------+ +--R | 2 +--R \|- 4a c + b +--R , +--R +---------+ +--R | 2 +--R (2a x + b)\|4a c - b +--R 2atan(----------------------) +--R 2 +--R 4a c - b +--R -----------------------------] +--R +---------+ +--R | 2 +--R \|4a c - b +--R Type: Union(List Expression Integer,...) +--E + +--S 36 +bb1:=b/(2*c^2)*log((a*x^2+b*x+c)/x^2)-1/(c*x)+(b^2-2*a*c)/(2*c^2)*t1.1 +--R +--R +--R (3) +--R 2 +--R (- 2a c + b )x +--R * +--R log +--R +-----------+ +--R 2 2 2 | 2 2 2 +--R (2a x + 2a b x - 2a c + b )\|- 4a c + b + (8a c - 2a b )x +--R + +--R 3 +--R 4a b c - b +--R / +--R 2 +--R a x + b x + c +--R + +--R 2 +-----------+ +--R a x + b x + c | 2 +--R (b x log(--------------) - 2c)\|- 4a c + b +--R 2 +--R x +--R / +--R +-----------+ +--R 2 | 2 +--R 2c x\|- 4a c + b +--R Type: Expression Integer +--E + +--S 37 +bb2:=b/(2*c^2)*log((a*x^2+b*x+c)/x^2)-1/(c*x)+(b^2-2*a*c)/(2*c^2)*t1.2 +--R +--R +--R (4) +--R +---------+ +--R | 2 +--R 2 (2a x + b)\|4a c - b +--R (- 4a c + 2b )x atan(----------------------) +--R 2 +--R 4a c - b +--R + +--R 2 +---------+ +--R a x + b x + c | 2 +--R (b x log(--------------) - 2c)\|4a c - b +--R 2 +--R x +--R / +--R +---------+ +--R 2 | 2 +--R 2c x\|4a c - b +--R Type: Expression Integer +--E + +--S 38 +cc1:=bb1-aa.1 +--R +--R (5) +--R 2 +--R (- 2a c + b ) +--R * +--R log +--R +-----------+ +--R 2 2 2 | 2 2 2 +--R (2a x + 2a b x - 2a c + b )\|- 4a c + b + (8a c - 2a b )x +--R + +--R 3 +--R 4a b c - b +--R / +--R 2 +--R a x + b x + c +--R + +--R 2 +--R (- 2a c + b ) +--R * +--R log +--R +-----------+ +--R 2 2 2 | 2 2 2 +--R (2a x + 2a b x - 2a c + b )\|- 4a c + b + (- 8a c + 2a b )x +--R + +--R 3 +--R - 4a b c + b +--R / +--R 2 +--R a x + b x + c +--R + +--R 2 +-----------+ +--R 2 a x + b x + c | 2 +--R (- b log(a x + b x + c) + 2b log(x) + b log(--------------))\|- 4a c + b +--R 2 +--R x +--R / +--R +-----------+ +--R 2 | 2 +--R 2c \|- 4a c + b +--R Type: Expression Integer +--E + +--S 39 +dd1:=expandLog cc1 +--R +--R (6) +--R 2 +--R (- 2a c + b ) +--R * +--R log +--R +-----------+ +--R 2 2 2 | 2 2 2 +--R (2a x + 2a b x - 2a c + b )\|- 4a c + b + (8a c - 2a b )x +--R + +--R 3 +--R 4a b c - b +--R + +--R 2 +--R (- 2a c + b ) +--R * +--R log +--R +-----------+ +--R 2 2 2 | 2 2 2 +--R (2a x + 2a b x - 2a c + b )\|- 4a c + b + (- 8a c + 2a b )x +--R + +--R 3 +--R - 4a b c + b +--R + +--R 2 2 +--R (4a c - 2b )log(a x + b x + c) +--R / +--R +-----------+ +--R 2 | 2 +--R 2c \|- 4a c + b +--R Type: Expression Integer +--E + +--S 40 14:270 Schaums and Axiom differ by a constant +ee1:=complexNormalize dd1 +--R +--R 2 3 2 2 +--R (- 2a c + b )log(- 16a c + 4a b ) +--R (7) --------------------------------- +--R +-----------+ +--R 2 | 2 +--R 2c \|- 4a c + b +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.271~~~~~$\displaystyle +\int{\frac{dx}{x^n(ax^2+bx+c)}}$} +$$\int{\frac{1}{x^n(ax^2+bx+c)}}= +-\frac{1}{(n-1)cx^{n-1}}- +\frac{b}{c}\int{\frac{1}{x^{n-1}(ax^2+bx+c)}}- +\frac{a}{c}\int{\frac{1}{x^{x-2}(ax^2+bx+c)}} +$$ +<<*>>= +)clear all + +--S 41 14:271 Axiom cannot compute this integral +aa:=integrate(1/(x^n*(a*x^2+b*x+c)),x) +--R +--R +--R x +--R ++ 1 +--I (1) | -------------------- d%N +--R ++ 2 n +--I (c + %N b + %N a)%N +--R Type: Union(Expression Integer,...) +--E + +@ + +\section{\cite{1}:14.272~~~~~$\displaystyle +\int{\frac{dx}{(ax^2+bx+c)^2}}$} +$$\int{\frac{1}{(ax^2+bx+c)^2}}= +\frac{2xa+b}{(4ac-b^2)(ax^2+bx+c)}+ +\frac{2a}{4ac-b^2}\int{\frac{1}{ax^2+bx+c}} +$$ +<<*>>= +)clear all + +--S 42 +aa:=integrate(1/(a*x^2+b*x+c)^2,x) +--R +--R +--R (1) +--R [ +--R 2 2 +--R (2a x + 2a b x + 2a c) +--R * +--R log +--R +-----------+ +--R 2 2 2 | 2 2 2 +--R (2a x + 2a b x - 2a c + b )\|- 4a c + b + (8a c - 2a b )x +--R + +--R 3 +--R 4a b c - b +--R / +--R 2 +--R a x + b x + c +--R + +--R +-----------+ +--R | 2 +--R (2a x + b)\|- 4a c + b +--R / +--R +-----------+ +--R 2 2 2 3 2 2 | 2 +--R ((4a c - a b )x + (4a b c - b )x + 4a c - b c)\|- 4a c + b +--R , +--R +---------+ +--R | 2 +---------+ +--R 2 2 (2a x + b)\|4a c - b | 2 +--R (4a x + 4a b x + 4a c)atan(----------------------) + (2a x + b)\|4a c - b +--R 2 +--R 4a c - b +--R ---------------------------------------------------------------------------- +--R +---------+ +--R 2 2 2 3 2 2 | 2 +--R ((4a c - a b )x + (4a b c - b )x + 4a c - b c)\|4a c - b +--R ] +--R Type: Union(List Expression Integer,...) +--E + +--S 43 +t1:=integrate(1/(a*x^2+b*x+c),x) +--R +--R +--R (2) +--R [ +--R log +--R +-----------+ +--R 2 2 2 | 2 2 2 +--R (2a x + 2a b x - 2a c + b )\|- 4a c + b + (8a c - 2a b )x +--R + +--R 3 +--R 4a b c - b +--R / +--R 2 +--R a x + b x + c +--R / +--R +-----------+ +--R | 2 +--R \|- 4a c + b +--R , +--R +---------+ +--R | 2 +--R (2a x + b)\|4a c - b +--R 2atan(----------------------) +--R 2 +--R 4a c - b +--R -----------------------------] +--R +---------+ +--R | 2 +--R \|4a c - b +--R Type: Union(List Expression Integer,...) +--E + +--S 44 +bb1:=(2*a*x+b)/((4*a*c-b^2)*(a*x^2+b*x+c))+(2*a)/(4*a*c-b^2)*t1.1 +--R +--R (3) +--R 2 2 +--R (2a x + 2a b x + 2a c) +--R * +--R log +--R +-----------+ +--R 2 2 2 | 2 2 2 +--R (2a x + 2a b x - 2a c + b )\|- 4a c + b + (8a c - 2a b )x +--R + +--R 3 +--R 4a b c - b +--R / +--R 2 +--R a x + b x + c +--R + +--R +-----------+ +--R | 2 +--R (2a x + b)\|- 4a c + b +--R / +--R +-----------+ +--R 2 2 2 3 2 2 | 2 +--R ((4a c - a b )x + (4a b c - b )x + 4a c - b c)\|- 4a c + b +--R Type: Expression Integer +--E + +--S 45 +bb2:=(2*a*x+b)/((4*a*c-b^2)*(a*x^2+b*x+c))+(2*a)/(4*a*c-b^2)*t1.2 +--R +--R (4) +--R +---------+ +--R | 2 +---------+ +--R 2 2 (2a x + b)\|4a c - b | 2 +--R (4a x + 4a b x + 4a c)atan(----------------------) + (2a x + b)\|4a c - b +--R 2 +--R 4a c - b +--R ---------------------------------------------------------------------------- +--R +---------+ +--R 2 2 2 3 2 2 | 2 +--R ((4a c - a b )x + (4a b c - b )x + 4a c - b c)\|4a c - b +--R Type: Expression Integer +--E + +--S 46 +cc1:=aa.1-bb1 +--R +--R (5) 0 +--R Type: Expression Integer +--E + +--S 47 +cc2:=aa.2-bb1 +--R +--R (6) +--R - +--R +---------+ +--R | 2 +--R 2a\|4a c - b +--R * +--R log +--R +-----------+ +--R 2 2 2 | 2 2 2 +--R (2a x + 2a b x - 2a c + b )\|- 4a c + b + (8a c - 2a b )x +--R + +--R 3 +--R 4a b c - b +--R / +--R 2 +--R a x + b x + c +--R + +--R +---------+ +--R +-----------+ | 2 +--R | 2 (2a x + b)\|4a c - b +--R 4a\|- 4a c + b atan(----------------------) +--R 2 +--R 4a c - b +--R / +--R +-----------+ +---------+ +--R 2 | 2 | 2 +--R (4a c - b )\|- 4a c + b \|4a c - b +--R Type: Expression Integer +--E + +--S 48 +cc3:=aa.1-bb2 +--R +--R (7) +--R +---------+ +--R | 2 +--R 2a\|4a c - b +--R * +--R log +--R +-----------+ +--R 2 2 2 | 2 2 2 +--R (2a x + 2a b x - 2a c + b )\|- 4a c + b + (8a c - 2a b )x +--R + +--R 3 +--R 4a b c - b +--R / +--R 2 +--R a x + b x + c +--R + +--R +---------+ +--R +-----------+ | 2 +--R | 2 (2a x + b)\|4a c - b +--R - 4a\|- 4a c + b atan(----------------------) +--R 2 +--R 4a c - b +--R / +--R +-----------+ +---------+ +--R 2 | 2 | 2 +--R (4a c - b )\|- 4a c + b \|4a c - b +--R Type: Expression Integer +--E + +--S 49 14:272 Schaums and Axiom agree +cc4:=aa.2-bb2 +--R +--R (8) 0 +--R Type: Expression Integer +--E + +@ + +\section{\cite{1}:14.273~~~~~$\displaystyle +\int{\frac{x~dx}{(ax^2+bx+c)^2}}$} +$$\int{\frac{x}{(ax^2+bx+c)^2}}= +-\frac{bx+2c}{(4ac-b^2)(ax^2+bx+c)}- +\frac{b}{4ac-b^2}\int{\frac{1}{ax^2+bx+c}} +$$ +<<*>>= +)clear all + +--S 50 +aa:=integrate(x/(a*x^2+b*x+c)^2,x) +--R +--R +--R (1) +--R [ +--R 2 2 +--R (a b x + b x + b c) +--R * +--R log +--R +-----------+ +--R 2 2 2 | 2 2 2 +--R (2a x + 2a b x - 2a c + b )\|- 4a c + b + (- 8a c + 2a b )x +--R + +--R 3 +--R - 4a b c + b +--R / +--R 2 +--R a x + b x + c +--R + +--R +-----------+ +--R | 2 +--R (- b x - 2c)\|- 4a c + b +--R / +--R +-----------+ +--R 2 2 2 3 2 2 | 2 +--R ((4a c - a b )x + (4a b c - b )x + 4a c - b c)\|- 4a c + b +--R , +--R +--R +---------+ +--R | 2 +--R 2 2 (2a x + b)\|4a c - b +--R (- 2a b x - 2b x - 2b c)atan(----------------------) +--R 2 +--R 4a c - b +--R + +--R +---------+ +--R | 2 +--R (- b x - 2c)\|4a c - b +--R / +--R +---------+ +--R 2 2 2 3 2 2 | 2 +--R ((4a c - a b )x + (4a b c - b )x + 4a c - b c)\|4a c - b +--R ] +--R Type: Union(List Expression Integer,...) +--E + +--S 51 +t1:=integrate(1/(a*x^2+b*x+c),x) +--R +--R +--R (2) +--R [ +--R log +--R +-----------+ +--R 2 2 2 | 2 2 2 +--R (2a x + 2a b x - 2a c + b )\|- 4a c + b + (8a c - 2a b )x +--R + +--R 3 +--R 4a b c - b +--R / +--R 2 +--R a x + b x + c +--R / +--R +-----------+ +--R | 2 +--R \|- 4a c + b +--R , +--R +---------+ +--R | 2 +--R (2a x + b)\|4a c - b +--R 2atan(----------------------) +--R 2 +--R 4a c - b +--R -----------------------------] +--R +---------+ +--R | 2 +--R \|4a c - b +--R Type: Union(List Expression Integer,...) +--E + +--S 52 +bb1:=-(b*x+2*c)/((4*a*c-b^2)*(a*x^2+b*x+c))-b/(4*a*c-b^2)*t1.1 +--R +--R +--R (3) +--R 2 2 +--R (- a b x - b x - b c) +--R * +--R log +--R +-----------+ +--R 2 2 2 | 2 2 2 +--R (2a x + 2a b x - 2a c + b )\|- 4a c + b + (8a c - 2a b )x +--R + +--R 3 +--R 4a b c - b +--R / +--R 2 +--R a x + b x + c +--R + +--R +-----------+ +--R | 2 +--R (- b x - 2c)\|- 4a c + b +--R / +--R +-----------+ +--R 2 2 2 3 2 2 | 2 +--R ((4a c - a b )x + (4a b c - b )x + 4a c - b c)\|- 4a c + b +--R Type: Expression Integer +--E + +--S 53 +bb2:=-(b*x+2*c)/((4*a*c-b^2)*(a*x^2+b*x+c))-b/(4*a*c-b^2)*t1.2 +--R +--R +--R (4) +--R +---------+ +--R | 2 +--R 2 2 (2a x + b)\|4a c - b +--R (- 2a b x - 2b x - 2b c)atan(----------------------) +--R 2 +--R 4a c - b +--R + +--R +---------+ +--R | 2 +--R (- b x - 2c)\|4a c - b +--R / +--R +---------+ +--R 2 2 2 3 2 2 | 2 +--R ((4a c - a b )x + (4a b c - b )x + 4a c - b c)\|4a c - b +--R Type: Expression Integer +--E + +--S 54 +cc1:=bb1-aa.1 +--R +--R (5) +--R - +--R b +--R * +--R log +--R +-----------+ +--R 2 2 2 | 2 2 2 +--R (2a x + 2a b x - 2a c + b )\|- 4a c + b + (8a c - 2a b )x +--R + +--R 3 +--R 4a b c - b +--R / +--R 2 +--R a x + b x + c +--R + +--R - +--R b +--R * +--R log +--R +-----------+ +--R 2 2 2 | 2 +--R (2a x + 2a b x - 2a c + b )\|- 4a c + b +--R + +--R 2 2 3 +--R (- 8a c + 2a b )x - 4a b c + b +--R / +--R 2 +--R a x + b x + c +--R / +--R +-----------+ +--R 2 | 2 +--R (4a c - b )\|- 4a c + b +--R Type: Expression Integer +--E + +--S 55 +dd1:=expandLog cc1 +--R +--R (6) +--R - +--R b +--R * +--R log +--R +-----------+ +--R 2 2 2 | 2 2 2 +--R (2a x + 2a b x - 2a c + b )\|- 4a c + b + (8a c - 2a b )x +--R + +--R 3 +--R 4a b c - b +--R + +--R - +--R b +--R * +--R log +--R +-----------+ +--R 2 2 2 | 2 2 2 +--R (2a x + 2a b x - 2a c + b )\|- 4a c + b + (- 8a c + 2a b )x +--R + +--R 3 +--R - 4a b c + b +--R + +--R 2 +--R 2b log(a x + b x + c) +--R / +--R +-----------+ +--R 2 | 2 +--R (4a c - b )\|- 4a c + b +--R Type: Expression Integer +--E + +--S 56 14:273 Schaums and Axiom differ by a constant +ee1:=complexNormalize dd1 +--R +--R 3 2 2 +--R b log(- 16a c + 4a b ) +--R (7) - ------------------------- +--R +-----------+ +--R 2 | 2 +--R (4a c - b )\|- 4a c + b +--R Type: Expression Integer +--E + +@ + +\section{\cite{1}:14.274~~~~~$\displaystyle +\int{\frac{x^2~dx}{(ax^2+bx+c)^2}}$} +$$\int{\frac{x^2}{(ax^2+bx+c)^2}}= +\frac{(b^2-2ac)x+bc}{a(4ac-b^2)(ax^2+bx+c)}+ +\frac{2c}{4ac-b^2}\int{\frac{1}{ax^2+bx+c}} +$$ +<<*>>= +)clear all + +--S 57 +aa:=integrate(x^2/(a*x^2+b*x+c)^2,x) +--R +--R +--R (1) +--R [ +--R 2 2 2 +--R (2a c x + 2a b c x + 2a c ) +--R * +--R log +--R +-----------+ +--R 2 2 2 | 2 2 2 +--R (2a x + 2a b x - 2a c + b )\|- 4a c + b + (8a c - 2a b )x +--R + +--R 3 +--R 4a b c - b +--R / +--R 2 +--R a x + b x + c +--R + +--R +-----------+ +--R 2 | 2 +--R ((- 2a c + b )x + b c)\|- 4a c + b +--R / +--R +-----------+ +--R 3 2 2 2 2 3 2 2 2 | 2 +--R ((4a c - a b )x + (4a b c - a b )x + 4a c - a b c)\|- 4a c + b +--R , +--R +--R +---------+ +--R | 2 +--R 2 2 2 (2a x + b)\|4a c - b +--R (4a c x + 4a b c x + 4a c )atan(----------------------) +--R 2 +--R 4a c - b +--R + +--R +---------+ +--R 2 | 2 +--R ((- 2a c + b )x + b c)\|4a c - b +--R / +--R +---------+ +--R 3 2 2 2 2 3 2 2 2 | 2 +--R ((4a c - a b )x + (4a b c - a b )x + 4a c - a b c)\|4a c - b +--R ] +--R Type: Union(List Expression Integer,...) +--E + +--S 58 +t1:=integrate(1/(a*x^2+b*x+c),x) +--R +--R +--R (2) +--R [ +--R log +--R +-----------+ +--R 2 2 2 | 2 2 2 +--R (2a x + 2a b x - 2a c + b )\|- 4a c + b + (8a c - 2a b )x +--R + +--R 3 +--R 4a b c - b +--R / +--R 2 +--R a x + b x + c +--R / +--R +-----------+ +--R | 2 +--R \|- 4a c + b +--R , +--R +---------+ +--R | 2 +--R (2a x + b)\|4a c - b +--R 2atan(----------------------) +--R 2 +--R 4a c - b +--R -----------------------------] +--R +---------+ +--R | 2 +--R \|4a c - b +--R Type: Union(List Expression Integer,...) +--E + +--S 59 +bb1:=((b^2-2*a*c)*x+b*c)/(a*(4*a*c-b^2)*(a*x^2+b*x+c))+(2*c)/(4*a*c-b^2)*t1.1 +--R +--R (3) +--R 2 2 2 +--R (2a c x + 2a b c x + 2a c ) +--R * +--R log +--R +-----------+ +--R 2 2 2 | 2 2 2 +--R (2a x + 2a b x - 2a c + b )\|- 4a c + b + (8a c - 2a b )x +--R + +--R 3 +--R 4a b c - b +--R / +--R 2 +--R a x + b x + c +--R + +--R +-----------+ +--R 2 | 2 +--R ((- 2a c + b )x + b c)\|- 4a c + b +--R / +--R +-----------+ +--R 3 2 2 2 2 3 2 2 2 | 2 +--R ((4a c - a b )x + (4a b c - a b )x + 4a c - a b c)\|- 4a c + b +--R Type: Expression Integer +--E + +--S 60 +bb2:=((b^2-2*a*c)*x+b*c)/(a*(4*a*c-b^2)*(a*x^2+b*x+c))+(2*c)/(4*a*c-b^2)*t1.2 +--R +--R (4) +--R +---------+ +--R | 2 +--R 2 2 2 (2a x + b)\|4a c - b +--R (4a c x + 4a b c x + 4a c )atan(----------------------) +--R 2 +--R 4a c - b +--R + +--R +---------+ +--R 2 | 2 +--R ((- 2a c + b )x + b c)\|4a c - b +--R / +--R +---------+ +--R 3 2 2 2 2 3 2 2 2 | 2 +--R ((4a c - a b )x + (4a b c - a b )x + 4a c - a b c)\|4a c - b +--R Type: Expression Integer +--E + +--S 61 14:274 Schaums and Axiom agree +cc1:=aa.1-bb1 +--R +--R (5) 0 +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.275~~~~~$\displaystyle +\int{\frac{x^m~dx}{(ax^2+bx+c)^n}}$} +$$ +\begin{array}{lr} +\displaystyle\int{\frac{x^m}{(ax^2+bx+c)^n}}= +&\displaystyle-\frac{x^{m-1}}{(2n-m-1)a(ax^2+bx+c)^{n-1}}\\ +&\\ +&\displaystyle+\frac{(m-1)c}{(2n-m-1)a}\int{\frac{x^{m-2}}{(ax^2+bx+2)^n}}\\ +&\\ +&\displaystyle-\frac{(n-m)b}{(2n-m-1)a}\int{\frac{x^{m-1}}{(ax^2+bx+c)^n}} +\end{array} +$$ +<<*>>= +)clear all + +--S 62 14:275 Axiom cannot compute this integral +aa:=integrate(x^m/(a*x^2+b*x+c)^n,x) +--R +--R +--R x m +--I ++ %N +--I (1) | ------------------ d%N +--R ++ 2 n +--I (c + %N b + %N a) +--R Type: Union(Expression Integer,...) +--E + +@ + +\section{\cite{1}:14.276~~~~~$\displaystyle +\int{\frac{x^{2n-1}~dx}{(ax^2+bx+c)^n}}$} +$$\begin{array}{lr} +\displaystyle\int{\frac{x^{2n-1}}{(ax^2+bx+c)^n}}= +&\displaystyle\frac{1}{a}\int{\frac{x^{2n-3}}{(ax^2+bx+c)^{n-1}}}\\ +&\\ +&\displaystyle-\frac{c}{a}\int{\frac{x^{2n-3}}{(ax^2+bx+c)^n}}\\ +&\\ +&\displaystyle-\frac{b}{a}\int{\frac{x^{2n-2}}{(ax^2+bx+c)^n}} +\end{array} +$$ +<<*>>= +)clear all + +--S 63 14:276 Axiom cannot compute this integral +aa:=integrate(x^(2*n-1)/(a*x^2+b*x+c)^n,x) +--R +--R +--R x 2n - 1 +--I ++ %N +--I (1) | ------------------ d%N +--R ++ 2 n +--I (c + %N b + %N a) +--R Type: Union(Expression Integer,...) +--E + +@ + +\section{\cite{1}:14.277~~~~~$\displaystyle +\int{\frac{dx}{x(ax^2+bx+c)^2}}$} +$$\begin{array}{lr} +\displaystyle\int{\frac{1}{x(ax^2+bx+c)^2}}= +&\displaystyle\frac{1}{2c(ax^2+bx+2)}\\ +&\\ +&\displaystyle-\frac{b}{2c}\int{\frac{1}{(ax^2+bx+c)^2}}\\ +&\\ +&\displaystyle+\frac{1}{c}\int{\frac{1}{x(ax^2+bx+c)}} +\end{array} +$$ +<<*>>= +)clear all + +--S 64 +aa:=integrate(1/(x*(a*x^2+b*x+c)^2),x) +--R +--R +--R (1) +--R [ +--R 2 3 2 2 4 2 3 +--R ((6a b c - a b )x + (6a b c - b )x + 6a b c - b c) +--R * +--R log +--R +-----------+ +--R 2 2 2 | 2 2 2 +--R (2a x + 2a b x - 2a c + b )\|- 4a c + b + (- 8a c + 2a b )x +--R + +--R 3 +--R - 4a b c + b +--R / +--R 2 +--R a x + b x + c +--R + +--R 2 2 2 3 2 2 +--R ((- 4a c + a b )x + (- 4a b c + b )x - 4a c + b c) +--R * +--R 2 +--R log(a x + b x + c) +--R + +--R 2 2 2 3 2 2 +--R ((8a c - 2a b )x + (8a b c - 2b )x + 8a c - 2b c)log(x) +--R + +--R 2 2 +--R - 2a b c x + 4a c - 2b c +--R * +--R +-----------+ +--R | 2 +--R \|- 4a c + b +--R / +--R +-----------+ +--R 2 3 2 2 2 3 3 2 4 2 3 | 2 +--R ((8a c - 2a b c )x + (8a b c - 2b c )x + 8a c - 2b c )\|- 4a c + b +--R , +--R +--R 2 3 2 2 4 2 3 +--R ((- 12a b c + 2a b )x + (- 12a b c + 2b )x - 12a b c + 2b c) +--R * +--R +---------+ +--R | 2 +--R (2a x + b)\|4a c - b +--R atan(----------------------) +--R 2 +--R 4a c - b +--R + +--R 2 2 2 3 2 2 +--R ((- 4a c + a b )x + (- 4a b c + b )x - 4a c + b c) +--R * +--R 2 +--R log(a x + b x + c) +--R + +--R 2 2 2 3 2 2 +--R ((8a c - 2a b )x + (8a b c - 2b )x + 8a c - 2b c)log(x) +--R + +--R 2 2 +--R - 2a b c x + 4a c - 2b c +--R * +--R +---------+ +--R | 2 +--R \|4a c - b +--R / +--R +---------+ +--R 2 3 2 2 2 3 3 2 4 2 3 | 2 +--R ((8a c - 2a b c )x + (8a b c - 2b c )x + 8a c - 2b c )\|4a c - b +--R ] +--R Type: Union(List Expression Integer,...) +--E + +--S 65 +t1:=integrate(1/(a*x^2+b*x+c)^2,x) +--R +--R +--R (2) +--R [ +--R 2 2 +--R (2a x + 2a b x + 2a c) +--R * +--R log +--R +-----------+ +--R 2 2 2 | 2 2 2 +--R (2a x + 2a b x - 2a c + b )\|- 4a c + b + (8a c - 2a b )x +--R + +--R 3 +--R 4a b c - b +--R / +--R 2 +--R a x + b x + c +--R + +--R +-----------+ +--R | 2 +--R (2a x + b)\|- 4a c + b +--R / +--R +-----------+ +--R 2 2 2 3 2 2 | 2 +--R ((4a c - a b )x + (4a b c - b )x + 4a c - b c)\|- 4a c + b +--R , +--R +---------+ +--R | 2 +---------+ +--R 2 2 (2a x + b)\|4a c - b | 2 +--R (4a x + 4a b x + 4a c)atan(----------------------) + (2a x + b)\|4a c - b +--R 2 +--R 4a c - b +--R ---------------------------------------------------------------------------- +--R +---------+ +--R 2 2 2 3 2 2 | 2 +--R ((4a c - a b )x + (4a b c - b )x + 4a c - b c)\|4a c - b +--R ] +--R Type: Union(List Expression Integer,...) +--E + +--S 66 +t2:=integrate(1/(x*(a*x^2+b*x+c)),x) +--R +--R +--R (3) +--R [ +--R b +--R * +--R log +--R +-----------+ +--R 2 2 2 | 2 2 2 +--R (2a x + 2a b x - 2a c + b )\|- 4a c + b + (- 8a c + 2a b )x +--R + +--R 3 +--R - 4a b c + b +--R / +--R 2 +--R a x + b x + c +--R + +--R +-----------+ +--R 2 | 2 +--R (- log(a x + b x + c) + 2log(x))\|- 4a c + b +--R / +--R +-----------+ +--R | 2 +--R 2c\|- 4a c + b +--R , +--R +--R +---------+ +--R | 2 +--R (2a x + b)\|4a c - b +--R - 2b atan(----------------------) +--R 2 +--R 4a c - b +--R + +--R +---------+ +--R 2 | 2 +--R (- log(a x + b x + c) + 2log(x))\|4a c - b +--R / +--R +---------+ +--R | 2 +--R 2c\|4a c - b +--R ] +--R Type: Union(List Expression Integer,...) +--E + +--S 67 +bb1:=1/(2*c*(a*x^2+b*x+c))-b/(2*c)*t1.1+1/c*t2.1 +--R +--R +--R (4) +--R 2 2 2 2 +--R (- 2a b c x - 2a b c x - 2a b c ) +--R * +--R log +--R +-----------+ +--R 2 2 2 | 2 2 2 +--R (2a x + 2a b x - 2a c + b )\|- 4a c + b + (8a c - 2a b )x +--R + +--R 3 +--R 4a b c - b +--R / +--R 2 +--R a x + b x + c +--R + +--R 2 3 2 2 4 2 3 +--R ((4a b c - a b )x + (4a b c - b )x + 4a b c - b c) +--R * +--R log +--R +-----------+ +--R 2 2 2 | 2 2 2 +--R (2a x + 2a b x - 2a c + b )\|- 4a c + b + (- 8a c + 2a b )x +--R + +--R 3 +--R - 4a b c + b +--R / +--R 2 +--R a x + b x + c +--R + +--R 2 2 2 3 2 2 +--R ((- 4a c + a b )x + (- 4a b c + b )x - 4a c + b c) +--R * +--R 2 +--R log(a x + b x + c) +--R + +--R 2 2 2 3 2 2 +--R ((8a c - 2a b )x + (8a b c - 2b )x + 8a c - 2b c)log(x) - 2a b c x +--R + +--R 2 2 +--R 4a c - 2b c +--R * +--R +-----------+ +--R | 2 +--R \|- 4a c + b +--R / +--R +-----------+ +--R 2 3 2 2 2 3 3 2 4 2 3 | 2 +--R ((8a c - 2a b c )x + (8a b c - 2b c )x + 8a c - 2b c )\|- 4a c + b +--R Type: Expression Integer +--E + +--S 68 +bb2:=1/(2*c*(a*x^2+b*x+c))-b/(2*c)*t1.2+1/c*t2.1 +--R +--R +--R (5) +--R +---------+ +--R 2 3 2 2 4 2 3 | 2 +--R ((4a b c - a b )x + (4a b c - b )x + 4a b c - b c)\|4a c - b +--R * +--R log +--R +-----------+ +--R 2 2 2 | 2 2 2 +--R (2a x + 2a b x - 2a c + b )\|- 4a c + b + (- 8a c + 2a b )x +--R + +--R 3 +--R - 4a b c + b +--R / +--R 2 +--R a x + b x + c +--R + +--R +-----------+ +--R 2 2 2 2 | 2 +--R (- 4a b c x - 4a b c x - 4a b c )\|- 4a c + b +--R * +--R +---------+ +--R | 2 +--R (2a x + b)\|4a c - b +--R atan(----------------------) +--R 2 +--R 4a c - b +--R + +--R 2 2 2 3 2 2 +--R ((- 4a c + a b )x + (- 4a b c + b )x - 4a c + b c) +--R * +--R 2 +--R log(a x + b x + c) +--R + +--R 2 2 2 3 2 2 +--R ((8a c - 2a b )x + (8a b c - 2b )x + 8a c - 2b c)log(x) - 2a b c x +--R + +--R 2 2 +--R 4a c - 2b c +--R * +--R +-----------+ +---------+ +--R | 2 | 2 +--R \|- 4a c + b \|4a c - b +--R / +--R +-----------+ +--R 2 3 2 2 2 3 3 2 4 2 3 | 2 +--R ((8a c - 2a b c )x + (8a b c - 2b c )x + 8a c - 2b c )\|- 4a c + b +--R * +--R +---------+ +--R | 2 +--R \|4a c - b +--R Type: Expression Integer +--E + +--S 69 +bb3:=1/(2*c*(a*x^2+b*x+c))-b/(2*c)*t1.1+1/c*t2.2 +--R +--R +--R (6) +--R +---------+ +--R 2 2 2 2 | 2 +--R (- 2a b c x - 2a b c x - 2a b c )\|4a c - b +--R * +--R log +--R +-----------+ +--R 2 2 2 | 2 2 2 +--R (2a x + 2a b x - 2a c + b )\|- 4a c + b + (8a c - 2a b )x +--R + +--R 3 +--R 4a b c - b +--R / +--R 2 +--R a x + b x + c +--R + +--R 2 3 2 2 4 2 3 +--R ((- 8a b c + 2a b )x + (- 8a b c + 2b )x - 8a b c + 2b c) +--R * +--R +---------+ +--R +-----------+ | 2 +--R | 2 (2a x + b)\|4a c - b +--R \|- 4a c + b atan(----------------------) +--R 2 +--R 4a c - b +--R + +--R 2 2 2 3 2 2 +--R ((- 4a c + a b )x + (- 4a b c + b )x - 4a c + b c) +--R * +--R 2 +--R log(a x + b x + c) +--R + +--R 2 2 2 3 2 2 +--R ((8a c - 2a b )x + (8a b c - 2b )x + 8a c - 2b c)log(x) - 2a b c x +--R + +--R 2 2 +--R 4a c - 2b c +--R * +--R +-----------+ +---------+ +--R | 2 | 2 +--R \|- 4a c + b \|4a c - b +--R / +--R +-----------+ +--R 2 3 2 2 2 3 3 2 4 2 3 | 2 +--R ((8a c - 2a b c )x + (8a b c - 2b c )x + 8a c - 2b c )\|- 4a c + b +--R * +--R +---------+ +--R | 2 +--R \|4a c - b +--R Type: Expression Integer +--E + +--S 70 +bb4:=1/(2*c*(a*x^2+b*x+c))-b/(2*c)*t1.2+1/c*t2.2 +--R +--R +--R (7) +--R 2 3 2 2 4 2 3 +--R ((- 12a b c + 2a b )x + (- 12a b c + 2b )x - 12a b c + 2b c) +--R * +--R +---------+ +--R | 2 +--R (2a x + b)\|4a c - b +--R atan(----------------------) +--R 2 +--R 4a c - b +--R + +--R 2 2 2 3 2 2 +--R ((- 4a c + a b )x + (- 4a b c + b )x - 4a c + b c) +--R * +--R 2 +--R log(a x + b x + c) +--R + +--R 2 2 2 3 2 2 +--R ((8a c - 2a b )x + (8a b c - 2b )x + 8a c - 2b c)log(x) - 2a b c x +--R + +--R 2 2 +--R 4a c - 2b c +--R * +--R +---------+ +--R | 2 +--R \|4a c - b +--R / +--R +---------+ +--R 2 3 2 2 2 3 3 2 4 2 3 | 2 +--R ((8a c - 2a b c )x + (8a b c - 2b c )x + 8a c - 2b c )\|4a c - b +--R Type: Expression Integer +--E + +--S 71 +cc1:=aa.1-bb1 +--R +--R (8) +--R a b +--R * +--R log +--R +-----------+ +--R 2 2 2 | 2 2 2 +--R (2a x + 2a b x - 2a c + b )\|- 4a c + b + (8a c - 2a b )x +--R + +--R 3 +--R 4a b c - b +--R / +--R 2 +--R a x + b x + c +--R + +--R a b +--R * +--R log +--R +-----------+ +--R 2 2 2 | 2 2 2 +--R (2a x + 2a b x - 2a c + b )\|- 4a c + b + (- 8a c + 2a b )x +--R + +--R 3 +--R - 4a b c + b +--R / +--R 2 +--R a x + b x + c +--R / +--R +-----------+ +--R 2 2 | 2 +--R (4a c - b c)\|- 4a c + b +--R Type: Expression Integer +--E + +--S 72 +dd1:=expandLog cc1 +--R +--R (9) +--R a b +--R * +--R log +--R +-----------+ +--R 2 2 2 | 2 2 2 +--R (2a x + 2a b x - 2a c + b )\|- 4a c + b + (8a c - 2a b )x +--R + +--R 3 +--R 4a b c - b +--R + +--R a b +--R * +--R log +--R +-----------+ +--R 2 2 2 | 2 2 2 +--R (2a x + 2a b x - 2a c + b )\|- 4a c + b + (- 8a c + 2a b )x +--R + +--R 3 +--R - 4a b c + b +--R + +--R 2 +--R - 2a b log(a x + b x + c) +--R / +--R +-----------+ +--R 2 2 | 2 +--R (4a c - b c)\|- 4a c + b +--R Type: Expression Integer +--E + +--S 73 14:277 Schaums and Axiom differ by a constant +ee1:=complexNormalize dd1 +--R +--R 3 2 2 +--R a b log(- 16a c + 4a b ) +--R (10) --------------------------- +--R +-----------+ +--R 2 2 | 2 +--R (4a c - b c)\|- 4a c + b +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.278~~~~~$\displaystyle +\int{\frac{dx}{x^2(ax^2+bx+c)^2}}$} +$$\begin{array}{lr} +\displaystyle\int{\frac{1}{x^2(ax^2+bx+c)^2}}= +&\displaystyle-\frac{1}{cx(ax^2+bx+c)}\\ +&\\ +&\displaystyle-\frac{3a}{c}\int{\frac{1}{(ax^2+bx+c)^2}}\\ +&\\ +&\displaystyle-\frac{2b}{c}\int{\frac{1}{x(ax^2+bx+c)^2}} +\end{array} +$$ +<<*>>= +)clear all + +--S 74 +aa:=integrate(1/(x^2*(a*x^2+b*x+c)^2),x) +--R +--R +--R (1) +--R [ +--R 3 2 2 2 4 3 2 2 3 5 2 +--R (6a c - 6a b c + a b )x + (6a b c - 6a b c + b )x +--R + +--R 2 3 2 2 4 +--R (6a c - 6a b c + b c)x +--R * +--R log +--R +-----------+ +--R 2 2 2 | 2 2 2 +--R (2a x + 2a b x - 2a c + b )\|- 4a c + b + (- 8a c + 2a b )x +--R + +--R 3 +--R - 4a b c + b +--R / +--R 2 +--R a x + b x + c +--R + +--R 2 3 3 2 4 2 2 3 +--R ((4a b c - a b )x + (4a b c - b )x + (4a b c - b c)x) +--R * +--R 2 +--R log(a x + b x + c) +--R + +--R 2 3 3 2 4 2 2 3 +--R ((- 8a b c + 2a b )x + (- 8a b c + 2b )x + (- 8a b c + 2b c)x) +--R * +--R log(x) +--R + +--R 2 2 2 2 2 3 3 2 2 +--R (- 6a c + 2a b c)x + (- 7a b c + 2b c)x - 4a c + b c +--R * +--R +-----------+ +--R | 2 +--R \|- 4a c + b +--R / +--R +-----------+ +--R 2 4 2 3 3 4 3 3 2 5 2 4 | 2 +--R ((4a c - a b c )x + (4a b c - b c )x + (4a c - b c )x)\|- 4a c + b +--R , +--R +--R 3 2 2 2 4 3 2 2 3 5 2 +--R (- 12a c + 12a b c - 2a b )x + (- 12a b c + 12a b c - 2b )x +--R + +--R 2 3 2 2 4 +--R (- 12a c + 12a b c - 2b c)x +--R * +--R +---------+ +--R | 2 +--R (2a x + b)\|4a c - b +--R atan(----------------------) +--R 2 +--R 4a c - b +--R + +--R 2 3 3 2 4 2 2 3 +--R ((4a b c - a b )x + (4a b c - b )x + (4a b c - b c)x) +--R * +--R 2 +--R log(a x + b x + c) +--R + +--R 2 3 3 2 4 2 2 3 +--R ((- 8a b c + 2a b )x + (- 8a b c + 2b )x + (- 8a b c + 2b c)x) +--R * +--R log(x) +--R + +--R 2 2 2 2 2 3 3 2 2 +--R (- 6a c + 2a b c)x + (- 7a b c + 2b c)x - 4a c + b c +--R * +--R +---------+ +--R | 2 +--R \|4a c - b +--R / +--R +---------+ +--R 2 4 2 3 3 4 3 3 2 5 2 4 | 2 +--R ((4a c - a b c )x + (4a b c - b c )x + (4a c - b c )x)\|4a c - b +--R ] +--R Type: Union(List Expression Integer,...) +--E + +--S 75 +t1:=integrate(1/(a*x^2+b*x+c)^2,x) +--R +--R +--R (2) +--R [ +--R 2 2 +--R (2a x + 2a b x + 2a c) +--R * +--R log +--R +-----------+ +--R 2 2 2 | 2 2 2 +--R (2a x + 2a b x - 2a c + b )\|- 4a c + b + (8a c - 2a b )x +--R + +--R 3 +--R 4a b c - b +--R / +--R 2 +--R a x + b x + c +--R + +--R +-----------+ +--R | 2 +--R (2a x + b)\|- 4a c + b +--R / +--R +-----------+ +--R 2 2 2 3 2 2 | 2 +--R ((4a c - a b )x + (4a b c - b )x + 4a c - b c)\|- 4a c + b +--R , +--R +---------+ +--R | 2 +---------+ +--R 2 2 (2a x + b)\|4a c - b | 2 +--R (4a x + 4a b x + 4a c)atan(----------------------) + (2a x + b)\|4a c - b +--R 2 +--R 4a c - b +--R ---------------------------------------------------------------------------- +--R +---------+ +--R 2 2 2 3 2 2 | 2 +--R ((4a c - a b )x + (4a b c - b )x + 4a c - b c)\|4a c - b +--R ] +--R Type: Union(List Expression Integer,...) +--E + +--S 76 +t2:=integrate(1/(x*(a*x^2+b*x+c)^2),x) +--R +--R +--R (3) +--R [ +--R 2 3 2 2 4 2 3 +--R ((6a b c - a b )x + (6a b c - b )x + 6a b c - b c) +--R * +--R log +--R +-----------+ +--R 2 2 2 | 2 2 2 +--R (2a x + 2a b x - 2a c + b )\|- 4a c + b + (- 8a c + 2a b )x +--R + +--R 3 +--R - 4a b c + b +--R / +--R 2 +--R a x + b x + c +--R + +--R 2 2 2 3 2 2 +--R ((- 4a c + a b )x + (- 4a b c + b )x - 4a c + b c) +--R * +--R 2 +--R log(a x + b x + c) +--R + +--R 2 2 2 3 2 2 +--R ((8a c - 2a b )x + (8a b c - 2b )x + 8a c - 2b c)log(x) +--R + +--R 2 2 +--R - 2a b c x + 4a c - 2b c +--R * +--R +-----------+ +--R | 2 +--R \|- 4a c + b +--R / +--R +-----------+ +--R 2 3 2 2 2 3 3 2 4 2 3 | 2 +--R ((8a c - 2a b c )x + (8a b c - 2b c )x + 8a c - 2b c )\|- 4a c + b +--R , +--R +--R 2 3 2 2 4 2 3 +--R ((- 12a b c + 2a b )x + (- 12a b c + 2b )x - 12a b c + 2b c) +--R * +--R +---------+ +--R | 2 +--R (2a x + b)\|4a c - b +--R atan(----------------------) +--R 2 +--R 4a c - b +--R + +--R 2 2 2 3 2 2 +--R ((- 4a c + a b )x + (- 4a b c + b )x - 4a c + b c) +--R * +--R 2 +--R log(a x + b x + c) +--R + +--R 2 2 2 3 2 2 +--R ((8a c - 2a b )x + (8a b c - 2b )x + 8a c - 2b c)log(x) +--R + +--R 2 2 +--R - 2a b c x + 4a c - 2b c +--R * +--R +---------+ +--R | 2 +--R \|4a c - b +--R / +--R +---------+ +--R 2 3 2 2 2 3 3 2 4 2 3 | 2 +--R ((8a c - 2a b c )x + (8a b c - 2b c )x + 8a c - 2b c )\|4a c - b +--R ] +--R Type: Union(List Expression Integer,...) +--E + +--S 77 +bb1:=-1/(c*x*(a*x^2+b*x+c))-((3*a)/c)*t1.1-(2*b)/c*t2.1 +--R +--R +--R (4) +--R 3 2 3 2 2 2 2 3 +--R (- 6a c x - 6a b c x - 6a c x) +--R * +--R log +--R +-----------+ +--R 2 2 2 | 2 2 2 +--R (2a x + 2a b x - 2a c + b )\|- 4a c + b + (8a c - 2a b )x +--R + +--R 3 +--R 4a b c - b +--R / +--R 2 +--R a x + b x + c +--R + +--R 2 2 4 3 3 5 2 2 2 4 +--R ((- 6a b c + a b )x + (- 6a b c + b )x + (- 6a b c + b c)x) +--R * +--R log +--R +-----------+ +--R 2 2 2 | 2 2 2 +--R (2a x + 2a b x - 2a c + b )\|- 4a c + b + (- 8a c + 2a b )x +--R + +--R 3 +--R - 4a b c + b +--R / +--R 2 +--R a x + b x + c +--R + +--R 2 3 3 2 4 2 2 3 +--R ((4a b c - a b )x + (4a b c - b )x + (4a b c - b c)x) +--R * +--R 2 +--R log(a x + b x + c) +--R + +--R 2 3 3 2 4 2 2 3 +--R ((- 8a b c + 2a b )x + (- 8a b c + 2b )x + (- 8a b c + 2b c)x) +--R * +--R log(x) +--R + +--R 2 2 2 2 2 3 3 2 2 +--R (- 6a c + 2a b c)x + (- 7a b c + 2b c)x - 4a c + b c +--R * +--R +-----------+ +--R | 2 +--R \|- 4a c + b +--R / +--R +-----------+ +--R 2 4 2 3 3 4 3 3 2 5 2 4 | 2 +--R ((4a c - a b c )x + (4a b c - b c )x + (4a c - b c )x)\|- 4a c + b +--R Type: Expression Integer +--E + +--S 78 +bb2:=-1/(c*x*(a*x^2+b*x+c))-((3*a)/c)*t1.2-(2*b)/c*t2.1 +--R +--R +--R (5) +--R 2 2 4 3 3 5 2 2 2 4 +--R ((- 6a b c + a b )x + (- 6a b c + b )x + (- 6a b c + b c)x) +--R * +--R +---------+ +--R | 2 +--R \|4a c - b +--R * +--R log +--R +-----------+ +--R 2 2 2 | 2 2 2 +--R (2a x + 2a b x - 2a c + b )\|- 4a c + b + (- 8a c + 2a b )x +--R + +--R 3 +--R - 4a b c + b +--R / +--R 2 +--R a x + b x + c +--R + +--R +-----------+ +--R 3 2 3 2 2 2 2 3 | 2 +--R (- 12a c x - 12a b c x - 12a c x)\|- 4a c + b +--R * +--R +---------+ +--R | 2 +--R (2a x + b)\|4a c - b +--R atan(----------------------) +--R 2 +--R 4a c - b +--R + +--R 2 3 3 2 4 2 2 3 +--R ((4a b c - a b )x + (4a b c - b )x + (4a b c - b c)x) +--R * +--R 2 +--R log(a x + b x + c) +--R + +--R 2 3 3 2 4 2 2 3 +--R ((- 8a b c + 2a b )x + (- 8a b c + 2b )x + (- 8a b c + 2b c)x) +--R * +--R log(x) +--R + +--R 2 2 2 2 2 3 3 2 2 +--R (- 6a c + 2a b c)x + (- 7a b c + 2b c)x - 4a c + b c +--R * +--R +-----------+ +---------+ +--R | 2 | 2 +--R \|- 4a c + b \|4a c - b +--R / +--R +-----------+ +--R 2 4 2 3 3 4 3 3 2 5 2 4 | 2 +--R ((4a c - a b c )x + (4a b c - b c )x + (4a c - b c )x)\|- 4a c + b +--R * +--R +---------+ +--R | 2 +--R \|4a c - b +--R Type: Expression Integer +--E + +--S 79 +bb3:=-1/(c*x*(a*x^2+b*x+c))-((3*a)/c)*t1.1-(2*b)/c*t2.2 +--R +--R +--R (6) +--R +---------+ +--R 3 2 3 2 2 2 2 3 | 2 +--R (- 6a c x - 6a b c x - 6a c x)\|4a c - b +--R * +--R log +--R +-----------+ +--R 2 2 2 | 2 2 2 +--R (2a x + 2a b x - 2a c + b )\|- 4a c + b + (8a c - 2a b )x +--R + +--R 3 +--R 4a b c - b +--R / +--R 2 +--R a x + b x + c +--R + +--R 2 2 4 3 3 5 2 2 2 4 +--R ((12a b c - 2a b )x + (12a b c - 2b )x + (12a b c - 2b c)x) +--R * +--R +---------+ +--R +-----------+ | 2 +--R | 2 (2a x + b)\|4a c - b +--R \|- 4a c + b atan(----------------------) +--R 2 +--R 4a c - b +--R + +--R 2 3 3 2 4 2 2 3 +--R ((4a b c - a b )x + (4a b c - b )x + (4a b c - b c)x) +--R * +--R 2 +--R log(a x + b x + c) +--R + +--R 2 3 3 2 4 2 2 3 +--R ((- 8a b c + 2a b )x + (- 8a b c + 2b )x + (- 8a b c + 2b c)x) +--R * +--R log(x) +--R + +--R 2 2 2 2 2 3 3 2 2 +--R (- 6a c + 2a b c)x + (- 7a b c + 2b c)x - 4a c + b c +--R * +--R +-----------+ +---------+ +--R | 2 | 2 +--R \|- 4a c + b \|4a c - b +--R / +--R +-----------+ +--R 2 4 2 3 3 4 3 3 2 5 2 4 | 2 +--R ((4a c - a b c )x + (4a b c - b c )x + (4a c - b c )x)\|- 4a c + b +--R * +--R +---------+ +--R | 2 +--R \|4a c - b +--R Type: Expression Integer +--E + +--S 80 +bb4:=-1/(c*x*(a*x^2+b*x+c))-((3*a)/c)*t1.2-(2*b)/c*t2.2 +--R +--R +--R (7) +--R 3 2 2 2 4 3 2 2 3 5 2 +--R (- 12a c + 12a b c - 2a b )x + (- 12a b c + 12a b c - 2b )x +--R + +--R 2 3 2 2 4 +--R (- 12a c + 12a b c - 2b c)x +--R * +--R +---------+ +--R | 2 +--R (2a x + b)\|4a c - b +--R atan(----------------------) +--R 2 +--R 4a c - b +--R + +--R 2 3 3 2 4 2 2 3 +--R ((4a b c - a b )x + (4a b c - b )x + (4a b c - b c)x) +--R * +--R 2 +--R log(a x + b x + c) +--R + +--R 2 3 3 2 4 2 2 3 +--R ((- 8a b c + 2a b )x + (- 8a b c + 2b )x + (- 8a b c + 2b c)x) +--R * +--R log(x) +--R + +--R 2 2 2 2 2 3 3 2 2 +--R (- 6a c + 2a b c)x + (- 7a b c + 2b c)x - 4a c + b c +--R * +--R +---------+ +--R | 2 +--R \|4a c - b +--R / +--R +---------+ +--R 2 4 2 3 3 4 3 3 2 5 2 4 | 2 +--R ((4a c - a b c )x + (4a b c - b c )x + (4a c - b c )x)\|4a c - b +--R Type: Expression Integer +--E + +--S 81 +cc1:=aa.1-bb1 +--R +--R (8) +--R 2 +--R 6a +--R * +--R log +--R +-----------+ +--R 2 2 2 | 2 2 2 +--R (2a x + 2a b x - 2a c + b )\|- 4a c + b + (8a c - 2a b )x +--R + +--R 3 +--R 4a b c - b +--R / +--R 2 +--R a x + b x + c +--R + +--R 2 +--R 6a +--R * +--R log +--R +-----------+ +--R 2 2 2 | 2 2 2 +--R (2a x + 2a b x - 2a c + b )\|- 4a c + b + (- 8a c + 2a b )x +--R + +--R 3 +--R - 4a b c + b +--R / +--R 2 +--R a x + b x + c +--R / +--R +-----------+ +--R 2 2 | 2 +--R (4a c - b c)\|- 4a c + b +--R Type: Expression Integer +--E + +--S 82 +dd1:=expandLog cc1 +--R +--R (9) +--R 2 +--R 6a +--R * +--R log +--R +-----------+ +--R 2 2 2 | 2 2 2 +--R (2a x + 2a b x - 2a c + b )\|- 4a c + b + (8a c - 2a b )x +--R + +--R 3 +--R 4a b c - b +--R + +--R 2 +--R 6a +--R * +--R log +--R +-----------+ +--R 2 2 2 | 2 2 2 +--R (2a x + 2a b x - 2a c + b )\|- 4a c + b + (- 8a c + 2a b )x +--R + +--R 3 +--R - 4a b c + b +--R + +--R 2 2 +--R - 12a log(a x + b x + c) +--R / +--R +-----------+ +--R 2 2 | 2 +--R (4a c - b c)\|- 4a c + b +--R Type: Expression Integer +--E + +--S 83 14:278 Schaums and Axiom differ by a constant +ee1:=complexNormalize dd1 +--R +--R 2 3 2 2 +--R 6a log(- 16a c + 4a b ) +--R (10) --------------------------- +--R +-----------+ +--R 2 2 | 2 +--R (4a c - b c)\|- 4a c + b +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.279~~~~~$\displaystyle +\int{\frac{dx}{x^m(ax^2+bx+c)^n}}$} +$$\begin{array}{lr} +\displaystyle\int{\frac{1}{x^m(ax^2+bx+c)^n}}= +&\displaystyle-\frac{1}{(m-1)cx^{m-1}(ax^2+bx+c)^{n-1}}\\ +&\\ +&\displaystyle-\frac{(m+2n-3)a}{(m-1)c}\int{\frac{1}{x^{m-2}(ax^2+bx+c)^n}}\\ +&\\ +&\displaystyle-\frac{(m+n-2)b}{(m-1)c}\int{\frac{1}{x^{m-1}(ax^2+bx+c)^n}} +\end{array} +$$ +<<*>>= +)clear all + +--S 84 14:279 Axiom cannot compute this integral +aa:=integrate(1/(x^m*(a*x^2+b*x+c)^n),x) +--R +--R +--R x +--R ++ 1 +--I (1) | --------------------- d%N +--R ++ m 2 n +--I %N (c + %N b + %N a) +--R Type: Union(Expression Integer,...) +--E + +)spool +)lisp (bye) +@ + +\eject +\begin{thebibliography}{99} +\bibitem{1} Spiegel, Murray R. +{\sl Mathematical Handbook of Formulas and Tables}\\ +Schaum's Outline Series McGraw-Hill 1968 p71 +\end{thebibliography} +\end{document} diff --git 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n +0000152715 00000 n +0000152632 00000 n +0000155227 00000 n +0000153344 00000 n +0000153875 00000 n +0000154406 00000 n +0000155128 00000 n +0000155045 00000 n +0000157831 00000 n +0000155757 00000 n +0000156479 00000 n +0000156396 00000 n +0000157010 00000 n +0000157732 00000 n +0000157649 00000 n +0000158216 00000 n +0000158239 00000 n +0000158261 00000 n +trailer +<< +/Size 361 +/Root 2 0 R +/Info 1 0 R +>> +startxref +158382 +%%EOF diff --git a/src/axiom-website/CATS/schaum13.input.pamphlet b/src/axiom-website/CATS/schaum13.input.pamphlet new file mode 100644 index 0000000..678e41c --- /dev/null +++ b/src/axiom-website/CATS/schaum13.input.pamphlet @@ -0,0 +1,5196 @@ +\documentclass{article} +\usepackage{axiom} +\begin{document} +\title{\$SPAD/input schaum13.input} +\author{Timothy Daly} +\maketitle +\eject +\tableofcontents +\eject +\section{\cite{1}:14.280~~~~~$\displaystyle +\int{\frac{dx}{\sqrt{ax^2+bx+c}}}$} +$$\int{\frac{1}{\sqrt{ax^2+bx+c}}}= +\left\{ +\begin{array}{l} +\displaystyle +\frac{1}{\sqrt{a}}\ln\left(2\sqrt{a}\sqrt{ax^2+bx+c}+2ax+b\right)\\ +\\ +\displaystyle +-\frac{1}{\sqrt{-a}}\sin{-1}\left(\frac{2ax+b}{\sqrt{b^2-4ac}}\right)\\ +\\ +\displaystyle +\frac{1}{\sqrt{a}}\sinh^{-1}\left(\frac{2ax+b}{\sqrt{4ac-b^2}}\right) +\end{array} +\right. +$$ +<<*>>= +)spool schaum13.output +)set message test on +)set message auto off +)clear all + +--S 1 +aa:=integrate(1/sqrt(a*x^2+b*x+c),x) +--R +--R (1) +--R [ +--R log +--R +--------------+ +--R +-+ +-+ | 2 +-+ +--R (2\|a \|c - 2a x)\|a x + b x + c + 2a x\|c +--R + +--R 2 +-+ +--R (- 2a x - b x - 2c)\|a +--R / +--R +--------------+ +--R +-+ | 2 +--R 2\|c \|a x + b x + c - b x - 2c +--R / +--R +-+ +--R \|a +--R , +--R +--------------+ +--R +---+ | 2 +---+ +-+ +--R \|- a \|a x + b x + c - \|- a \|c +--R 2atan(------------------------------------) +--R a x +--R -------------------------------------------] +--R +---+ +--R \|- a +--R Type: Union(List Expression Integer,...) +--E + +--S 2 +bb1:=1/sqrt(a)*log(2*sqrt(a)*sqrt(a*x^2+b*x+c)+2*a*x+b) +--R +--R +--------------+ +--R +-+ | 2 +--R log(2\|a \|a x + b x + c + 2a x + b) +--R (2) -------------------------------------- +--R +-+ +--R \|a +--R Type: Expression Integer +--E + +--S 3 +bb2:=-1/sqrt(-a)*asin((2*a*x+b)/sqrt(b^2-4*a*c)) +--R +--R 2a x + b +--R asin(--------------) +--R +-----------+ +--R | 2 +--R \|- 4a c + b +--R (3) - -------------------- +--R +---+ +--R \|- a +--R Type: Expression Integer +--E + +--S 4 +bb3:=1/sqrt(a)*asinh((2*a*x+b)/sqrt(4*a*c-b^2)) +--R +--R 2a x + b +--R asinh(------------) +--R +---------+ +--R | 2 +--R \|4a c - b +--R (4) ------------------- +--R +-+ +--R \|a +--R Type: Expression Integer +--E + +--S 5 +cc1:=bb1-aa.1 +--R +--R (5) +--R +--------------+ +--R +-+ | 2 +--R log(2\|a \|a x + b x + c + 2a x + b) +--R + +--R - +--R log +--R +--------------+ +--R +-+ +-+ | 2 +-+ +--R (2\|a \|c - 2a x)\|a x + b x + c + 2a x\|c +--R + +--R 2 +-+ +--R (- 2a x - b x - 2c)\|a +--R / +--R +--------------+ +--R +-+ | 2 +--R 2\|c \|a x + b x + c - b x - 2c +--R / +--R +-+ +--R \|a +--R Type: Expression Integer +--E + +--S 6 +cc2:=bb1-aa.2 +--R +--R (6) +--R +--------------+ +--R +---+ +-+ | 2 +--R \|- a log(2\|a \|a x + b x + c + 2a x + b) +--R + +--R +--------------+ +--R +---+ | 2 +---+ +-+ +--R +-+ \|- a \|a x + b x + c - \|- a \|c +--R - 2\|a atan(------------------------------------) +--R a x +--R / +--R +---+ +-+ +--R \|- a \|a +--R Type: Expression Integer +--E + +--S 7 +cc3:=bb2-aa.1 +--R +--R (7) +--R - +--R +---+ +--R \|- a +--R * +--R log +--R +--------------+ +--R +-+ +-+ | 2 +-+ +--R (2\|a \|c - 2a x)\|a x + b x + c + 2a x\|c +--R + +--R 2 +-+ +--R (- 2a x - b x - 2c)\|a +--R / +--R +--------------+ +--R +-+ | 2 +--R 2\|c \|a x + b x + c - b x - 2c +--R + +--R +-+ 2a x + b +--R - \|a asin(--------------) +--R +-----------+ +--R | 2 +--R \|- 4a c + b +--R / +--R +---+ +-+ +--R \|- a \|a +--R Type: Expression Integer +--E + +--S 8 +cc4:=bb2-aa.2 +--R +--R +--------------+ +--R +---+ | 2 +---+ +-+ +--R \|- a \|a x + b x + c - \|- a \|c 2a x + b +--R - 2atan(------------------------------------) - asin(--------------) +--R a x +-----------+ +--R | 2 +--R \|- 4a c + b +--R (8) -------------------------------------------------------------------- +--R +---+ +--R \|- a +--R Type: Expression Integer +--E + +--S 9 +cc5:=bb3-aa.1 +--R +--R (9) +--R - +--R log +--R +--------------+ +--R +-+ +-+ | 2 +-+ +--R (2\|a \|c - 2a x)\|a x + b x + c + 2a x\|c +--R + +--R 2 +-+ +--R (- 2a x - b x - 2c)\|a +--R / +--R +--------------+ +--R +-+ | 2 +--R 2\|c \|a x + b x + c - b x - 2c +--R + +--R 2a x + b +--R asinh(------------) +--R +---------+ +--R | 2 +--R \|4a c - b +--R / +--R +-+ +--R \|a +--R Type: Expression Integer +--E + +--S 10 +cc6:=bb3-aa.2 +--R +--R (10) +--R +--------------+ +--R +---+ | 2 +---+ +-+ +--R +-+ \|- a \|a x + b x + c - \|- a \|c +---+ 2a x + b +--R - 2\|a atan(------------------------------------) + \|- a asinh(------------) +--R a x +---------+ +--R | 2 +--R \|4a c - b +--R ----------------------------------------------------------------------------- +--R +---+ +-+ +--R \|- a \|a +--R Type: Expression Integer +--E + +--S 11 +dd1:=simplifyLog cc1 +--R +--R (11) +--R log +--R +--------------+ +--R +-+ +-+ | 2 +--R ((4a x + 2b)\|c + (- 2b x - 4c)\|a )\|a x + b x + c +--R + +--R 2 +-+ +-+ 2 2 +--R (4a x + 4b x + 4c)\|a \|c - 2a b x + (- 4a c - b )x - 2b c +--R / +--R +--------------+ +--R +-+ +-+ | 2 +-+ +--R (2\|a \|c - 2a x)\|a x + b x + c + 2a x\|c +--R + +--R 2 +-+ +--R (- 2a x - b x - 2c)\|a +--R / +--R +-+ +--R \|a +--R Type: Expression Integer +--E + +--S 12 14:280 Schaums and Axiom differ by a constant +ee1:=ratDenom dd1 +--R +--R +-+ +-+ +--R +-+ - 2a\|c + b\|a +--R \|a log(----------------) +--R a +--R (12) ------------------------- +--R a +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.281~~~~~$\displaystyle +\int{\frac{x~dx}{\sqrt{ax^2+bx+c}}}$} +$$\int{\frac{x}{\sqrt{ax^2+bx+c}}}= +\frac{\sqrt{ax^2+bx+c}}{a}-\frac{b}{2a}\int{\frac{1}{\sqrt{ax^2+bx+c}}} +$$ +<<*>>= +)clear all + +--S 11 +aa:=integrate(x/sqrt(a*x^2+b*x+c),x) +--R +--R +--R (1) +--R [ +--R +--------------+ +--R +-+ | 2 2 +--R (2b\|c \|a x + b x + c - b x - 2b c) +--R * +--R log +--R +--------------+ +--R +-+ +-+ | 2 +-+ +--R (2\|a \|c + 2a x)\|a x + b x + c - 2a x\|c +--R + +--R 2 +-+ +--R (- 2a x - b x - 2c)\|a +--R / +--R +--------------+ +--R +-+ | 2 +--R 2\|c \|a x + b x + c - b x - 2c +--R + +--R +--------------+ +--R +-+ | 2 2 +-+ +-+ +--R - 2b x\|a \|a x + b x + c + (4a x + 2b x)\|a \|c +--R / +--R +--------------+ +--R +-+ +-+ | 2 +-+ +--R 4a\|a \|c \|a x + b x + c + (- 2a b x - 4a c)\|a +--R , +--R +--R +--------------+ +--R +-+ | 2 2 +--R (- 2b\|c \|a x + b x + c + b x + 2b c) +--R * +--R +--------------+ +--R +---+ | 2 +---+ +-+ +--R \|- a \|a x + b x + c - \|- a \|c +--R atan(------------------------------------) +--R a x +--R + +--R +--------------+ +--R +---+ | 2 2 +---+ +-+ +--R - b x\|- a \|a x + b x + c + (2a x + b x)\|- a \|c +--R / +--R +--------------+ +--R +---+ +-+ | 2 +---+ +--R 2a\|- a \|c \|a x + b x + c + (- a b x - 2a c)\|- a +--R ] +--R Type: Union(List Expression Integer,...) +--E + +--S 12 +t1:=integrate(1/sqrt(a*x^2+b*x+c),x) +--R +--R (2) +--R [ +--R log +--R +--------------+ +--R +-+ +-+ | 2 +-+ +--R (2\|a \|c - 2a x)\|a x + b x + c + 2a x\|c +--R + +--R 2 +-+ +--R (- 2a x - b x - 2c)\|a +--R / +--R +--------------+ +--R +-+ | 2 +--R 2\|c \|a x + b x + c - b x - 2c +--R / +--R +-+ +--R \|a +--R , +--R +--------------+ +--R +---+ | 2 +---+ +-+ +--R \|- a \|a x + b x + c - \|- a \|c +--R 2atan(------------------------------------) +--R a x +--R -------------------------------------------] +--R +---+ +--R \|- a +--R Type: Union(List Expression Integer,...) +--E + +--S 13 +bb1:=sqrt(a*x^2+b*x+c)/a-b/(2*a)*t1.1 +--R +--R (3) +--R - +--R b +--R * +--R log +--R +--------------+ +--R +-+ +-+ | 2 +-+ +--R (2\|a \|c - 2a x)\|a x + b x + c + 2a x\|c +--R + +--R 2 +-+ +--R (- 2a x - b x - 2c)\|a +--R / +--R +--------------+ +--R +-+ | 2 +--R 2\|c \|a x + b x + c - b x - 2c +--R + +--R +--------------+ +--R +-+ | 2 +--R 2\|a \|a x + b x + c +--R / +--R +-+ +--R 2a\|a +--R Type: Expression Integer +--E + +--S 14 +bb2:=sqrt(a*x^2+b*x+c)/a-b/(2*a)*t1.2 +--R +--R (4) +--R +--------------+ +--R +---+ | 2 +---+ +-+ +--------------+ +--R \|- a \|a x + b x + c - \|- a \|c +---+ | 2 +--R - b atan(------------------------------------) + \|- a \|a x + b x + c +--R a x +--R ------------------------------------------------------------------------ +--R +---+ +--R a\|- a +--R Type: Expression Integer +--E + +--S 15 +cc1:=bb1-aa.1 +--R +--R (5) +--R +--------------+ +--R +-+ | 2 2 +--R (- 2b\|c \|a x + b x + c + b x + 2b c) +--R * +--R log +--R +--------------+ +--R +-+ +-+ | 2 +-+ +--R (2\|a \|c + 2a x)\|a x + b x + c - 2a x\|c +--R + +--R 2 +-+ +--R (- 2a x - b x - 2c)\|a +--R / +--R +--------------+ +--R +-+ | 2 +--R 2\|c \|a x + b x + c - b x - 2c +--R + +--R +--------------+ +--R +-+ | 2 2 +--R (- 2b\|c \|a x + b x + c + b x + 2b c) +--R * +--R log +--R +--------------+ +--R +-+ +-+ | 2 +-+ +--R (2\|a \|c - 2a x)\|a x + b x + c + 2a x\|c +--R + +--R 2 +-+ +--R (- 2a x - b x - 2c)\|a +--R / +--R +--------------+ +--R +-+ | 2 +--R 2\|c \|a x + b x + c - b x - 2c +--R + +--R +--------------+ +--R +-+ | 2 +-+ +-+ +--R - 4c\|a \|a x + b x + c + (2b x + 4c)\|a \|c +--R / +--R +--------------+ +--R +-+ +-+ | 2 +-+ +--R 4a\|a \|c \|a x + b x + c + (- 2a b x - 4a c)\|a +--R Type: Expression Integer +--E + +--S 16 +cc2:=bb1-aa.2 +--R +--R (6) +--R +--------------+ +--R +---+ +-+ | 2 2 +---+ +--R (- 2b\|- a \|c \|a x + b x + c + (b x + 2b c)\|- a ) +--R * +--R log +--R +--------------+ +--R +-+ +-+ | 2 +-+ +--R (2\|a \|c - 2a x)\|a x + b x + c + 2a x\|c +--R + +--R 2 +-+ +--R (- 2a x - b x - 2c)\|a +--R / +--R +--------------+ +--R +-+ | 2 +--R 2\|c \|a x + b x + c - b x - 2c +--R + +--R +--------------+ +--R +-+ +-+ | 2 2 +-+ +--R (4b\|a \|c \|a x + b x + c + (- 2b x - 4b c)\|a ) +--R * +--R +--------------+ +--R +---+ | 2 +---+ +-+ +--R \|- a \|a x + b x + c - \|- a \|c +--R atan(------------------------------------) +--R a x +--R + +--R +--------------+ +--R +---+ +-+ | 2 +---+ +-+ +-+ +--R - 4c\|- a \|a \|a x + b x + c + (2b x + 4c)\|- a \|a \|c +--R / +--R +--------------+ +--R +---+ +-+ +-+ | 2 +---+ +-+ +--R 4a\|- a \|a \|c \|a x + b x + c + (- 2a b x - 4a c)\|- a \|a +--R Type: Expression Integer +--E + +--S 17 +cc3:=bb2-aa.1 +--R +--R (7) +--R +--------------+ +--R +---+ +-+ | 2 2 +---+ +--R (- 2b\|- a \|c \|a x + b x + c + (b x + 2b c)\|- a ) +--R * +--R log +--R +--------------+ +--R +-+ +-+ | 2 +-+ +--R (2\|a \|c + 2a x)\|a x + b x + c - 2a x\|c +--R + +--R 2 +-+ +--R (- 2a x - b x - 2c)\|a +--R / +--R +--------------+ +--R +-+ | 2 +--R 2\|c \|a x + b x + c - b x - 2c +--R + +--R +--------------+ +--R +-+ +-+ | 2 2 +-+ +--R (- 4b\|a \|c \|a x + b x + c + (2b x + 4b c)\|a ) +--R * +--R +--------------+ +--R +---+ | 2 +---+ +-+ +--R \|- a \|a x + b x + c - \|- a \|c +--R atan(------------------------------------) +--R a x +--R + +--R +--------------+ +--R +---+ +-+ | 2 +---+ +-+ +-+ +--R - 4c\|- a \|a \|a x + b x + c + (2b x + 4c)\|- a \|a \|c +--R / +--R +--------------+ +--R +---+ +-+ +-+ | 2 +---+ +-+ +--R 4a\|- a \|a \|c \|a x + b x + c + (- 2a b x - 4a c)\|- a \|a +--R Type: Expression Integer +--E + +--S 18 +cc4:=bb2-aa.2 +--R +--R +--------------+ +--R | 2 +-+ +--R - 2c\|a x + b x + c + (b x + 2c)\|c +--R (8) -------------------------------------- +--R +--------------+ +--R +-+ | 2 +--R 2a\|c \|a x + b x + c - a b x - 2a c +--R Type: Expression Integer +--E + +--S 19 14:281 Schaums and Axiom differ by a constant +dd1:=ratDenom cc4 +--R +--R +-+ +--R \|c +--R (9) - ---- +--R a +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.282~~~~~$\displaystyle +\int{\frac{x^2dx}{\sqrt{ax^2+bx+c}}}$} +$$\int{\frac{x^2}{\sqrt{ax^2+bx+c}}}= +\frac{2ax-3b}{4a^2}\sqrt{ax^2+bx+c}+\frac{3b^2-4ac}{8a^2} +\int{\frac{1}{\sqrt{ax^2+bx+c}}} +$$ +<<*>>= +)clear all + +--S 19 +aa:=integrate(x^2/sqrt(a*x^2+b*x+c),x) +--R +--R (1) +--R [ +--R +--------------+ +--R 3 2 2 +-+ | 2 +--R ((16a b c - 12b )x + 32a c - 24b c)\|c \|a x + b x + c +--R + +--R 2 2 2 4 2 2 3 3 2 2 +--R (- 16a c + 8a b c + 3b )x + (- 32a b c + 24b c)x - 32a c + 24b c +--R * +--R log +--R +--------------+ +--R +-+ +-+ | 2 +-+ +--R (2\|a \|c + 2a x)\|a x + b x + c - 2a x\|c +--R + +--R 2 +-+ +--R (- 2a x - b x - 2c)\|a +--R / +--R +--------------+ +--R +-+ | 2 +--R 2\|c \|a x + b x + c - b x - 2c +--R + +--R 2 2 3 3 2 2 2 +-+ +--R ((- 16a c - 4a b )x + (- 8a b c + 6b )x + (- 32a c + 24b c)x)\|a +--R * +--R +--------------+ +--R | 2 +--R \|a x + b x + c +--R + +--R 2 4 2 2 3 3 2 +--R 16a b x + (32a c - 8a b )x + (24a b c - 18b )x +--R + +--R 2 2 +--R (32a c - 24b c)x +--R * +--R +-+ +-+ +--R \|a \|c +--R / +--R +--------------+ +--R 2 2 +-+ +-+ | 2 +--R (32a b x + 64a c)\|a \|c \|a x + b x + c +--R + +--R 3 2 2 2 2 2 2 +-+ +--R ((- 32a c - 8a b )x - 64a b c x - 64a c )\|a +--R , +--R +--R +--------------+ +--R 3 2 2 +-+ | 2 +--R ((- 16a b c + 12b )x - 32a c + 24b c)\|c \|a x + b x + c +--R + +--R 2 2 2 4 2 2 3 3 2 2 +--R (16a c - 8a b c - 3b )x + (32a b c - 24b c)x + 32a c - 24b c +--R * +--R +--------------+ +--R +---+ | 2 +---+ +-+ +--R \|- a \|a x + b x + c - \|- a \|c +--R atan(------------------------------------) +--R a x +--R + +--R 2 2 3 3 2 2 2 +---+ +--R ((- 8a c - 2a b )x + (- 4a b c + 3b )x + (- 16a c + 12b c)x)\|- a +--R * +--R +--------------+ +--R | 2 +--R \|a x + b x + c +--R + +--R 2 4 2 2 3 3 2 2 2 +--R (8a b x + (16a c - 4a b )x + (12a b c - 9b )x + (16a c - 12b c)x) +--R * +--R +---+ +-+ +--R \|- a \|c +--R / +--R +--------------+ +--R 2 2 +---+ +-+ | 2 +--R (16a b x + 32a c)\|- a \|c \|a x + b x + c +--R + +--R 3 2 2 2 2 2 2 +---+ +--R ((- 16a c - 4a b )x - 32a b c x - 32a c )\|- a +--R ] +--R Type: Union(List Expression Integer,...) +--E + +--S 20 +t1:=integrate(1/sqrt(a*x^2+b*x+c),x) +--R +--R (2) +--R [ +--R log +--R +--------------+ +--R +-+ +-+ | 2 +-+ +--R (2\|a \|c - 2a x)\|a x + b x + c + 2a x\|c +--R + +--R 2 +-+ +--R (- 2a x - b x - 2c)\|a +--R / +--R +--------------+ +--R +-+ | 2 +--R 2\|c \|a x + b x + c - b x - 2c +--R / +--R +-+ +--R \|a +--R , +--R +--------------+ +--R +---+ | 2 +---+ +-+ +--R \|- a \|a x + b x + c - \|- a \|c +--R 2atan(------------------------------------) +--R a x +--R -------------------------------------------] +--R +---+ +--R \|- a +--R Type: Union(List Expression Integer,...) +--E + +--S 21 +bb1:=(2*a*x-3*b)/(4*a^2)*sqrt(a*x^2+b*x+c)+(3*b^2-4*a*c)/(8*a^2)*t1.1 +--R +--R (3) +--R 2 +--R (- 4a c + 3b ) +--R * +--R log +--R +--------------+ +--R +-+ +-+ | 2 +-+ +--R (2\|a \|c - 2a x)\|a x + b x + c + 2a x\|c +--R + +--R 2 +-+ +--R (- 2a x - b x - 2c)\|a +--R / +--R +--------------+ +--R +-+ | 2 +--R 2\|c \|a x + b x + c - b x - 2c +--R + +--R +--------------+ +--R +-+ | 2 +--R (4a x - 6b)\|a \|a x + b x + c +--R / +--R 2 +-+ +--R 8a \|a +--R Type: Expression Integer +--E + +--S 22 +bb2:=(2*a*x-3*b)/(4*a^2)*sqrt(a*x^2+b*x+c)+(3*b^2-4*a*c)/(8*a^2)*t1.2 +--R +--R (4) +--R +--------------+ +--R +---+ | 2 +---+ +-+ +--R 2 \|- a \|a x + b x + c - \|- a \|c +--R (- 4a c + 3b )atan(------------------------------------) +--R a x +--R + +--R +--------------+ +--R +---+ | 2 +--R (2a x - 3b)\|- a \|a x + b x + c +--R / +--R 2 +---+ +--R 4a \|- a +--R Type: Expression Integer +--E + +--S 23 +cc1:=aa.1-bb1 +--R +--R (5) +--R +--------------+ +--R 3 2 2 +-+ | 2 +--R ((16a b c - 12b )x + 32a c - 24b c)\|c \|a x + b x + c +--R + +--R 2 2 2 4 2 2 3 3 2 2 +--R (- 16a c + 8a b c + 3b )x + (- 32a b c + 24b c)x - 32a c + 24b c +--R * +--R log +--R +--------------+ +--R +-+ +-+ | 2 +-+ +--R (2\|a \|c + 2a x)\|a x + b x + c - 2a x\|c +--R + +--R 2 +-+ +--R (- 2a x - b x - 2c)\|a +--R / +--R +--------------+ +--R +-+ | 2 +--R 2\|c \|a x + b x + c - b x - 2c +--R + +--R +--------------+ +--R 3 2 2 +-+ | 2 +--R ((16a b c - 12b )x + 32a c - 24b c)\|c \|a x + b x + c +--R + +--R 2 2 2 4 2 2 3 3 2 2 +--R (- 16a c + 8a b c + 3b )x + (- 32a b c + 24b c)x - 32a c + 24b c +--R * +--R log +--R +--------------+ +--R +-+ +-+ | 2 +-+ +--R (2\|a \|c - 2a x)\|a x + b x + c + 2a x\|c +--R + +--R 2 +-+ +--R (- 2a x - b x - 2c)\|a +--R / +--R +--------------+ +--R +-+ | 2 +--R 2\|c \|a x + b x + c - b x - 2c +--R + +--R +--------------+ +--R 2 2 +-+ | 2 +--R (- 24b c x - 48b c )\|a \|a x + b x + c +--R + +--R 3 2 2 2 +-+ +-+ +--R ((24a b c + 6b )x + 48b c x + 48b c )\|a \|c +--R / +--R +--------------+ +--R 2 2 +-+ +-+ | 2 +--R (32a b x + 64a c)\|a \|c \|a x + b x + c +--R + +--R 3 2 2 2 2 2 2 +-+ +--R ((- 32a c - 8a b )x - 64a b c x - 64a c )\|a +--R Type: Expression Integer +--E + +--S 24 +cc2:=aa.2-bb1 +--R +--R (6) +--R +--------------+ +--R 3 2 2 +---+ +-+ | 2 +--R ((16a b c - 12b )x + 32a c - 24b c)\|- a \|c \|a x + b x + c +--R + +--R 2 2 2 4 2 2 3 3 +--R (- 16a c + 8a b c + 3b )x + (- 32a b c + 24b c)x - 32a c +--R + +--R 2 2 +--R 24b c +--R * +--R +---+ +--R \|- a +--R * +--R log +--R +--------------+ +--R +-+ +-+ | 2 +-+ +--R (2\|a \|c - 2a x)\|a x + b x + c + 2a x\|c +--R + +--R 2 +-+ +--R (- 2a x - b x - 2c)\|a +--R / +--R +--------------+ +--R +-+ | 2 +--R 2\|c \|a x + b x + c - b x - 2c +--R + +--R +--------------+ +--R 3 2 2 +-+ +-+ | 2 +--R ((- 32a b c + 24b )x - 64a c + 48b c)\|a \|c \|a x + b x + c +--R + +--R 2 2 2 4 2 2 3 3 +--R (32a c - 16a b c - 6b )x + (64a b c - 48b c)x + 64a c +--R + +--R 2 2 +--R - 48b c +--R * +--R +-+ +--R \|a +--R * +--R +--------------+ +--R +---+ | 2 +---+ +-+ +--R \|- a \|a x + b x + c - \|- a \|c +--R atan(------------------------------------) +--R a x +--R + +--R +--------------+ +--R 2 2 +---+ +-+ | 2 +--R (- 24b c x - 48b c )\|- a \|a \|a x + b x + c +--R + +--R 3 2 2 2 +---+ +-+ +-+ +--R ((24a b c + 6b )x + 48b c x + 48b c )\|- a \|a \|c +--R / +--R +--------------+ +--R 2 2 +---+ +-+ +-+ | 2 +--R (32a b x + 64a c)\|- a \|a \|c \|a x + b x + c +--R + +--R 3 2 2 2 2 2 2 +---+ +-+ +--R ((- 32a c - 8a b )x - 64a b c x - 64a c )\|- a \|a +--R Type: Expression Integer +--E + +--S 25 +cc3:=aa.2-bb1 +--R +--R (7) +--R +--------------+ +--R 3 2 2 +---+ +-+ | 2 +--R ((16a b c - 12b )x + 32a c - 24b c)\|- a \|c \|a x + b x + c +--R + +--R 2 2 2 4 2 2 3 3 +--R (- 16a c + 8a b c + 3b )x + (- 32a b c + 24b c)x - 32a c +--R + +--R 2 2 +--R 24b c +--R * +--R +---+ +--R \|- a +--R * +--R log +--R +--------------+ +--R +-+ +-+ | 2 +-+ +--R (2\|a \|c - 2a x)\|a x + b x + c + 2a x\|c +--R + +--R 2 +-+ +--R (- 2a x - b x - 2c)\|a +--R / +--R +--------------+ +--R +-+ | 2 +--R 2\|c \|a x + b x + c - b x - 2c +--R + +--R +--------------+ +--R 3 2 2 +-+ +-+ | 2 +--R ((- 32a b c + 24b )x - 64a c + 48b c)\|a \|c \|a x + b x + c +--R + +--R 2 2 2 4 2 2 3 3 +--R (32a c - 16a b c - 6b )x + (64a b c - 48b c)x + 64a c +--R + +--R 2 2 +--R - 48b c +--R * +--R +-+ +--R \|a +--R * +--R +--------------+ +--R +---+ | 2 +---+ +-+ +--R \|- a \|a x + b x + c - \|- a \|c +--R atan(------------------------------------) +--R a x +--R + +--R +--------------+ +--R 2 2 +---+ +-+ | 2 +--R (- 24b c x - 48b c )\|- a \|a \|a x + b x + c +--R + +--R 3 2 2 2 +---+ +-+ +-+ +--R ((24a b c + 6b )x + 48b c x + 48b c )\|- a \|a \|c +--R / +--R +--------------+ +--R 2 2 +---+ +-+ +-+ | 2 +--R (32a b x + 64a c)\|- a \|a \|c \|a x + b x + c +--R + +--R 3 2 2 2 2 2 2 +---+ +-+ +--R ((- 32a c - 8a b )x - 64a b c x - 64a c )\|- a \|a +--R Type: Expression Integer +--E + +--S 26 +cc4:=aa.2-bb2 +--R +--R (8) +--R +--------------+ +--R 2 2 | 2 +--R (- 12b c x - 24b c )\|a x + b x + c +--R + +--R 3 2 2 2 +-+ +--R ((12a b c + 3b )x + 24b c x + 24b c )\|c +--R / +--R +--------------+ +--R 2 2 +-+ | 2 3 2 2 2 2 +--R (16a b x + 32a c)\|c \|a x + b x + c + (- 16a c - 4a b )x - 32a b c x +--R + +--R 2 2 +--R - 32a c +--R Type: Expression Integer +--E + +--S 27 14:282 Schaums and Axiom differ by a constant +dd4:=ratDenom cc4 +--R +--R +-+ +--R 3b\|c +--R (9) - ------ +--R 2 +--R 4a +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.283~~~~~$\displaystyle +\int{\frac{dx}{x\sqrt{ax^2+bx+c}}}$} +$$\int{\frac{1}{x\sqrt{ax^2+bx+c}}}= +\left\{ +\begin{array}{l} +\displaystyle +-\frac{1}{\sqrt{c}} +\ln\left(\frac{2\sqrt{c}{\sqrt{ax^2+bx+c}}+bx+2c}{x}\right)\\ +\\ +\displaystyle +\frac{1}{\sqrt{-c}}\sin^{-1}\left(\frac{bx+2c}{|x|\sqrt{b^2-4ac}}\right)\\ +\\ +\displaystyle +-\frac{1}{\sqrt{c}}\sinh^{-1}\left(\frac{bx+2c}{|x|\sqrt{4ac-b^2}}\right) +\end{array} +\right. +$$ +<<*>>= +)clear all + +--S 27 +aa:=integrate(1/(x*sqrt(a*x^2+b*x+c)),x) +--R +--R +--------------+ +--R +-+ | 2 +--R 2\|c \|a x + b x + c - b x - 2c +--R log(---------------------------------) +--R x +--R (1) -------------------------------------- +--R +-+ +--R \|c +--R Type: Union(Expression Integer,...) +--E + +--S 28 +bb1:=-1/sqrt(c)*log((2*sqrt(c)*sqrt(a*x^2+b*x+c)+b*x+2*c)/x) +--R +--R +--------------+ +--R +-+ | 2 +--R 2\|c \|a x + b x + c + b x + 2c +--R log(---------------------------------) +--R x +--R (2) - -------------------------------------- +--R +-+ +--R \|c +--R Type: Expression Integer +--E + +--S 29 +bb2:=1/sqrt(-c)*asin((b*x+2*c)/(x*sqrt(b^2-4*a*c))) +--R +--R b x + 2c +--R asin(---------------) +--R +-----------+ +--R | 2 +--R x\|- 4a c + b +--R (3) --------------------- +--R +---+ +--R \|- c +--R Type: Expression Integer +--E + +--S 30 +bb3:=-1/sqrt(c)*asinh((b*x+2*c)/(x*sqrt(4*a*c-b^2))) +--R +--R b x + 2c +--R asinh(-------------) +--R +---------+ +--R | 2 +--R x\|4a c - b +--R (4) - -------------------- +--R +-+ +--R \|c +--R Type: Expression Integer +--E + +--S 31 +cc1:=aa-bb1 +--R +--R (5) +--R +--------------+ +--R +-+ | 2 +--R 2\|c \|a x + b x + c + b x + 2c +--R log(---------------------------------) +--R x +--R + +--R +--------------+ +--R +-+ | 2 +--R 2\|c \|a x + b x + c - b x - 2c +--R log(---------------------------------) +--R x +--R / +--R +-+ +--R \|c +--R Type: Expression Integer +--E + +--S 32 +cc2:=aa-bb2 +--R +--R (6) +--R +--------------+ +--R +-+ | 2 +--R +---+ 2\|c \|a x + b x + c - b x - 2c +-+ b x + 2c +--R \|- c log(---------------------------------) - \|c asin(---------------) +--R x +-----------+ +--R | 2 +--R x\|- 4a c + b +--R ------------------------------------------------------------------------ +--R +---+ +-+ +--R \|- c \|c +--R Type: Expression Integer +--E + +--S 33 +cc3:=aa-bb3 +--R +--R +--------------+ +--R +-+ | 2 +--R 2\|c \|a x + b x + c - b x - 2c b x + 2c +--R log(---------------------------------) + asinh(-------------) +--R x +---------+ +--R | 2 +--R x\|4a c - b +--R (7) ------------------------------------------------------------- +--R +-+ +--R \|c +--R Type: Expression Integer +--E + +--S 34 +dd1:=expandLog cc1 +--R +--R (8) +--R +--------------+ +--R +-+ | 2 +--R log(2\|c \|a x + b x + c + b x + 2c) +--R + +--R +--------------+ +--R +-+ | 2 +--R log(2\|c \|a x + b x + c - b x - 2c) - 2log(x) +--R / +--R +-+ +--R \|c +--R Type: Expression Integer +--E + +--S 35 +ee1:=ratDenom dd1 +--R +--R (9) +--R +--------------+ +--R +-+ +-+ | 2 +--R \|c log(2\|c \|a x + b x + c + b x + 2c) +--R + +--R +--------------+ +--R +-+ +-+ | 2 +-+ +--R \|c log(2\|c \|a x + b x + c - b x - 2c) - 2log(x)\|c +--R / +--R c +--R Type: Expression Integer +--E + +--S 36 14:283 Schaums and Axiom differ by a constant +ff1:=complexNormalize ee1 +--R +--R 2 +-+ +--R log(4a c - b )\|c +--R (10) ------------------ +--R c +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.284~~~~~$\displaystyle +\int{\frac{dx}{x^2\sqrt{ax^2+bx+c}}}$} +$$\int{\frac{1}{x^2\sqrt{ax^2+bx+c}}}= +-\frac{\sqrt{ax^2+bx+c}}{cx}-\frac{b}{2c}\int{\frac{1}{x\sqrt{ax^2+bx+c}}} +$$ +<<*>>= +)clear all + +--S 37 +aa:=integrate(1/(x^2*sqrt(a*x^2+b*x+c)),x) +--R +--R (1) +--R +--------------+ +--R +-+ | 2 2 2 +--R (- 4b x\|c \|a x + b x + c + 2b x + 4b c x) +--R * +--R +--------------+ +--R | 2 +-+ +--R 2c\|a x + b x + c + (- b x - 2c)\|c +--R log(--------------------------------------) +--R 2c x +--R + +--R +--------------+ +--R +-+ | 2 2 2 2 +--R (2b x + 8c)\|c \|a x + b x + c + (- 8a c + b )x - 6b c x - 8c +--R / +--R +--------------+ +--R 2 | 2 2 2 +-+ +--R 8c x\|a x + b x + c + (- 4b c x - 8c x)\|c +--R Type: Union(Expression Integer,...) +--E + +--S 38 +t1:=integrate(1/(x*sqrt(a*x^2+b*x+c)),x) +--R +--R +--------------+ +--R +-+ | 2 +--R 2\|c \|a x + b x + c - b x - 2c +--R log(---------------------------------) +--R x +--R (2) -------------------------------------- +--R +-+ +--R \|c +--R Type: Union(Expression Integer,...) +--E + +--S 39 +bb:=-sqrt(a*x^2+b*x+c)/(c*x)-b/(2*c)*t1 +--R +--R +--------------+ +--R +-+ | 2 +--------------+ +--R 2\|c \|a x + b x + c - b x - 2c +-+ | 2 +--R - b x log(---------------------------------) - 2\|c \|a x + b x + c +--R x +--R (3) --------------------------------------------------------------------- +--R +-+ +--R 2c x\|c +--R Type: Expression Integer +--E + +--S 40 +cc:=aa-bb +--R +--R (4) +--R +--------------+ +--R | 2 2 +-+ +--R (4b c\|a x + b x + c + (- 2b x - 4b c)\|c ) +--R * +--R +--------------+ +--R +-+ | 2 +--R 2\|c \|a x + b x + c - b x - 2c +--R log(---------------------------------) +--R x +--R + +--R +--------------+ +--R | 2 2 +-+ +--R (- 4b c\|a x + b x + c + (2b x + 4b c)\|c ) +--R * +--R +--------------+ +--R | 2 +-+ +--R 2c\|a x + b x + c + (- b x - 2c)\|c +--R log(--------------------------------------) +--R 2c x +--R + +--R +--------------+ +--R | 2 2 +-+ +--R - 2b c\|a x + b x + c + (b x + 2b c)\|c +--R / +--R +--------------+ +--R 2 +-+ | 2 2 3 +--R 8c \|c \|a x + b x + c - 4b c x - 8c +--R Type: Expression Integer +--E + +--S 41 +dd:=expandLog cc +--R +--R (5) +--R +--------------+ +--R | 2 2 +-+ +--R (4b c\|a x + b x + c + (- 2b x - 4b c)\|c ) +--R * +--R +--------------+ +--R +-+ | 2 +--R log(2\|c \|a x + b x + c - b x - 2c) +--R + +--R +--------------+ +--R | 2 2 +-+ +--R (- 4b c\|a x + b x + c + (2b x + 4b c)\|c ) +--R * +--R +--------------+ +--R | 2 +-+ +--R log(2c\|a x + b x + c + (- b x - 2c)\|c ) +--R + +--R +--------------+ +--R | 2 +--R (4b c log(c) + 4b c log(2) - 2b c)\|a x + b x + c +--R + +--R 2 2 2 +-+ +--R ((- 2b x - 4b c)log(c) + (- 2b x - 4b c)log(2) + b x + 2b c)\|c +--R / +--R +--------------+ +--R 2 +-+ | 2 2 3 +--R 8c \|c \|a x + b x + c - 4b c x - 8c +--R Type: Expression Integer +--E + +--S 42 +ee:=ratDenom dd +--R +--R (6) +--R +--------------+ +--R +-+ +-+ | 2 +--R 2b\|c log(2\|c \|a x + b x + c - b x - 2c) +--R + +--R +--------------+ +--R +-+ | 2 +-+ +--R - 2b\|c log(2c\|a x + b x + c + (- b x - 2c)\|c ) +--R + +--R +-+ +--R (2b log(c) + 2b log(2) - b)\|c +--R / +--R 2 +--R 4c +--R Type: Expression Integer +--E + +--S 43 14:284 Schaums and Axiom differ by a constant +ff:=complexNormalize ee +--R +--R +-+ +--R (b log(c) + 2b log(2) - b)\|c +--R (7) ------------------------------ +--R 2 +--R 4c +--R Type: Expression Integer +--E + +@ + +\section{\cite{1}:14.285~~~~~$\displaystyle +\int{\sqrt{ax^2+bx+c}}~dx$} +$$\int{\sqrt{ax^2+bx+c}}= +\frac{(2ax+b)\sqrt{ax^2+bx+c}}{4a}+ +\frac{4ac-b^2}{8a}\int{\frac{1}{\sqrt{ax^2+bx+c}}} +$$ +<<*>>= +)clear all + +--S 44 +aa:=integrate(sqrt(a*x^2+b*x+c),x) +--R +--R +--R (1) +--R [ +--R +--------------+ +--R 3 2 2 +-+ | 2 +--R ((16a b c - 4b )x + 32a c - 8b c)\|c \|a x + b x + c +--R + +--R 2 2 4 2 2 3 3 2 2 +--R (- 16a c + b )x + (- 32a b c + 8b c)x - 32a c + 8b c +--R * +--R log +--R +--------------+ +--R +-+ +-+ | 2 +-+ +--R (2\|a \|c - 2a x)\|a x + b x + c + 2a x\|c +--R + +--R 2 +-+ +--R (- 2a x - b x - 2c)\|a +--R / +--R +--------------+ +--R +-+ | 2 +--R 2\|c \|a x + b x + c - b x - 2c +--R + +--R 2 2 3 3 2 2 2 +-+ +--R ((- 16a c - 4a b )x + (- 40a b c - 2b )x + (- 32a c - 8b c)x)\|a +--R * +--R +--------------+ +--R | 2 +--R \|a x + b x + c +--R + +--R 2 4 2 2 3 3 2 +--R 16a b x + (32a c + 24a b )x + (56a b c + 6b )x +--R + +--R 2 2 +--R (32a c + 8b c)x +--R * +--R +-+ +-+ +--R \|a \|c +--R / +--R +--------------+ +--R +-+ +-+ | 2 +--R (32a b x + 64a c)\|a \|c \|a x + b x + c +--R + +--R 2 2 2 2 +-+ +--R ((- 32a c - 8a b )x - 64a b c x - 64a c )\|a +--R , +--R +--R +--------------+ +--R 3 2 2 +-+ | 2 +--R ((16a b c - 4b )x + 32a c - 8b c)\|c \|a x + b x + c +--R + +--R 2 2 4 2 2 3 3 2 2 +--R (- 16a c + b )x + (- 32a b c + 8b c)x - 32a c + 8b c +--R * +--R +--------------+ +--R +---+ | 2 +---+ +-+ +--R \|- a \|a x + b x + c - \|- a \|c +--R atan(------------------------------------) +--R a x +--R + +--R 2 2 3 3 2 2 2 +---+ +--R ((- 8a c - 2a b )x + (- 20a b c - b )x + (- 16a c - 4b c)x)\|- a +--R * +--R +--------------+ +--R | 2 +--R \|a x + b x + c +--R + +--R 2 4 2 2 3 3 2 2 2 +--R (8a b x + (16a c + 12a b )x + (28a b c + 3b )x + (16a c + 4b c)x) +--R * +--R +---+ +-+ +--R \|- a \|c +--R / +--R +--------------+ +--R +---+ +-+ | 2 +--R (16a b x + 32a c)\|- a \|c \|a x + b x + c +--R + +--R 2 2 2 2 +---+ +--R ((- 16a c - 4a b )x - 32a b c x - 32a c )\|- a +--R ] +--R Type: Union(List Expression Integer,...) +--E + +--S 45 +t1:=integrate(1/sqrt(a*x^2+b*x+c),x) +--R +--R (2) +--R [ +--R log +--R +--------------+ +--R +-+ +-+ | 2 +-+ +--R (2\|a \|c - 2a x)\|a x + b x + c + 2a x\|c +--R + +--R 2 +-+ +--R (- 2a x - b x - 2c)\|a +--R / +--R +--------------+ +--R +-+ | 2 +--R 2\|c \|a x + b x + c - b x - 2c +--R / +--R +-+ +--R \|a +--R , +--R +--------------+ +--R +---+ | 2 +---+ +-+ +--R \|- a \|a x + b x + c - \|- a \|c +--R 2atan(------------------------------------) +--R a x +--R -------------------------------------------] +--R +---+ +--R \|- a +--R Type: Union(List Expression Integer,...) +--E + +--S 46 +bb1:=((2*a*x+b)*sqrt(a*x^2+b*x+c))/(4*a)+(4*a*c-b^2)/(8*a)*t1.1 +--R +--R (3) +--R 2 +--R (4a c - b ) +--R * +--R log +--R +--------------+ +--R +-+ +-+ | 2 +-+ +--R (2\|a \|c - 2a x)\|a x + b x + c + 2a x\|c +--R + +--R 2 +-+ +--R (- 2a x - b x - 2c)\|a +--R / +--R +--------------+ +--R +-+ | 2 +--R 2\|c \|a x + b x + c - b x - 2c +--R + +--R +--------------+ +--R +-+ | 2 +--R (4a x + 2b)\|a \|a x + b x + c +--R / +--R +-+ +--R 8a\|a +--R Type: Expression Integer +--E + +--S 47 +bb2:=((2*a*x+b)*sqrt(a*x^2+b*x+c))/(4*a)+(4*a*c-b^2)/(8*a)*t1.2 +--R +--R (4) +--R +--------------+ +--R +---+ | 2 +---+ +-+ +--R 2 \|- a \|a x + b x + c - \|- a \|c +--R (4a c - b )atan(------------------------------------) +--R a x +--R + +--R +--------------+ +--R +---+ | 2 +--R (2a x + b)\|- a \|a x + b x + c +--R / +--R +---+ +--R 4a\|- a +--R Type: Expression Integer +--E + +--S 48 +cc1:=aa.1-bb1 +--R +--R (5) +--R +--------------+ +--R 2 2 | 2 +--R (4b c x + 8b c )\|a x + b x + c +--R + +--R 3 2 2 2 +-+ +--R ((- 4a b c - b )x - 8b c x - 8b c )\|c +--R / +--R +--------------+ +--R +-+ | 2 2 2 2 +--R (16a b x + 32a c)\|c \|a x + b x + c + (- 16a c - 4a b )x - 32a b c x +--R + +--R 2 +--R - 32a c +--R Type: Expression Integer +--E + +--S 49 +cc2:=aa.2-bb1 +--R +--R (6) +--R +--------------+ +--R 3 2 2 +---+ +-+ | 2 +--R ((- 16a b c + 4b )x - 32a c + 8b c)\|- a \|c \|a x + b x + c +--R + +--R 2 2 4 2 2 3 3 2 2 +---+ +--R ((16a c - b )x + (32a b c - 8b c)x + 32a c - 8b c )\|- a +--R * +--R log +--R +--------------+ +--R +-+ +-+ | 2 +-+ +--R (2\|a \|c - 2a x)\|a x + b x + c + 2a x\|c +--R + +--R 2 +-+ +--R (- 2a x - b x - 2c)\|a +--R / +--R +--------------+ +--R +-+ | 2 +--R 2\|c \|a x + b x + c - b x - 2c +--R + +--R +--------------+ +--R 3 2 2 +-+ +-+ | 2 +--R ((32a b c - 8b )x + 64a c - 16b c)\|a \|c \|a x + b x + c +--R + +--R 2 2 4 2 2 3 3 2 2 +-+ +--R ((- 32a c + 2b )x + (- 64a b c + 16b c)x - 64a c + 16b c )\|a +--R * +--R +--------------+ +--R +---+ | 2 +---+ +-+ +--R \|- a \|a x + b x + c - \|- a \|c +--R atan(------------------------------------) +--R a x +--R + +--R +--------------+ +--R 2 2 +---+ +-+ | 2 +--R (8b c x + 16b c )\|- a \|a \|a x + b x + c +--R + +--R 3 2 2 2 +---+ +-+ +-+ +--R ((- 8a b c - 2b )x - 16b c x - 16b c )\|- a \|a \|c +--R / +--R +--------------+ +--R +---+ +-+ +-+ | 2 +--R (32a b x + 64a c)\|- a \|a \|c \|a x + b x + c +--R + +--R 2 2 2 2 +---+ +-+ +--R ((- 32a c - 8a b )x - 64a b c x - 64a c )\|- a \|a +--R Type: Expression Integer +--E + +--S 50 +cc3:=aa.1-bb2 +--R +--R (7) +--R +--------------+ +--R 3 2 2 +---+ +-+ | 2 +--R ((16a b c - 4b )x + 32a c - 8b c)\|- a \|c \|a x + b x + c +--R + +--R 2 2 4 2 2 3 3 2 2 +---+ +--R ((- 16a c + b )x + (- 32a b c + 8b c)x - 32a c + 8b c )\|- a +--R * +--R log +--R +--------------+ +--R +-+ +-+ | 2 +-+ +--R (2\|a \|c - 2a x)\|a x + b x + c + 2a x\|c +--R + +--R 2 +-+ +--R (- 2a x - b x - 2c)\|a +--R / +--R +--------------+ +--R +-+ | 2 +--R 2\|c \|a x + b x + c - b x - 2c +--R + +--R +--------------+ +--R 3 2 2 +-+ +-+ | 2 +--R ((- 32a b c + 8b )x - 64a c + 16b c)\|a \|c \|a x + b x + c +--R + +--R 2 2 4 2 2 3 3 2 2 +-+ +--R ((32a c - 2b )x + (64a b c - 16b c)x + 64a c - 16b c )\|a +--R * +--R +--------------+ +--R +---+ | 2 +---+ +-+ +--R \|- a \|a x + b x + c - \|- a \|c +--R atan(------------------------------------) +--R a x +--R + +--R +--------------+ +--R 2 2 +---+ +-+ | 2 +--R (8b c x + 16b c )\|- a \|a \|a x + b x + c +--R + +--R 3 2 2 2 +---+ +-+ +-+ +--R ((- 8a b c - 2b )x - 16b c x - 16b c )\|- a \|a \|c +--R / +--R +--------------+ +--R +---+ +-+ +-+ | 2 +--R (32a b x + 64a c)\|- a \|a \|c \|a x + b x + c +--R + +--R 2 2 2 2 +---+ +-+ +--R ((- 32a c - 8a b )x - 64a b c x - 64a c )\|- a \|a +--R Type: Expression Integer +--E + +--S 51 +cc4:=aa.2-bb2 +--R +--R (8) +--R +--------------+ +--R 2 2 | 2 +--R (4b c x + 8b c )\|a x + b x + c +--R + +--R 3 2 2 2 +-+ +--R ((- 4a b c - b )x - 8b c x - 8b c )\|c +--R / +--R +--------------+ +--R +-+ | 2 2 2 2 +--R (16a b x + 32a c)\|c \|a x + b x + c + (- 16a c - 4a b )x - 32a b c x +--R + +--R 2 +--R - 32a c +--R Type: Expression Integer +--E + +--S 52 14:285 Schaums and Axiom differ by a constant +dd4:=ratDenom cc4 +--R +--R +-+ +--R b\|c +--R (9) ----- +--R 4a +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.286~~~~~$\displaystyle +\int{x\sqrt{ax^2+bx+c}}~dx$} +$$ +\begin{array}{rl} +\displaystyle +\int{x\sqrt{ax^2+bx+c}}=&\displaystyle\frac{(ax^2+bx+c)^{3/2}}{3a}\\ +&\\ +&\displaystyle-\frac{b(2ax+b)}{8a^2}\sqrt{ax^2+bx+c}\\ +&\\ +&\displaystyle-\frac{b(4ac-b^2)}{16a^2}\int{\frac{1}{\sqrt{ax^2+bx+c}}} +\end{array} +$$ +<<*>>= +)clear all + +--S 53 +aa:=integrate(x*sqrt(a*x^2+b*x+c),x) +--R +--R +--R (1) +--R [ +--R 2 2 3 5 2 2 2 4 +--R (96a b c + 48a b c - 18b )x + (384a b c - 96b c)x +--R + +--R 3 3 2 +--R 384a b c - 96b c +--R * +--R +--------------+ +--R +-+ | 2 +--R \|c \|a x + b x + c +--R + +--R 2 2 2 4 6 3 +--R (- 144a b c + 24a b c + 3b )x +--R + +--R 2 3 3 2 5 2 2 3 4 2 +--R (- 288a b c - 144a b c + 54b c)x + (- 576a b c + 144b c )x +--R + +--R 4 3 3 +--R - 384a b c + 96b c +--R * +--R log +--R +--------------+ +--R +-+ +-+ | 2 +-+ +--R (2\|a \|c + 2a x)\|a x + b x + c - 2a x\|c +--R + +--R 2 +-+ +--R (- 2a x - b x - 2c)\|a +--R / +--R +--------------+ +--R +-+ | 2 +--R 2\|c \|a x + b x + c - b x - 2c +--R + +--R 3 2 3 5 3 2 2 2 4 4 +--R (- 192a b c - 16a b )x + (- 384a c - 336a b c - 4a b )x +--R + +--R 2 2 3 5 3 +--R (- 1056a b c - 16a b c + 6b )x +--R + +--R 2 3 2 2 4 2 3 3 2 +--R (- 768a c - 288a b c + 72b c)x + (- 384a b c + 96b c )x +--R * +--R +--------------+ +--R +-+ | 2 +--R \|a \|a x + b x + c +--R + +--R 4 3 2 6 3 2 3 5 +--R (128a c + 96a b )x + (672a b c + 120a b )x +--R + +--R 3 2 2 2 4 4 2 2 3 5 3 +--R (768a c + 816a b c - 12a b )x + (1632a b c + 64a b c - 30b )x +--R + +--R 2 3 2 2 4 2 3 3 2 +--R (768a c + 480a b c - 120b c)x + (384a b c - 96b c )x +--R * +--R +-+ +-+ +--R \|a \|c +--R / +--R 3 2 2 2 2 2 2 +-+ +-+ +--R ((384a c + 288a b )x + 1536a b c x + 1536a c )\|a \|c +--R * +--R +--------------+ +--R | 2 +--R \|a x + b x + c +--R + +--R 3 2 3 3 3 2 2 2 2 2 2 +--R (- 576a b c - 48a b )x + (- 1152a c - 864a b c)x - 2304a b c x +--R + +--R 2 3 +--R - 1536a c +--R * +--R +-+ +--R \|a +--R , +--R +--R 2 2 3 5 2 2 2 4 +--R (- 96a b c - 48a b c + 18b )x + (- 384a b c + 96b c)x +--R + +--R 3 3 2 +--R - 384a b c + 96b c +--R * +--R +--------------+ +--R +-+ | 2 +--R \|c \|a x + b x + c +--R + +--R 2 2 2 4 6 3 2 3 3 2 5 2 +--R (144a b c - 24a b c - 3b )x + (288a b c + 144a b c - 54b c)x +--R + +--R 2 3 4 2 4 3 3 +--R (576a b c - 144b c )x + 384a b c - 96b c +--R * +--R +--------------+ +--R +---+ | 2 +---+ +-+ +--R \|- a \|a x + b x + c - \|- a \|c +--R atan(------------------------------------) +--R a x +--R + +--R 3 2 3 5 3 2 2 2 4 4 +--R (- 96a b c - 8a b )x + (- 192a c - 168a b c - 2a b )x +--R + +--R 2 2 3 5 3 2 3 2 2 4 2 +--R (- 528a b c - 8a b c + 3b )x + (- 384a c - 144a b c + 36b c)x +--R + +--R 3 3 2 +--R (- 192a b c + 48b c )x +--R * +--R +--------------+ +--R +---+ | 2 +--R \|- a \|a x + b x + c +--R + +--R 4 3 2 6 3 2 3 5 +--R (64a c + 48a b )x + (336a b c + 60a b )x +--R + +--R 3 2 2 2 4 4 2 2 3 5 3 +--R (384a c + 408a b c - 6a b )x + (816a b c + 32a b c - 15b )x +--R + +--R 2 3 2 2 4 2 3 3 2 +--R (384a c + 240a b c - 60b c)x + (192a b c - 48b c )x +--R * +--R +---+ +-+ +--R \|- a \|c +--R / +--R 3 2 2 2 2 2 2 +---+ +-+ +--R ((192a c + 144a b )x + 768a b c x + 768a c )\|- a \|c +--R * +--R +--------------+ +--R | 2 +--R \|a x + b x + c +--R + +--R 3 2 3 3 3 2 2 2 2 2 2 +--R (- 288a b c - 24a b )x + (- 576a c - 432a b c)x - 1152a b c x +--R + +--R 2 3 +--R - 768a c +--R * +--R +---+ +--R \|- a +--R ] +--R Type: Union(List Expression Integer,...) +--E + +--S 54 +t1:=integrate(1/sqrt(a*x^2+b*x+c),x) +--R +--R (2) +--R [ +--R log +--R +--------------+ +--R +-+ +-+ | 2 +-+ +--R (2\|a \|c - 2a x)\|a x + b x + c + 2a x\|c +--R + +--R 2 +-+ +--R (- 2a x - b x - 2c)\|a +--R / +--R +--------------+ +--R +-+ | 2 +--R 2\|c \|a x + b x + c - b x - 2c +--R / +--R +-+ +--R \|a +--R , +--R +--------------+ +--R +---+ | 2 +---+ +-+ +--R \|- a \|a x + b x + c - \|- a \|c +--R 2atan(------------------------------------) +--R a x +--R -------------------------------------------] +--R +---+ +--R \|- a +--R Type: Union(List Expression Integer,...) +--E + +--S 55 +bb1:=(a*x^2+b*x+c)^(3/2)/(3*a)-(b*(2*a*x+b))/(8*a^2)*sqrt(a*x^2+b*x+c)-(b*(4*a*c-b^2))/(16*a^2)*t1.1 +--R +--R (3) +--R 3 +--R (- 12a b c + 3b ) +--R * +--R log +--R +--------------+ +--R +-+ +-+ | 2 +-+ +--R (2\|a \|c - 2a x)\|a x + b x + c + 2a x\|c +--R + +--R 2 +-+ +--R (- 2a x - b x - 2c)\|a +--R / +--R +--------------+ +--R +-+ | 2 +--R 2\|c \|a x + b x + c - b x - 2c +--R + +--R +--------------+ +--R 2 2 2 +-+ | 2 +--R (16a x + 4a b x + 16a c - 6b )\|a \|a x + b x + c +--R / +--R 2 +-+ +--R 48a \|a +--R Type: Expression Integer +--E + +--S 56 +bb2:=(a*x^2+b*x+c)^(3/2)/(3*a)-(b*(2*a*x+b))/(8*a^2)*sqrt(a*x^2+b*x+c)-(b*(4*a*c-b^2))/(16*a^2)*t1.2 +--R +--R (4) +--R +--------------+ +--R +---+ | 2 +---+ +-+ +--R 3 \|- a \|a x + b x + c - \|- a \|c +--R (- 12a b c + 3b )atan(------------------------------------) +--R a x +--R + +--R +--------------+ +--R 2 2 2 +---+ | 2 +--R (8a x + 2a b x + 8a c - 3b )\|- a \|a x + b x + c +--R / +--R 2 +---+ +--R 24a \|- a +--R Type: Expression Integer +--E + +--S 57 +cc1:=aa.1-bb1 +--R +--R (5) +--R 2 2 3 5 2 2 2 4 3 +--R (96a b c + 48a b c - 18b )x + (384a b c - 96b c)x + 384a b c +--R + +--R 3 2 +--R - 96b c +--R * +--R +--------------+ +--R +-+ | 2 +--R \|c \|a x + b x + c +--R + +--R 2 2 2 4 6 3 2 3 3 2 5 2 +--R (- 144a b c + 24a b c + 3b )x + (- 288a b c - 144a b c + 54b c)x +--R + +--R 2 3 4 2 4 3 3 +--R (- 576a b c + 144b c )x - 384a b c + 96b c +--R * +--R log +--R +--------------+ +--R +-+ +-+ | 2 +-+ +--R (2\|a \|c + 2a x)\|a x + b x + c - 2a x\|c +--R + +--R 2 +-+ +--R (- 2a x - b x - 2c)\|a +--R / +--R +--------------+ +--R +-+ | 2 +--R 2\|c \|a x + b x + c - b x - 2c +--R + +--R 2 2 3 5 2 2 2 4 3 +--R (96a b c + 48a b c - 18b )x + (384a b c - 96b c)x + 384a b c +--R + +--R 3 2 +--R - 96b c +--R * +--R +--------------+ +--R +-+ | 2 +--R \|c \|a x + b x + c +--R + +--R 2 2 2 4 6 3 2 3 3 2 5 2 +--R (- 144a b c + 24a b c + 3b )x + (- 288a b c - 144a b c + 54b c)x +--R + +--R 2 3 4 2 4 3 3 +--R (- 576a b c + 144b c )x - 384a b c + 96b c +--R * +--R log +--R +--------------+ +--R +-+ +-+ | 2 +-+ +--R (2\|a \|c - 2a x)\|a x + b x + c + 2a x\|c +--R + +--R 2 +-+ +--R (- 2a x - b x - 2c)\|a +--R / +--R +--------------+ +--R +-+ | 2 +--R 2\|c \|a x + b x + c - b x - 2c +--R + +--R 2 3 2 2 4 2 3 3 2 4 +--R (128a c + 48a b c - 36b c)x + (512a b c - 192b c )x + 512a c +--R + +--R 2 3 +--R - 192b c +--R * +--R +--------------+ +--R +-+ | 2 +--R \|a \|a x + b x + c +--R + +--R 2 2 3 5 3 2 3 2 2 4 2 +--R (- 192a b c + 56a b c + 6b )x + (- 384a c - 144a b c + 108b c)x +--R + +--R 3 3 2 4 2 3 +--R (- 768a b c + 288b c )x - 512a c + 192b c +--R * +--R +-+ +-+ +--R \|a \|c +--R / +--R +--------------+ +--R 3 2 2 2 2 2 2 +-+ +-+ | 2 +--R ((384a c + 288a b )x + 1536a b c x + 1536a c )\|a \|c \|a x + b x + c +--R + +--R 3 2 3 3 3 2 2 2 2 2 2 +--R (- 576a b c - 48a b )x + (- 1152a c - 864a b c)x - 2304a b c x +--R + +--R 2 3 +--R - 1536a c +--R * +--R +-+ +--R \|a +--R Type: Expression Integer +--E + +--S 58 +cc2:=aa.2-bb1 +--R +--R (6) +--R 2 2 3 5 2 2 2 4 3 +--R (96a b c + 48a b c - 18b )x + (384a b c - 96b c)x + 384a b c +--R + +--R 3 2 +--R - 96b c +--R * +--R +--------------+ +--R +---+ +-+ | 2 +--R \|- a \|c \|a x + b x + c +--R + +--R 2 2 2 4 6 3 +--R (- 144a b c + 24a b c + 3b )x +--R + +--R 2 3 3 2 5 2 2 3 4 2 +--R (- 288a b c - 144a b c + 54b c)x + (- 576a b c + 144b c )x +--R + +--R 4 3 3 +--R - 384a b c + 96b c +--R * +--R +---+ +--R \|- a +--R * +--R log +--R +--------------+ +--R +-+ +-+ | 2 +-+ +--R (2\|a \|c - 2a x)\|a x + b x + c + 2a x\|c +--R + +--R 2 +-+ +--R (- 2a x - b x - 2c)\|a +--R / +--R +--------------+ +--R +-+ | 2 +--R 2\|c \|a x + b x + c - b x - 2c +--R + +--R 2 2 3 5 2 2 2 4 +--R (- 192a b c - 96a b c + 36b )x + (- 768a b c + 192b c)x +--R + +--R 3 3 2 +--R - 768a b c + 192b c +--R * +--R +--------------+ +--R +-+ +-+ | 2 +--R \|a \|c \|a x + b x + c +--R + +--R 2 2 2 4 6 3 +--R (288a b c - 48a b c - 6b )x +--R + +--R 2 3 3 2 5 2 2 3 4 2 +--R (576a b c + 288a b c - 108b c)x + (1152a b c - 288b c )x +--R + +--R 4 3 3 +--R 768a b c - 192b c +--R * +--R +-+ +--R \|a +--R * +--R +--------------+ +--R +---+ | 2 +---+ +-+ +--R \|- a \|a x + b x + c - \|- a \|c +--R atan(------------------------------------) +--R a x +--R + +--R 2 3 2 2 4 2 3 3 2 4 +--R (128a c + 48a b c - 36b c)x + (512a b c - 192b c )x + 512a c +--R + +--R 2 3 +--R - 192b c +--R * +--R +--------------+ +--R +---+ +-+ | 2 +--R \|- a \|a \|a x + b x + c +--R + +--R 2 2 3 5 3 2 3 2 2 4 2 +--R (- 192a b c + 56a b c + 6b )x + (- 384a c - 144a b c + 108b c)x +--R + +--R 3 3 2 4 2 3 +--R (- 768a b c + 288b c )x - 512a c + 192b c +--R * +--R +---+ +-+ +-+ +--R \|- a \|a \|c +--R / +--R 3 2 2 2 2 2 2 +---+ +-+ +-+ +--R ((384a c + 288a b )x + 1536a b c x + 1536a c )\|- a \|a \|c +--R * +--R +--------------+ +--R | 2 +--R \|a x + b x + c +--R + +--R 3 2 3 3 3 2 2 2 2 2 2 +--R (- 576a b c - 48a b )x + (- 1152a c - 864a b c)x - 2304a b c x +--R + +--R 2 3 +--R - 1536a c +--R * +--R +---+ +-+ +--R \|- a \|a +--R Type: Expression Integer +--E + +--S 59 +cc3:=aa.1-bb2 +--R +--R (7) +--R 2 2 3 5 2 2 2 4 3 +--R (96a b c + 48a b c - 18b )x + (384a b c - 96b c)x + 384a b c +--R + +--R 3 2 +--R - 96b c +--R * +--R +--------------+ +--R +---+ +-+ | 2 +--R \|- a \|c \|a x + b x + c +--R + +--R 2 2 2 4 6 3 +--R (- 144a b c + 24a b c + 3b )x +--R + +--R 2 3 3 2 5 2 2 3 4 2 +--R (- 288a b c - 144a b c + 54b c)x + (- 576a b c + 144b c )x +--R + +--R 4 3 3 +--R - 384a b c + 96b c +--R * +--R +---+ +--R \|- a +--R * +--R log +--R +--------------+ +--R +-+ +-+ | 2 +-+ +--R (2\|a \|c + 2a x)\|a x + b x + c - 2a x\|c +--R + +--R 2 +-+ +--R (- 2a x - b x - 2c)\|a +--R / +--R +--------------+ +--R +-+ | 2 +--R 2\|c \|a x + b x + c - b x - 2c +--R + +--R 2 2 3 5 2 2 2 4 +--R (192a b c + 96a b c - 36b )x + (768a b c - 192b c)x +--R + +--R 3 3 2 +--R 768a b c - 192b c +--R * +--R +--------------+ +--R +-+ +-+ | 2 +--R \|a \|c \|a x + b x + c +--R + +--R 2 2 2 4 6 3 +--R (- 288a b c + 48a b c + 6b )x +--R + +--R 2 3 3 2 5 2 2 3 4 2 +--R (- 576a b c - 288a b c + 108b c)x + (- 1152a b c + 288b c )x +--R + +--R 4 3 3 +--R - 768a b c + 192b c +--R * +--R +-+ +--R \|a +--R * +--R +--------------+ +--R +---+ | 2 +---+ +-+ +--R \|- a \|a x + b x + c - \|- a \|c +--R atan(------------------------------------) +--R a x +--R + +--R 2 3 2 2 4 2 3 3 2 4 +--R (128a c + 48a b c - 36b c)x + (512a b c - 192b c )x + 512a c +--R + +--R 2 3 +--R - 192b c +--R * +--R +--------------+ +--R +---+ +-+ | 2 +--R \|- a \|a \|a x + b x + c +--R + +--R 2 2 3 5 3 2 3 2 2 4 2 +--R (- 192a b c + 56a b c + 6b )x + (- 384a c - 144a b c + 108b c)x +--R + +--R 3 3 2 4 2 3 +--R (- 768a b c + 288b c )x - 512a c + 192b c +--R * +--R +---+ +-+ +-+ +--R \|- a \|a \|c +--R / +--R 3 2 2 2 2 2 2 +---+ +-+ +-+ +--R ((384a c + 288a b )x + 1536a b c x + 1536a c )\|- a \|a \|c +--R * +--R +--------------+ +--R | 2 +--R \|a x + b x + c +--R + +--R 3 2 3 3 3 2 2 2 2 2 2 +--R (- 576a b c - 48a b )x + (- 1152a c - 864a b c)x - 2304a b c x +--R + +--R 2 3 +--R - 1536a c +--R * +--R +---+ +-+ +--R \|- a \|a +--R Type: Expression Integer +--E + +--S 60 +cc4:=aa.2-bb2 +--R +--R (8) +--R 2 3 2 2 4 2 3 3 2 4 +--R (64a c + 24a b c - 18b c)x + (256a b c - 96b c )x + 256a c +--R + +--R 2 3 +--R - 96b c +--R * +--R +--------------+ +--R | 2 +--R \|a x + b x + c +--R + +--R 2 2 3 5 3 2 3 2 2 4 2 +--R (- 96a b c + 28a b c + 3b )x + (- 192a c - 72a b c + 54b c)x +--R + +--R 3 3 2 4 2 3 +--R (- 384a b c + 144b c )x - 256a c + 96b c +--R * +--R +-+ +--R \|c +--R / +--R +--------------+ +--R 3 2 2 2 2 2 2 +-+ | 2 +--R ((192a c + 144a b )x + 768a b c x + 768a c )\|c \|a x + b x + c +--R + +--R 3 2 3 3 3 2 2 2 2 2 2 2 3 +--R (- 288a b c - 24a b )x + (- 576a c - 432a b c)x - 1152a b c x - 768a c +--R Type: Expression Integer +--E + +--S 61 14:286 Schaums and Axiom differ by a constant +dd4:=ratDenom cc4 +--R +--R 2 +-+ +--R (8a c - 3b )\|c +--R (9) ---------------- +--R 2 +--R 24a +--R Type: Expression Integer +--E +@ + +\section{\cite{1}:14.287~~~~~$\displaystyle +\int{x^2\sqrt{ax^2+bx+c}}~dx$} +$$\int{x^2\sqrt{ax^2+bx+c}}= +\frac{6ax-5b}{24a^2}(ax^2+bx+c)^{3/2}+ +\frac{5b^2-4ac}{16a^2}\int{\sqrt{ax^2+bx+c}} +$$ +<<*>>= +)clear all + +--S 62 +aa:=integrate(x^2*sqrt(a*x^2+b*x+c),x) +--R +--R +--R (1) +--R [ +--R 3 3 2 3 2 5 7 3 +--R (1536a b c - 1920a b c - 96a b c + 120b )x +--R + +--R 3 4 2 2 3 4 2 6 2 +--R (3072a c - 768a b c - 4800a b c + 1200b c)x +--R + +--R 2 4 3 3 5 2 2 5 2 4 +--R (9216a b c - 13824a b c + 2880b c )x + 6144a c - 9216a b c +--R + +--R 4 3 +--R 1920b c +--R * +--R +--------------+ +--R +-+ | 2 +--R \|c \|a x + b x + c +--R + +--R 4 4 2 4 2 6 8 4 +--R (- 768a c + 1440a b c - 288a b c - 15b )x +--R + +--R 3 4 2 3 3 5 2 7 3 +--R (- 6144a b c + 7680a b c + 384a b c - 480b c)x +--R + +--R 3 5 2 2 4 4 3 6 2 2 +--R (- 6144a c + 1536a b c + 9600a b c - 2400b c )x +--R + +--R 2 5 3 4 5 3 2 6 2 5 +--R (- 12288a b c + 18432a b c - 3840b c )x - 6144a c + 9216a b c +--R + +--R 4 4 +--R - 1920b c +--R * +--R log +--R +--------------+ +--R +-+ +-+ | 2 +-+ +--R (2\|a \|c + 2a x)\|a x + b x + c - 2a x\|c +--R + +--R 2 +-+ +--R (- 2a x - b x - 2c)\|a +--R / +--R +--------------+ +--R +-+ | 2 +--R 2\|c \|a x + b x + c - b x - 2c +--R + +--R 5 2 4 2 3 4 7 +--R (- 1536a c - 2304a b c - 96a b )x +--R + +--R 4 2 3 3 2 5 6 +--R (- 12544a b c - 3456a b c - 16a b )x +--R + +--R 4 3 3 2 2 2 4 6 5 +--R (- 13056a c - 18240a b c - 80a b c + 20a b )x +--R + +--R 3 3 2 3 2 5 7 4 +--R (- 31104a b c + 480a b c + 24a b c - 30b )x +--R + +--R 3 4 2 2 3 4 2 6 3 +--R (- 18432a c + 768a b c + 2816a b c - 720b c)x +--R + +--R 2 4 3 3 5 2 2 +--R (- 7680a b c + 11520a b c - 2400b c )x +--R + +--R 2 5 2 4 4 3 +--R (- 6144a c + 9216a b c - 1920b c )x +--R * +--R +--------------+ +--R +-+ | 2 +--R \|a \|a x + b x + c +--R + +--R 5 4 3 8 5 2 4 2 3 4 7 +--R (3072a b c + 768a b )x + (6144a c + 11264a b c + 896a b )x +--R + +--R 4 2 3 3 2 5 6 +--R (30208a b c + 9984a b c - 32a b )x +--R + +--R 4 3 3 2 2 2 4 6 5 +--R (21504a c + 31488a b c - 320a b c + 80a b )x +--R + +--R 3 3 2 3 2 5 7 4 +--R (42624a b c - 4896a b c + 152a b c + 210b )x +--R + +--R 3 4 2 2 3 4 2 6 3 +--R (21504a c - 2304a b c - 6464a b c + 1680b c)x +--R + +--R 2 4 3 3 5 2 2 +--R (10752a b c - 16128a b c + 3360b c )x +--R + +--R 2 5 2 4 4 3 +--R (6144a c - 9216a b c + 1920b c )x +--R * +--R +-+ +-+ +--R \|a \|c +--R / +--R 4 3 3 3 4 2 3 2 2 +--R (12288a b c + 3072a b )x + (24576a c + 30720a b c)x +--R + +--R 3 2 3 3 +--R 73728a b c x + 49152a c +--R * +--R +--------------+ +--R +-+ +-+ | 2 +--R \|a \|c \|a x + b x + c +--R + +--R 5 2 4 2 3 4 4 +--R (- 6144a c - 9216a b c - 384a b )x +--R + +--R 4 2 3 3 3 4 3 3 2 2 2 +--R (- 49152a b c - 12288a b c)x + (- 49152a c - 61440a b c )x +--R + +--R 3 3 3 4 +--R - 98304a b c x - 49152a c +--R * +--R +-+ +--R \|a +--R , +--R +--R 3 3 2 3 2 5 7 3 +--R (- 1536a b c + 1920a b c + 96a b c - 120b )x +--R + +--R 3 4 2 2 3 4 2 6 2 +--R (- 3072a c + 768a b c + 4800a b c - 1200b c)x +--R + +--R 2 4 3 3 5 2 2 5 +--R (- 9216a b c + 13824a b c - 2880b c )x - 6144a c +--R + +--R 2 4 4 3 +--R 9216a b c - 1920b c +--R * +--R +--------------+ +--R +-+ | 2 +--R \|c \|a x + b x + c +--R + +--R 4 4 2 4 2 6 8 4 +--R (768a c - 1440a b c + 288a b c + 15b )x +--R + +--R 3 4 2 3 3 5 2 7 3 +--R (6144a b c - 7680a b c - 384a b c + 480b c)x +--R + +--R 3 5 2 2 4 4 3 6 2 2 +--R (6144a c - 1536a b c - 9600a b c + 2400b c )x +--R + +--R 2 5 3 4 5 3 2 6 2 5 +--R (12288a b c - 18432a b c + 3840b c )x + 6144a c - 9216a b c +--R + +--R 4 4 +--R 1920b c +--R * +--R +--------------+ +--R +---+ | 2 +---+ +-+ +--R \|- a \|a x + b x + c - \|- a \|c +--R atan(------------------------------------) +--R a x +--R + +--R 5 2 4 2 3 4 7 +--R (- 768a c - 1152a b c - 48a b )x +--R + +--R 4 2 3 3 2 5 6 +--R (- 6272a b c - 1728a b c - 8a b )x +--R + +--R 4 3 3 2 2 2 4 6 5 +--R (- 6528a c - 9120a b c - 40a b c + 10a b )x +--R + +--R 3 3 2 3 2 5 7 4 +--R (- 15552a b c + 240a b c + 12a b c - 15b )x +--R + +--R 3 4 2 2 3 4 2 6 3 +--R (- 9216a c + 384a b c + 1408a b c - 360b c)x +--R + +--R 2 4 3 3 5 2 2 +--R (- 3840a b c + 5760a b c - 1200b c )x +--R + +--R 2 5 2 4 4 3 +--R (- 3072a c + 4608a b c - 960b c )x +--R * +--R +--------------+ +--R +---+ | 2 +--R \|- a \|a x + b x + c +--R + +--R 5 4 3 8 5 2 4 2 3 4 7 +--R (1536a b c + 384a b )x + (3072a c + 5632a b c + 448a b )x +--R + +--R 4 2 3 3 2 5 6 +--R (15104a b c + 4992a b c - 16a b )x +--R + +--R 4 3 3 2 2 2 4 6 5 +--R (10752a c + 15744a b c - 160a b c + 40a b )x +--R + +--R 3 3 2 3 2 5 7 4 +--R (21312a b c - 2448a b c + 76a b c + 105b )x +--R + +--R 3 4 2 2 3 4 2 6 3 +--R (10752a c - 1152a b c - 3232a b c + 840b c)x +--R + +--R 2 4 3 3 5 2 2 +--R (5376a b c - 8064a b c + 1680b c )x +--R + +--R 2 5 2 4 4 3 +--R (3072a c - 4608a b c + 960b c )x +--R * +--R +---+ +-+ +--R \|- a \|c +--R / +--R 4 3 3 3 4 2 3 2 2 +--R (6144a b c + 1536a b )x + (12288a c + 15360a b c)x +--R + +--R 3 2 3 3 +--R 36864a b c x + 24576a c +--R * +--R +--------------+ +--R +---+ +-+ | 2 +--R \|- a \|c \|a x + b x + c +--R + +--R 5 2 4 2 3 4 4 +--R (- 3072a c - 4608a b c - 192a b )x +--R + +--R 4 2 3 3 3 4 3 3 2 2 2 +--R (- 24576a b c - 6144a b c)x + (- 24576a c - 30720a b c )x +--R + +--R 3 3 3 4 +--R - 49152a b c x - 24576a c +--R * +--R +---+ +--R \|- a +--R ] +--R Type: Union(List Expression Integer,...) +--E + +--S 63 +t1:=integrate(sqrt(a*x^2+b*x+c),x) +--R +--R (2) +--R [ +--R +--------------+ +--R 3 2 2 +-+ | 2 +--R ((16a b c - 4b )x + 32a c - 8b c)\|c \|a x + b x + c +--R + +--R 2 2 4 2 2 3 3 2 2 +--R (- 16a c + b )x + (- 32a b c + 8b c)x - 32a c + 8b c +--R * +--R log +--R +--------------+ +--R +-+ +-+ | 2 +-+ +--R (2\|a \|c - 2a x)\|a x + b x + c + 2a x\|c +--R + +--R 2 +-+ +--R (- 2a x - b x - 2c)\|a +--R / +--R +--------------+ +--R +-+ | 2 +--R 2\|c \|a x + b x + c - b x - 2c +--R + +--R 2 2 3 3 2 2 2 +-+ +--R ((- 16a c - 4a b )x + (- 40a b c - 2b )x + (- 32a c - 8b c)x)\|a +--R * +--R +--------------+ +--R | 2 +--R \|a x + b x + c +--R + +--R 2 4 2 2 3 3 2 +--R 16a b x + (32a c + 24a b )x + (56a b c + 6b )x +--R + +--R 2 2 +--R (32a c + 8b c)x +--R * +--R +-+ +-+ +--R \|a \|c +--R / +--R +--------------+ +--R +-+ +-+ | 2 +--R (32a b x + 64a c)\|a \|c \|a x + b x + c +--R + +--R 2 2 2 2 +-+ +--R ((- 32a c - 8a b )x - 64a b c x - 64a c )\|a +--R , +--R +--R +--------------+ +--R 3 2 2 +-+ | 2 +--R ((16a b c - 4b )x + 32a c - 8b c)\|c \|a x + b x + c +--R + +--R 2 2 4 2 2 3 3 2 2 +--R (- 16a c + b )x + (- 32a b c + 8b c)x - 32a c + 8b c +--R * +--R +--------------+ +--R +---+ | 2 +---+ +-+ +--R \|- a \|a x + b x + c - \|- a \|c +--R atan(------------------------------------) +--R a x +--R + +--R 2 2 3 3 2 2 2 +---+ +--R ((- 8a c - 2a b )x + (- 20a b c - b )x + (- 16a c - 4b c)x)\|- a +--R * +--R +--------------+ +--R | 2 +--R \|a x + b x + c +--R + +--R 2 4 2 2 3 3 2 2 2 +--R (8a b x + (16a c + 12a b )x + (28a b c + 3b )x + (16a c + 4b c)x) +--R * +--R +---+ +-+ +--R \|- a \|c +--R / +--R +--------------+ +--R +---+ +-+ | 2 +--R (16a b x + 32a c)\|- a \|c \|a x + b x + c +--R + +--R 2 2 2 2 +---+ +--R ((- 16a c - 4a b )x - 32a b c x - 32a c )\|- a +--R ] +--R Type: Union(List Expression Integer,...) +--E + +--S 64 +bb1:=(6*a*x-5*b)/(24*a^2)*(a*x^2+b*x+c)^(3/2)+(5*b^2-4*a*c)/(16*a^2)*t1.1 +--R +--R (3) +--R 2 2 3 5 2 3 2 2 4 +--R ((- 192a b c + 288a b c - 60b )x - 384a c + 576a b c - 120b c) +--R * +--R +---------