diff --git a/changelog b/changelog index 26d39fa..d3ccbf9 100644 --- a/changelog +++ b/changelog @@ -1,3 +1,8 @@ +20080414 tpd src/input/Makefile add integration regression testing +20080414 tpd src/input/schaum34.input integrals of csch(ax) +20080414 tpd src/input/schaum33.input integrals of csch(ax) +20080414 tpd src/input/schaum32.input integrals of sech(ax) +20080414 tpd src/input/schaum31.input integrals of coth(ax) 20080413 tpd src/input/Makefile add integration regression testing 20080413 tpd src/input/schaum30.input integrals of tanh(ax) 20080413 tpd src/input/schaum29.input integrals of sinh(ax) and cosh(ax) diff --git a/src/input/Makefile.pamphlet b/src/input/Makefile.pamphlet index 57ff1e9..fdf2fff 100644 --- a/src/input/Makefile.pamphlet +++ b/src/input/Makefile.pamphlet @@ -362,7 +362,8 @@ REGRES= algaggr.regress algbrbf.regress algfacob.regress alist.regress \ schaum17.regress schaum18.regress schaum19.regress schaum20.regress \ schaum21.regress schaum22.regress schaum23.regress schaum24.regress \ schaum25.regress schaum26.regress schaum27.regress schaum28.regress \ - schaum29.regress schaum30.regress \ + schaum29.regress schaum30.regress schaum31.regress schaum32.regress \ + schaum33.regress schaum34.regress \ scherk.regress scope.regress seccsc.regress \ segbind.regress seg.regress \ series2.regress series.regress sersolve.regress set.regress \ @@ -644,7 +645,8 @@ FILES= ${OUT}/algaggr.input ${OUT}/algbrbf.input ${OUT}/algfacob.input \ ${OUT}/schaum20.input ${OUT}/schaum21.input ${OUT}/schaum22.input \ ${OUT}/schaum23.input ${OUT}/schaum24.input ${OUT}/schaum25.input \ ${OUT}/schaum26.input ${OUT}/schaum27.input ${OUT}/schaum28.input \ - ${OUT}/schaum29.input ${OUT}/schaum30.input \ + ${OUT}/schaum29.input ${OUT}/schaum30.input ${OUT}/schaum31.input \ + ${OUT}/schaum32.input ${OUT}/schaum33.input ${OUT}/schaum34.input \ ${OUT}/saddle.input \ ${OUT}/scherk.input ${OUT}/scope.input ${OUT}/seccsc.input \ ${OUT}/segbind.input ${OUT}/seg.input ${OUT}/series2.input \ @@ -958,6 +960,8 @@ DOCFILES= \ ${DOC}/schaum25.input.dvi ${DOC}/schaum26.input.dvi \ ${DOC}/schaum27.input.dvi ${DOC}/schaum28.input.dvi \ ${DOC}/schaum29.input.dvi ${DOC}/schaum30.input.dvi \ + ${DOC}/schaum31.input.dvi ${DOC}/schaum32.input.dvi \ + ${DOC}/schaum33.input.dvi ${DOC}/schaum34.input.dvi \ ${DOC}/s01eaf.input.dvi ${DOC}/s13aaf.input.dvi \ ${DOC}/s13acf.input.dvi ${DOC}/s13adf.input.dvi \ ${DOC}/s14aaf.input.dvi ${DOC}/s14abf.input.dvi \ diff --git a/src/input/schaum31.input.pamphlet b/src/input/schaum31.input.pamphlet new file mode 100644 index 0000000..3e81bf2 --- /dev/null +++ b/src/input/schaum31.input.pamphlet @@ -0,0 +1,286 @@ +\documentclass{article} +\usepackage{axiom} +\begin{document} +\title{\$SPAD/input schaum31.input} +\author{Timothy Daly} +\maketitle +\eject +\tableofcontents +\eject +\section{\cite{1}:14.615~~~~~$\displaystyle +\int{\coth{ax}}~dx$} +$$\int{\coth{ax}}= +\frac{1}{a}\ln\sinh{ax} +$$ +<<*>>= +)spool schaum31.output +)set message test on +)set message auto off +)clear all + +--S 1 of 11 +aa:=integrate(coth(a*x),x) +--R +--R +--R 2sinh(a x) +--R log(- ---------------------) - a x +--R sinh(a x) - cosh(a x) +--R (1) ---------------------------------- +--R a +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.616~~~~~$\displaystyle +\int{\coth^2{ax}}~dx$} +$$\int{\coth^2{ax}}= +x-\frac{\coth{ax}}{a} +$$ +<<*>>= +)clear all + +--S 2 of 11 +aa:=integrate(coth(a*x)^2,x) +--R +--R +--R (a x + 1)sinh(a x) - cosh(a x) +--R (1) ------------------------------ +--R a sinh(a x) +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.617~~~~~$\displaystyle +\int{\coth^3{ax}}~dx$} +$$\int{\coth^3{ax}}= +\frac{1}{a}\ln\sinh{ax}-\frac{\coth^2{ax}}{2a} +$$ +<<*>>= +)clear all + +--S 3 of 11 +aa:=integrate(coth(a*x)^3,x) +--R +--R +--R (1) +--R 4 3 2 2 +--R sinh(a x) + 4cosh(a x)sinh(a x) + (6cosh(a x) - 2)sinh(a x) +--R + +--R 3 4 2 +--R (4cosh(a x) - 4cosh(a x))sinh(a x) + cosh(a x) - 2cosh(a x) + 1 +--R * +--R 2sinh(a x) +--R log(- ---------------------) +--R sinh(a x) - cosh(a x) +--R + +--R 4 3 +--R - a x sinh(a x) - 4a x cosh(a x)sinh(a x) +--R + +--R 2 2 +--R (- 6a x cosh(a x) + 2a x - 2)sinh(a x) +--R + +--R 3 4 +--R (- 4a x cosh(a x) + (4a x - 4)cosh(a x))sinh(a x) - a x cosh(a x) +--R + +--R 2 +--R (2a x - 2)cosh(a x) - a x +--R / +--R 4 3 2 2 +--R a sinh(a x) + 4a cosh(a x)sinh(a x) + (6a cosh(a x) - 2a)sinh(a x) +--R + +--R 3 4 2 +--R (4a cosh(a x) - 4a cosh(a x))sinh(a x) + a cosh(a x) - 2a cosh(a x) + a +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.618~~~~~$\displaystyle +\int{\coth^n{ax}{{\rm ~csch}^2{ax}}}~dx$} +$$\int{\coth^n{ax}{{\rm ~csch}^2{ax}}}= +-\frac{\coth^{n+1}{ax}}{(n+1)a} +$$ +<<*>>= +)clear all + +--S 4 of 11 +aa:=integrate(coth(a*x)^n*csch(a*x)^2,x) +--R +--R +--R cosh(a x) cosh(a x) +--R - cosh(a x)sinh(n log(---------)) - cosh(a x)cosh(n log(---------)) +--R sinh(a x) sinh(a x) +--R (1) ------------------------------------------------------------------- +--R (a n + a)sinh(a x) +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.619~~~~~$\displaystyle +\int{\frac{{\rm csch}^2{ax}}{\coth{ax}}}~dx$} +$$\int{\frac{{\rm csch}^2{ax}}{\coth{ax}}}= +-\frac{1}{a}\ln\coth{ax} +$$ +<<*>>= +)clear all + +--S 5 of 11 +aa:=integrate(csch(a*x)^2/coth(a*x),x) +--R +--R +--R 2cosh(a x) 2sinh(a x) +--R - log(- ---------------------) + log(- ---------------------) +--R sinh(a x) - cosh(a x) sinh(a x) - cosh(a x) +--R (1) ------------------------------------------------------------- +--R a +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.620~~~~~$\displaystyle +\int{\frac{dx}{\coth{ax}}}~dx$} +$$\int{\frac{1}{\coth{ax}}}= +\frac{1}{a}\ln\cosh{ax} +$$ +<<*>>= +)clear all + +--S 6 of 11 +aa:=integrate(1/coth(a*x),x) +--R +--R +--R 2cosh(a x) +--R log(- ---------------------) - a x +--R sinh(a x) - cosh(a x) +--R (1) ---------------------------------- +--R a +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.621~~~~~$\displaystyle +\int{x\coth{ax}}~dx$} +$$\int{x\coth{ax}}= +\frac{1}{a^2}\left\{ +ax+\frac{(ax)^3}{9}-\frac{(ax)^5}{225}+\cdots +\frac{(-1)^{n-1}2^{2n}B_n(ax)^{2n+1}}{(2n+1)!}+\cdots\right\} +$$ +<<*>>= +)clear all + +--S 7 of 11 +aa:=integrate(x*coth(a*x),x) +--R +--R +--R x +--R ++ +--I (1) | %O coth(%O a)d%O +--R ++ +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.622~~~~~$\displaystyle +\int{x\coth^2{ax}}~dx$} +$$\int{x\coth^2{ax}}= +\frac{x^2}{2}-\frac{x\coth{ax}}{a}+\frac{1}{a^2}\ln\sinh{ax} +$$ +<<*>>= +)clear all + +--S 8 of 11 +aa:=integrate(x*coth(a*x)^2,x) +--R +--R +--R (1) +--R 2 2 +--R (2sinh(a x) + 4cosh(a x)sinh(a x) + 2cosh(a x) - 2) +--R * +--R 2sinh(a x) +--R log(- ---------------------) +--R sinh(a x) - cosh(a x) +--R + +--R 2 2 2 2 2 +--R (a x - 4a x)sinh(a x) + (2a x - 8a x)cosh(a x)sinh(a x) +--R + +--R 2 2 2 2 2 +--R (a x - 4a x)cosh(a x) - a x +--R / +--R 2 2 2 2 2 2 +--R 2a sinh(a x) + 4a cosh(a x)sinh(a x) + 2a cosh(a x) - 2a +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.623~~~~~$\displaystyle +\int{\frac{\coth{ax}}{x}}~dx$} +$$\int{\frac{\coth{ax}}{x}}= +-\frac{1}{ax}+\frac{(ax)}{3}-\frac{(ax)^3}{135}+\cdots +\frac{(-1)^{n}2^{2n}B_n(ax)^{2n-1}}{(2n-1)(2n)!}+\cdots +$$ +<<*>>= +)clear all + +--S 9 of 11 +aa:=integrate(coth(a*x)/x,x) +--R +--R +--R x +--I ++ coth(%O a) +--I (1) | ---------- d%O +--I ++ %O +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.624~~~~~$\displaystyle +\int{\frac{dx}{p+q\coth{ax}}}~dx$} +$$\int{\frac{1}{p+q\coth{ax}}}= +\frac{px}{p^2-q^2}-\frac{q}{a(p^2-q^2)}\ln(p\sinh{ax}+q\cosh{ax}) +$$ +<<*>>= +)clear all + +--S 10 of 11 +aa:=integrate(1/(p+q*coth(a*x)),x) +--R +--R +--R - 2p sinh(a x) - 2q cosh(a x) +--R q log(-----------------------------) + (- a q - a p)x +--R sinh(a x) - cosh(a x) +--R (1) ----------------------------------------------------- +--R 2 2 +--R a q - a p +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.625~~~~~$\displaystyle +\int{\coth^n{ax}}~dx$} +$$\int{\coth^n{ax}}= +-\frac{\coth^{n-1}{ax}}{a(n-1)}+\int{\coth^{n-1}{ax}} +$$ +<<*>>= +)clear all + +--S 11 of 11 +aa:=integrate(coth(a*x)^n,x) +--R +--R +--R x +--R ++ n +--I (1) | coth(%O a) d%O +--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 pp90-91 +\end{thebibliography} +\end{document} diff --git a/src/input/schaum32.input.pamphlet b/src/input/schaum32.input.pamphlet new file mode 100644 index 0000000..085ddf0 --- /dev/null +++ b/src/input/schaum32.input.pamphlet @@ -0,0 +1,294 @@ +\documentclass{article} +\usepackage{axiom} +\begin{document} +\title{\$SPAD/input schaum32.input} +\author{Timothy Daly} +\maketitle +\eject +\tableofcontents +\eject +\section{\cite{1}:14.626~~~~~$\displaystyle +\int{{\rm sech~}{ax}}~dx$} +$$\int{{\rm sech~}{ax}}= +\frac{2}{a}\tanh^{-1}{e^{ax}} +$$ +<<*>>= +)spool schaum32.output +)set message test on +)set message auto off +)clear all + +--S 1 of 10 +aa:=integrate(sech(a*x),x) +--R +--R +--R 2atan(sinh(a x) + cosh(a x)) +--R (1) ---------------------------- +--R a +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.627~~~~~$\displaystyle +\int{{\rm sech}^2~{ax}}~dx$} +$$\int{{\rm sech}^2~{ax}}= +\frac{\tanh{ax}}{a} +$$ +<<*>>= +)clear all + +--S 2 of 10 +aa:=integrate(sech(a*x)^2,x) +--R +--R +--R 2 +--R (1) - ------------------------------------------------------- +--R 2 2 +--R a sinh(a x) + 2a cosh(a x)sinh(a x) + a cosh(a x) + a +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.628~~~~~$\displaystyle +\int{{\rm sech}^3~{ax}}~dx$} +$$\int{{\rm sech}^3~{ax}}= +\frac{{\rm sech}~{ax}~\tanh{ax}}{2a}+\frac{1}{2a}\tan^{-1}{\rm ~sech~}{ax} +$$ +<<*>>= +)clear all + +--S 3 of 10 +aa:=integrate(sech(a*x)^3,x) +--R +--R +--R (1) +--R 4 3 2 2 +--R sinh(a x) + 4cosh(a x)sinh(a x) + (6cosh(a x) + 2)sinh(a x) +--R + +--R 3 4 2 +--R (4cosh(a x) + 4cosh(a x))sinh(a x) + cosh(a x) + 2cosh(a x) + 1 +--R * +--R atan(sinh(a x) + cosh(a x)) +--R + +--R 3 2 2 +--R sinh(a x) + 3cosh(a x)sinh(a x) + (3cosh(a x) - 1)sinh(a x) +--R + +--R 3 +--R cosh(a x) - cosh(a x) +--R / +--R 4 3 2 2 +--R a sinh(a x) + 4a cosh(a x)sinh(a x) + (6a cosh(a x) + 2a)sinh(a x) +--R + +--R 3 4 2 +--R (4a cosh(a x) + 4a cosh(a x))sinh(a x) + a cosh(a x) + 2a cosh(a x) + a +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.629~~~~~$\displaystyle +\int{{\rm sech}^n~{ax}~{\tanh{ax}}}~dx$} +$$\int{{\rm sech~}^n{ax}~{\tanh{ax}}}= +-\frac{{\rm sech~}^{n}{ax}}{na} +$$ +<<*>>= +)clear all + +--S 4 of 10 +aa:=integrate(sech(a*x)^n*tanh(a*x),x) +--R +--R +--R (1) +--R 2sinh(a x) + 2cosh(a x) +--R - sinh(n log(-------------------------------------------------)) +--R 2 2 +--R sinh(a x) + 2cosh(a x)sinh(a x) + cosh(a x) + 1 +--R + +--R 2sinh(a x) + 2cosh(a x) +--R - cosh(n log(-------------------------------------------------)) +--R 2 2 +--R sinh(a x) + 2cosh(a x)sinh(a x) + cosh(a x) + 1 +--R / +--R a n +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.630~~~~~$\displaystyle +\int{\frac{dx}{{\rm sech~}{ax}}}~dx$} +$$\int{\frac{1}{{\rm sech~}{ax}}}= +\frac{{\rm sech}~{ax}}{a} +$$ +<<*>>= +)clear all + +--S 5 of 10 +aa:=integrate(1/sech(a*x),x) +--R +--R +--R sinh(a x) +--R (1) --------- +--R a +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.631~~~~~$\displaystyle +\int{x{\rm ~sech~}{ax}}~dx$} +$$\int{x{\rm ~sech~}{ax}}= +\frac{1}{a^2}\left\{ +\frac{(ax)^2}{2}-\frac{(ax)^4}{8}+\frac{5(ax)^6}{144}+\cdots +\frac{(-1)^{n}E_n(ax)^{2n+2}}{(2n+2)(2n)!}+\cdots\right\} +$$ +<<*>>= +)clear all + +--S 6 of 10 +aa:=integrate(x*sech(a*x),x) +--R +--R +--R x +--R ++ +--I (1) | %O sech(%O a)d%O +--R ++ +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.632~~~~~$\displaystyle +\int{x~{\rm sech}^2~{ax}}~dx$} +$$\int{x~{\rm sech}^2~{ax}}= +\frac{x\tanh{ax}}{a}-\frac{1}{a^2}\ln\cosh{ax} +$$ +<<*>>= +)clear all + +--S 7 of 10 +aa:=integrate(x*sech(a*x)^2,x) +--R +--R +--R (1) +--R 2 2 +--R (- sinh(a x) - 2cosh(a x)sinh(a x) - cosh(a x) - 1) +--R * +--R 2cosh(a x) +--R log(- ---------------------) +--R sinh(a x) - cosh(a x) +--R + +--R 2 2 +--R 2a x sinh(a x) + 4a x cosh(a x)sinh(a x) + 2a x cosh(a x) +--R / +--R 2 2 2 2 2 2 +--R a sinh(a x) + 2a cosh(a x)sinh(a x) + a cosh(a x) + a +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.633~~~~~$\displaystyle +\int{\frac{{\rm sech~}{ax}}{x}}~dx$} +$$\int{\frac{{\rm sech~}{ax}}{x}}= +\ln{x}-\frac{(ax)^2}{4}+\frac{5(ax)^4}{96}-\frac{61(ax)^6}{4320}+\cdots +\frac{(-1)^{n}E_n(ax)^{2n}}{2n(2n)!}+\cdots +$$ +<<*>>= +)clear all + +--S 8 of 10 +aa:=integrate(sech(a*x)/x,x) +--R +--R +--R x +--I ++ sech(%O a) +--I (1) | ---------- d%O +--I ++ %O +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.634~~~~~$\displaystyle +\int{\frac{dx}{q+p{\rm ~sech~}{ax}}}~dx$} +$$\int{\frac{1}{q+p{\rm ~sech~}{ax}}}= +\frac{x}{q}-\frac{p}{q}\int{\frac{dx}{p+q\cosh{ax}}} +$$ +<<*>>= +)clear all + +--S 9 of 10 +aa:=integrate(1/(q+p*sech(a*x)),x) +--R +--R +--R (1) +--R [ +--R p +--R * +--R log +--R 2 2 2 +--R q sinh(a x) + (2q cosh(a x) + 2p q)sinh(a x) +--R + +--R 2 2 2 2 +--R q cosh(a x) + 2p q cosh(a x) - q + 2p +--R * +--R +---------+ +--R | 2 2 +--R \|- q + p +--R + +--R 3 2 3 2 2 3 +--R (- 2q + 2p q)sinh(a x) + (- 2q + 2p q)cosh(a x) - 2p q + 2p +--R / +--R 2 2 +--R q sinh(a x) + (2q cosh(a x) + 2p)sinh(a x) + q cosh(a x) +--R + +--R 2p cosh(a x) + q +--R + +--R +---------+ +--R | 2 2 +--R a x\|- q + p +--R / +--R +---------+ +--R | 2 2 +--R a q\|- q + p +--R , +--R +-------+ +--R | 2 2 +-------+ +--R (q sinh(a x) + q cosh(a x) + p)\|q - p | 2 2 +--R - 2p atan(-----------------------------------------) + a x\|q - p +--R 2 2 +--R q - p +--R --------------------------------------------------------------------] +--R +-------+ +--R | 2 2 +--R a q\|q - p +--R Type: Union(List Expression Integer,...) +--E +@ + +\section{\cite{1}:14.635~~~~~$\displaystyle +\int{{\rm sech}^n~{ax}}~dx$} +$$\int{{\rm sech}^n~{ax}}= +\frac{{\rm sech}^{n-2}~{ax}~\tanh{ax}}{a(n-1)} ++\frac{n-2}{n-1}\int{{\rm sech}^{n-2}~{ax}} +$$ +<<*>>= +)clear all + +--S 10 of 10 +aa:=integrate(sech(a*x)^n,x) +--R +--R +--R x +--R ++ n +--I (1) | sech(%O a) d%O +--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 p91 +\end{thebibliography} +\end{document} diff --git a/src/input/schaum33.input.pamphlet b/src/input/schaum33.input.pamphlet new file mode 100644 index 0000000..c5ee6a2 --- /dev/null +++ b/src/input/schaum33.input.pamphlet @@ -0,0 +1,292 @@ +\documentclass{article} +\usepackage{axiom} +\begin{document} +\title{\$SPAD/input schaum33.input} +\author{Timothy Daly} +\maketitle +\eject +\tableofcontents +\eject +\section{\cite{1}:14.636~~~~~$\displaystyle +\int{{\rm csch~}{ax}}~dx$} +$$\int{{\rm csch~}{ax}}= +\frac{1}{a}\ln\tanh{\frac{ax}{2}} +$$ +<<*>>= +)spool schaum33.output +)set message test on +)set message auto off +)clear all + +--S 1 of 10 +aa:=integrate(csch(a*x),x) +--R +--R +--R - log(sinh(a x) + cosh(a x) + 1) + log(sinh(a x) + cosh(a x) - 1) +--R (1) ----------------------------------------------------------------- +--R a +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.637~~~~~$\displaystyle +\int{{\rm csch}^2~{ax}}~dx$} +$$\int{{\rm csch}^2~{ax}}= +-\frac{\coth{ax}}{a} +$$ +<<*>>= +)clear all + +--S 2 of 10 +aa:=integrate(csch(a*x)^2,x) +--R +--R +--R 2 +--R (1) - ------------------------------------------------------- +--R 2 2 +--R a sinh(a x) + 2a cosh(a x)sinh(a x) + a cosh(a x) - a +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.638~~~~~$\displaystyle +\int{{\rm csch}^3~{ax}}~dx$} +$$\int{{\rm csch}^3~{ax}}= +-\frac{{\rm csch~}{ax}\coth{ax}}{2a}-\frac{1}{2a}\ln\tanh\frac{ax}{2} +$$ +<<*>>= +)clear all + +--S 3 of 10 +aa:=integrate(csch(a*x)^3,x) +--R +--R +--R (1) +--R 4 3 2 2 +--R sinh(a x) + 4cosh(a x)sinh(a x) + (6cosh(a x) - 2)sinh(a x) +--R + +--R 3 4 2 +--R (4cosh(a x) - 4cosh(a x))sinh(a x) + cosh(a x) - 2cosh(a x) + 1 +--R * +--R log(sinh(a x) + cosh(a x) + 1) +--R + +--R 4 3 2 2 +--R - sinh(a x) - 4cosh(a x)sinh(a x) + (- 6cosh(a x) + 2)sinh(a x) +--R + +--R 3 4 2 +--R (- 4cosh(a x) + 4cosh(a x))sinh(a x) - cosh(a x) + 2cosh(a x) - 1 +--R * +--R log(sinh(a x) + cosh(a x) - 1) +--R + +--R 3 2 2 +--R - 2sinh(a x) - 6cosh(a x)sinh(a x) + (- 6cosh(a x) - 2)sinh(a x) +--R + +--R 3 +--R - 2cosh(a x) - 2cosh(a x) +--R / +--R 4 3 2 2 +--R 2a sinh(a x) + 8a cosh(a x)sinh(a x) + (12a cosh(a x) - 4a)sinh(a x) +--R + +--R 3 4 2 +--R (8a cosh(a x) - 8a cosh(a x))sinh(a x) + 2a cosh(a x) - 4a cosh(a x) +--R + +--R 2a +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.639~~~~~$\displaystyle +\int{{\rm csch}^n~{ax}~{\coth{ax}}}~dx$} +$$\int{{\rm csch~}^n{ax}~{\coth{ax}}}= +-\frac{{\rm csch~}^{n}{ax}}{na} +$$ +<<*>>= +)clear all + +--S 4 of 10 +aa:=integrate(csch(a*x)^n*coth(a*x),x) +--R +--R +--R (1) +--R 2sinh(a x) + 2cosh(a x) +--R - sinh(n log(-------------------------------------------------)) +--R 2 2 +--R sinh(a x) + 2cosh(a x)sinh(a x) + cosh(a x) - 1 +--R + +--R 2sinh(a x) + 2cosh(a x) +--R - cosh(n log(-------------------------------------------------)) +--R 2 2 +--R sinh(a x) + 2cosh(a x)sinh(a x) + cosh(a x) - 1 +--R / +--R a n +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.640~~~~~$\displaystyle +\int{\frac{dx}{{\rm csch~}{ax}}}~dx$} +$$\int{\frac{1}{{\rm csch~}{ax}}}= +\frac{1}{a}{\rm cosh}~{ax} +$$ +<<*>>= +)clear all + +--S 5 of 10 +aa:=integrate(1/csch(a*x),x) +--R +--R +--R cosh(a x) +--R (1) --------- +--R a +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.641~~~~~$\displaystyle +\int{x{\rm ~csch~}{ax}}~dx$} +$$\int{x{\rm ~csch~}{ax}}= +\frac{1}{a^2}\left\{ +ax-\frac{(ax)^3}{18}+\frac{7(ax)^5}{1800}+\cdots+ +\frac{2(-1)^n(2^{2n-1}-1)B_n(ax)^{2n+1}}{(2n+1)!}+\cdots\right\} +$$ +<<*>>= +)clear all + +--S 6 of 10 +aa:=integrate(x*csch(a*x),x) +--R +--R +--R x +--R ++ +--I (1) | %O csch(%O a)d%O +--R ++ +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.642~~~~~$\displaystyle +\int{x~{\rm csch}^2~{ax}}~dx$} +$$\int{x~{\rm csch}^2~{ax}}= +-\frac{x\coth{ax}}{a}+\frac{1}{a^2}\ln\sinh{ax} +$$ +<<*>>= +)clear all + +--S 7 of 10 +aa:=integrate(x*csch(a*x)^2,x) +--R +--R +--R (1) +--R 2 2 +--R (sinh(a x) + 2cosh(a x)sinh(a x) + cosh(a x) - 1) +--R * +--R 2sinh(a x) +--R log(- ---------------------) +--R sinh(a x) - cosh(a x) +--R + +--R 2 2 +--R - 2a x sinh(a x) - 4a x cosh(a x)sinh(a x) - 2a x cosh(a x) +--R / +--R 2 2 2 2 2 2 +--R a sinh(a x) + 2a cosh(a x)sinh(a x) + a cosh(a x) - a +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.643~~~~~$\displaystyle +\int{\frac{{\rm csch~}{ax}}{x}}~dx$} +$$\int{\frac{{\rm csch~}{ax}}{x}}= +-\frac{1}{ax}-\frac{ax}{6}+\frac{7(ax)^3}{1080}+\cdots +\frac{(-1)^n2(2^{2n-1}-1)B_n(ax)^{2n-1}}{(2n-1)(2n)!}+\cdots +$$ +<<*>>= +)clear all + +--S 8 of 10 +aa:=integrate(csch(a*x)/x,x) +--R +--R +--R x +--I ++ csch(%O a) +--I (1) | ---------- d%O +--I ++ %O +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.644~~~~~$\displaystyle +\int{\frac{dx}{q+p{\rm ~csch~}{ax}}}~dx$} +$$\int{\frac{1}{q+p{\rm ~csch~}{ax}}}= +\frac{x}{q}-\frac{p}{q}\int{\frac{dx}{p+q\sinh{ax}}} +$$ +<<*>>= +)clear all + +--S 9 of 10 +aa:=integrate(1/(q+p*csch(a*x)),x) +--R +--R +--R (1) +--R p +--R * +--R log +--R 2 2 2 2 2 +--R q sinh(a x) + (2q cosh(a x) + 2p q)sinh(a x) + q cosh(a x) +--R + +--R 2 2 +--R 2p q cosh(a x) + q + 2p +--R * +--R +-------+ +--R | 2 2 +--R \|q + p +--R + +--R 3 2 3 2 2 3 +--R (2q + 2p q)sinh(a x) + (2q + 2p q)cosh(a x) + 2p q + 2p +--R / +--R 2 2 +--R q sinh(a x) + (2q cosh(a x) + 2p)sinh(a x) + q cosh(a x) +--R + +--R 2p cosh(a x) - q +--R + +--R +-------+ +--R | 2 2 +--R a x\|q + p +--R / +--R +-------+ +--R | 2 2 +--R a q\|q + p +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.645~~~~~$\displaystyle +\int{{\rm csch}^n~{ax}}~dx$} +$$\int{{\rm csch}^n~{ax}}= +\frac{-{\rm csch}^{n-2}~{ax}~\coth{ax}}{a(n-1)} +-\frac{n-2}{n-1}\int{{\rm csch}^{n-2}~{ax}} +$$ +<<*>>= +)clear all + +--S 10 of 10 +aa:=integrate(csch(a*x)^n,x) +--R +--R +--R x +--R ++ n +--I (1) | csch(%O a) d%O +--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 pp91-92 +\end{thebibliography} +\end{document} diff --git a/src/input/schaum34.input.pamphlet b/src/input/schaum34.input.pamphlet new file mode 100644 index 0000000..1a5c359 --- /dev/null +++ b/src/input/schaum34.input.pamphlet @@ -0,0 +1,901 @@ +\documentclass{article} +\usepackage{axiom} +\begin{document} +\title{\$SPAD/input schaum34.input} +\author{Timothy Daly} +\maketitle +\eject +\tableofcontents +\eject +\section{\cite{1}:14.646~~~~~$\displaystyle +\int{\sinh^{-1}\frac{x}{a}}~dx$} +$$\int{\sinh^{-1}\frac{x}{a}}= +x\sinh^{-1}\frac{x}{a}-\sqrt{x^2+a^2} +$$ +<<*>>= +)spool schaum34.output +)set message test on +)set message auto off +)clear all + +--S 1 of 32 +aa:=integrate(asinh(x/a),x) +--R +--R +--R +-------+ +--R +-------+ | 2 2 +-------+ +--R | 2 2 2 \|x + a + x | 2 2 2 2 +--R (x\|x + a - x )log(--------------) + x\|x + a - x - a +--R a +--R (1) ------------------------------------------------------------- +--R +-------+ +--R | 2 2 +--R \|x + a - x +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.647~~~~~$\displaystyle +\int{x\sinh^{-1}\frac{x}{a}}~dx$} +$$\int{x\sinh^{-1}\frac{x}{a}}= +\left(\frac{x^2}{2}+\frac{a^2}{4}\right)\sinh^{-1}\frac{x}{a} +-\frac{x\sqrt{x^2+a^2}}{4} +$$ +<<*>>= +)clear all + +--S 2 of 32 +aa:=integrate(x*asinh(x/a),x) +--R +--R +--R (1) +--R +-------+ +--R +-------+ | 2 2 +--R 3 2 | 2 2 4 2 2 4 \|x + a + x +--R ((4x + 2a x)\|x + a - 4x - 4a x - a )log(--------------) +--R a +--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 8x\|x + a - 8x - 4a +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.648~~~~~$\displaystyle +\int{x^2\sinh^{-1}\frac{x}{a}}~dx$} +$$\int{x^2\sinh^{-1}\frac{x}{a}}= +\frac{x^3}{3}\sinh^{-1}\frac{x}{a}+\frac{(2a^2-x^2)\sqrt{x^2+a^2}}{9} +$$ +<<*>>= +)clear all + +--S 3 of 32 +aa:=integrate(x^2*asinh(x/a),x) +--R +--R +--R (1) +--R +-------+ +--R +-------+ | 2 2 +--R 5 2 3 | 2 2 6 2 4 \|x + a + x +--R ((12x + 3a x )\|x + a - 12x - 9a x )log(--------------) +--R a +--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 / +--R +-------+ +--R 2 2 | 2 2 3 2 +--R (36x + 9a )\|x + a - 36x - 27a x +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.649~~~~~$\displaystyle +\int{\frac{\sinh^{-1}(x/a)}{x}}~dx$} +$$\int{\frac{\sinh^{-1}(x/a)}{x}}= +\left\{ +\begin{array}{lr} +\displaystyle +\frac{x}{a}-\frac{(x/a)^3}{2\cdot 3\cdot 3} ++\frac{1\cdot 3(x/a)^5}{2\cdot 4\cdot 5\cdot 5} +-\frac{1\cdot 3\cdot 5(x/a)^7}{2\cdot 4\cdot 6\cdot 7\cdot 7}+\cdots& +|x| a\\ +\\ +\displaystyle +-\frac{\ln^2(-2x/a)}{2}+\frac{(a/x)^2}{2\cdot 2\cdot 2} +-\frac{1\cdot 3(a/x)^4}{2\cdot 4\cdot 4\cdot 4} ++\frac{1\cdot 3\cdot 5(a/x)^6}{2\cdot 4\cdot 6\cdot 6\cdot 6}-\cdots& +x<-a\\ +\end{array} +\right. +$$ +<<*>>= +)clear all + +--S 4 of 32 +aa:=integrate(asinh(x/a)/x,x) +--R +--R +--I %P +--R x asinh(--) +--R ++ a +--I (1) | --------- d%P +--I ++ %P +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.650~~~~~$\displaystyle +\int{\frac{\sinh^{-1}(x/a)}{x^2}}~dx$} +$$\int{\frac{\sinh^{-1}(x/a)}{x^2}}= +-\frac{\sinh^{-1}(x/a)}{x} +-\frac{1}{a}\ln\left(\frac{a+\sqrt{x^2+a^2}}{x}\right) +$$ +<<*>>= +)clear all + +--S 5 of 32 +aa:=integrate(asinh(x/a)/x^2,x) +--R +--R +--R (1) +--R +-------+ +-------+ +--R | 2 2 | 2 2 +--R - x log(\|x + a - x + a) + x log(\|x + a - x - a) +--R + +--R +-------+ +--R | 2 2 +--R \|x + a + x +--R - a log(--------------) +--R a +--R / +--R a x +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.651~~~~~$\displaystyle +\int{\cosh^{-1}\frac{x}{a}}~dx$} +$$\int{\cosh^{-1}\frac{x}{a}}= +\left\{ +\begin{array}{l} +\displaystyle +x\cosh^{-1}(x/a)-\sqrt{x^2-a^2},\quad\cosh^{-1}\frac{x}{a} > 0\\ +\\ +\displaystyle +x\cosh^{-1}(x/a)+\sqrt{x^2-a^2},\quad\cosh^{-1}\frac{x}{a} < 0\\ +\end{array} +\right. +$$ +<<*>>= +)clear all + +--S 6 of 32 +aa:=integrate(acosh(x/a),x) +--R +--R +--R +-------+ +--R +-------+ | 2 2 +-------+ +--R | 2 2 2 \|x - a + x | 2 2 2 2 +--R (x\|x - a - x )log(--------------) + x\|x - a - x + a +--R a +--R (1) ------------------------------------------------------------- +--R +-------+ +--R | 2 2 +--R \|x - a - x +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.652~~~~~$\displaystyle +\int{x\cosh^{-1}\frac{x}{a}}~dx$} +$$\int{x\cosh^{-1}\frac{x}{a}}= +\left\{ +\begin{array}{l} +\displaystyle +\frac{1}{4}(2x^2-a^2)\cosh^{-1}(x/a)-\frac{1}{4}x\sqrt{x^2-a^2}, +\quad\cosh^{-1}(x/a)>0\\ +\\ +\displaystyle +\frac{1}{4}(2x^2-a^2)\cosh^{-1}(x/a)+\frac{1}{4}x\sqrt{x^2-a^2}, +\quad\cosh^{-1}(x/a)<0\\ +\end{array} +\right. +$$ +<<*>>= +)clear all + +--S 7 of 32 +aa:=integrate(x*acosh(x/a),x) +--R +--R +--R (1) +--R +-------+ +--R +-------+ | 2 2 +--R 3 2 | 2 2 4 2 2 4 \|x - a + x +--R ((4x - 2a x)\|x - a - 4x + 4a x - a )log(--------------) +--R a +--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 8x\|x - a - 8x + 4a +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.653~~~~~$\displaystyle +\int{x^2\cosh^{-1}\frac{x}{a}}~dx$} +$$\int{x^2\cosh^{-1}\frac{x}{a}}= +\left\{ +\begin{array}{l} +\displaystyle +\frac{1}{3}x^3\cosh^{-1}(x/a)-\frac{1}{9}(x^2+2a^2)\sqrt{x^2-a^2}, +\quad\cosh^{-1}(x/a)>0\\ +\\ +\displaystyle +\frac{1}{3}x^3\cosh^{-1}(x/a)+\frac{1}{9}(x^2+2a^2)\sqrt{x^2-a^2}, +\quad\cosh^{-1}(x/a)<0\\ +\end{array} +\right. +$$ +<<*>>= +)clear all + +--S 8 of 32 +aa:=integrate(x^2*acosh(x/a),x) +--R +--R +--R (1) +--R +-------+ +--R +-------+ | 2 2 +--R 5 2 3 | 2 2 6 2 4 \|x - a + x +--R ((12x - 3a x )\|x - a - 12x + 9a x )log(--------------) +--R a +--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 / +--R +-------+ +--R 2 2 | 2 2 3 2 +--R (36x - 9a )\|x - a - 36x + 27a x +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.654~~~~~$\displaystyle +\int{\frac{\cosh^{-1}(x/a)}{x}}~dx$} +$$\int{\frac{\cosh^{-1}(x/a)}{x}}= +\begin{array}{l} +\displaystyle +\pm\left[\frac{1}{2}\ln^2(2x/a)+\frac{(a/x)^2}{2\cdot 2\cdot 2} ++\frac{1\cdot 3(a/x)^4}{2\cdot 4\cdot 4\cdot 4} ++\frac{1\cdot 3\cdot 5(a/x)^6}{2\cdot 4\cdot 6\cdot 6\cdot 6}+\cdots\right]\\ +\\ +\displaystyle +\hbox{\hskip 2cm}+{\rm if\ }\cosh^{-1}(x/a)>0, +\quad -{\rm if\ }\cosh^{-1}(x/a)<0, +\end{array} +$$ +<<*>>= +)clear all + +--S 9 of 32 +aa:=integrate(acosh(x/a)/x,x) +--R +--R +--I %P +--R x acosh(--) +--R ++ a +--I (1) | --------- d%P +--I ++ %P +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.655~~~~~$\displaystyle +\int{\frac{\cosh^{-1}(x/a)}{x^2}}~dx$} +$$\int{\frac{\cosh^{-1}(x/a)}{x^2}}= +\begin{array}{l} +\displaystyle +-\frac{\cosh^{-1}(x/a)}{x} +\mp\frac{1}{a}\ln\left(\frac{a+\sqrt{x^2+a^2}}{x}\right)\\ +\\ +\displaystyle +\hbox{\hskip 1cm}-{\rm if\ }\cosh^{-1}(x/a)>0, +\quad +{\rm if\ }\cosh^{-1}(x/a)<0, +\end{array} +$$ +<<*>>= +)clear all + +--S 10 of 32 +aa:=integrate(acosh(x/a)/x^2,x) +--R +--R +--R +-------+ +-------+ +--R | 2 2 | 2 2 +--R \|x - a + x \|x - a - x +--R - a log(--------------) + 2x atan(--------------) +--R a a +--R (1) ------------------------------------------------- +--R a x +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.656~~~~~$\displaystyle +\int{\tanh^{-1}\frac{x}{a}}~dx$} +$$\int{\tanh^{-1}\frac{x}{a}}= +x\tanh^{-1}\frac{x}{a}+\frac{a}{2}\ln(a^2-x^2) +$$ +<<*>>= +)clear all + +--S 11 of 32 +aa:=integrate(atanh(x/a),x) +--R +--R +--R 2 2 - x - a +--R a log(x - a ) + x log(-------) +--R x - a +--R (1) ------------------------------- +--R 2 +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.657~~~~~$\displaystyle +\int{x*\tanh^{-1}\frac{x}{a}}~dx$} +$$\int{x*\tanh^{-1}\frac{x}{a}}= +\frac{ax}{2}+\frac{1}{2}(x^2-a^2)\tanh^{-1}\frac{x}{a} +$$ +<<*>>= +)clear all + +--S 12 of 32 +aa:=integrate(x*atanh(x/a),x) +--R +--R +--R 2 2 - x - a +--R (x - a )log(-------) + 2a x +--R x - a +--R (1) ---------------------------- +--R 4 +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.658~~~~~$\displaystyle +\int{x^2\tanh^{-1}\frac{x}{a}}~dx$} +$$\int{x^2\tanh^{-1}\frac{x}{a}}= +\frac{ax^2}{6}+\frac{x^3}{3}\tanh^{-1}\frac{x}{a} ++\frac{a^3}{6}\ln(a^2-x^2) +$$ +<<*>>= +)clear all + +--S 13 of 32 +aa:=integrate(x^2*atanh(x/a),x) +--R +--R +--R 3 2 2 3 - x - a 2 +--R a log(x - a ) + x log(-------) + a x +--R x - a +--R (1) -------------------------------------- +--R 6 +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.659~~~~~$\displaystyle +\int{\frac{\tanh^{-1}(x/a)}{a}}~dx$} +$$\int{\frac{\tanh^{-1}(x/a)}{a}}= +\frac{x}{a}+\frac{(x/a)^3}{3^2}+\frac{(x/a)^5}{5^2}+\cdots +$$ +<<*>>= +)clear all + +--S 14 of 32 +aa:=integrate(atanh(x/a)/x,x) +--R +--R +--I %P +--R x atanh(--) +--R ++ a +--I (1) | --------- d%P +--I ++ %P +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.660~~~~~$\displaystyle +\int{\frac{tanh^{-1}(x/a)}{x^2}}~dx$} +$$\int{\frac{tanh^{-1}(x/a)}{x^2}}= +-\frac{\tanh^{-1}(x/a)}{x}+\frac{1}{2a}\ln\left(\frac{x^2}{a^2-x^2}\right) +$$ +<<*>>= +)clear all + +--S 15 of 32 +aa:=integrate(atanh(x/a)/x^2,x) +--R +--R +--R 2 2 - x - a +--R - x log(x - a ) + 2x log(x) - a log(-------) +--R x - a +--R (1) --------------------------------------------- +--R 2a x +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.661~~~~~$\displaystyle +\int{\coth^{-1}\frac{x}{a}}~dx$} +$$\int{\coth^{-1}\frac{x}{a}}= +x\coth^{-1}{x}+\frac{a}{2}\ln(x^2-a^2) +$$ +<<*>>= +)clear all + +--S 16 of 32 +aa:=integrate(acoth(x/a),x) +--R +--R +--R 2 2 x + a +--R a log(x - a ) + x log(-----) +--R x - a +--R (1) ----------------------------- +--R 2 +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.662~~~~~$\displaystyle +\int{x\coth^{-1}\frac{x}{a}}~dx$} +$$\int{x\coth^{-1}\frac{x}{a}}= +\frac{ax}{2}+\frac{1}{2}(x^2-a^2)\coth^{-1}\frac{x}{a} +$$ +<<*>>= +)clear all + +--S 17 of 32 +aa:=integrate(x*acoth(x/a),x) +--R +--R +--R 2 2 x + a +--R (x - a )log(-----) + 2a x +--R x - a +--R (1) -------------------------- +--R 4 +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.663~~~~~$\displaystyle +\int{x^2\coth^{-1}\frac{x}{a}}~dx$} +$$\int{x^2\coth^{-1}\frac{x}{a}}= +\frac{ax^2}{6}+\frac{x^3}{3}\coth^{-1}\frac{x}{a} ++\frac{a^3}{6}\ln(x^2-a^2) +$$ +<<*>>= +)clear all + +--S 18 of 32 +aa:=integrate(x^2*acoth(x/a),x) +--R +--R +--R 3 2 2 3 x + a 2 +--R a log(x - a ) + x log(-----) + a x +--R x - a +--R (1) ------------------------------------ +--R 6 +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.664~~~~~$\displaystyle +\int{\frac{\coth^{-1}(x/a)}{x}}~dx$} +$$\int{\frac{\coth^{-1}(x/a)}{x}}= +-\left(\frac{a}{x}+\frac{(a/x)^3}{3^2}+\frac{(a/x)^5}{5^2}+\cdots\right) +$$ +<<*>>= +)clear all + +--S 19 of 32 +aa:=integrate(acoth(x/a)/x,x) +--R +--R +--I %P +--R x acoth(--) +--R ++ a +--I (1) | --------- d%P +--I ++ %P +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.665~~~~~$\displaystyle +\int{\frac{\coth^{-1}(x/a)}{x^2}}~dx$} +$$\int{\frac{\coth^{-1}(x/a)}{x^2}}= +-\frac{\coth^{-1}(x/a)}{x}+\frac{1}{2a}\ln\left(\frac{x^2}{x^2-a^2}\right) +$$ +<<*>>= +)clear all + +--S 20 of 32 +aa:=integrate(acoth(x/a)/x^2,x) +--R +--R +--R 2 2 x + a +--R - x log(x - a ) + 2x log(x) - a log(-----) +--R x - a +--R (1) ------------------------------------------- +--R 2a x +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.666~~~~~$\displaystyle +\int{{\rm sech}^{-1}\frac{x}{a}}~dx$} +$$\int{{\rm sech}^{-1}\frac{x}{a}}= +\left\{ +\begin{array}{l} +\displaystyle +x{\rm ~sech}^{-1}(x/a)+a\sin^{-1}(x/a),\quad{\rm sech}^{-1}(x/a)>0\\ +\\ +\displaystyle +x{\rm ~sech}^{-1}(x/a)-a\sin^{-1}(x/a),\quad{\rm sech}^{-1}(x/a)<0\\ +\end{array} +\right. +$$ +<<*>>= +)clear all + +--S 21 of 32 +aa:=integrate(asech(x/a),x) +--R +--R +--R +---------+ +---------+ +--R | 2 2 | 2 2 +--R \|- x + a + a \|- x + a - a +--R (1) x log(----------------) - 2a atan(----------------) +--R x x +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.667~~~~~$\displaystyle +\int{x{\rm ~sech}^{-1}\frac{x}{a}}~dx$} +$$\int{x{\rm ~sech}^{-1}\frac{x}{a}}= +\left\{ +\begin{array}{l} +\displaystyle +\frac{1}{2}x^2{\rm ~sech}^{-1}(x/a)-\frac{1}{2}a\sqrt{a^2-x^2}, +\quad{\rm sech}^{-1}(x/a)>0\\ +\\ +\displaystyle +\frac{1}{2}x^2{\rm ~sech}^{-1}(x/a)+\frac{1}{2}a\sqrt{a^2-x^2}, +\quad{\rm sech}^{-1}(x/a)<0\\ +\end{array} +\right. +$$ +<<*>>= +)clear all + +--S 22 of 32 +aa:=integrate(x*asech(x/a),x) +--R +--R +--R +---------+ +--R +---------+ | 2 2 +--R 2 | 2 2 2 \|- x + a + a 2 +--R (x \|- x + a - a x )log(----------------) + a x +--R x +--R (1) --------------------------------------------------- +--R +---------+ +--R | 2 2 +--R 2\|- x + a - 2a +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.668~~~~~$\displaystyle +\int{\frac{{\rm sech}^{-1}(x/a)}{x}}~dx$} +$$\int{\frac{{\rm sech}^{-1}(x/a)}{x}}= +\left\{ +\begin{array}{l} +\displaystyle +-\frac{1}{2}\ln(a/x)\ln(4a/x)-\frac{(x/a)^2}{2\cdot 2\cdot 2} +-\frac{1\cdot 3(x/a)^4}{2\cdot 4\cdot 4\cdot 4} +-\cdots,\quad{\rm sech}^{-1}(x/a)>0\\ +\\ +\displaystyle +\frac{1}{2}\ln(a/x)\ln(4a/x)+\frac{(x/a)^2}{2\cdot 2\cdot 2} ++\frac{1\cdot 3(x/a)^4}{2\cdot 4\cdot 4\cdot 4} ++\cdots,\quad{\rm sech}^{-1}(x/a)<0\\ +\end{array} +\right. +$$ +<<*>>= +)clear all + +--S 23 of 32 +aa:=integrate(asech(x/a)/x,x) +--R +--R +--R +---------+ 2 +--R +---------+ | 2 2 +--R | 2 2 \|- x + a + a +--R \|- x + a log(----------------) +--R x +--R (1) - ---------------------------------- +--R 2a +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.669~~~~~$\displaystyle +\int{{\rm csch}^{-1}\frac{x}{a}}~dx$} +$$\int{{\rm csch}^{-1}\frac{x}{a}}= +x{\rm ~csch}^{-1}\frac{x}{a}\pm a\sinh^{-1}\frac{x}{a} +\quad +{\rm if\ }x>0, -{\rm if\ }x<0 +$$ +<<*>>= +)clear all + +--S 24 of 32 +aa:=integrate(acsch(x/a),x) +--R +--R +--R +-------+ +--R +-------+ | 2 2 +--R | 2 2 \|x + a + a +--R (1) - a log(\|x + a - x) + x log(--------------) +--R x +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.670~~~~~$\displaystyle +\int{x{\rm ~csch}^{-1}\frac{x}{a}}~dx$} +$$\int{x{\rm ~csch}^{-1}\frac{x}{a}}= +\frac{x^2}{2}{\rm ~csch}^{-1}\frac{x}{a}\pm \frac{a\sqrt{x^2+a^2}}{2} +\quad +{\rm if\ }x>0, -{\rm if\ }x<0 +$$ +<<*>>= +)clear all + +--S 25 of 32 +aa:=integrate(x*acsch(x/a),x) +--R +--R +--R +-------+ +--R +-------+ | 2 2 +-------+ +--R 2 | 2 2 3 \|x + a + a | 2 2 2 3 +--R (x \|x + a - x )log(--------------) - a x\|x + a + a x + a +--R x +--R (1) ------------------------------------------------------------------ +--R +-------+ +--R | 2 2 +--R 2\|x + a - 2x +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.671~~~~~$\displaystyle +\int{\frac{{\rm csch}^{-1}(x/a)}{x}}~dx$} +$$\int{\frac{{\rm csch}^{-1}(x/a)}{x}}= +\left\{ +\begin{array}{lr} +\displaystyle +\frac{1}{2}\ln(x/a)\ln(4a/x)+\frac{1(x/a)^2}{2\cdot 2\cdot 2} +-\frac{1\cdot 3(x/a)^4}{2\cdot 4\cdot 4\cdot 4}+\cdots& +0a +\end{array} +\right. +$$ +<<*>>= +)clear all + +--S 26 of 32 +aa:=integrate(acsch(x/a)/x,x) +--R +--R +--R +-------+ 2 +--R +-------+ | 2 2 +--R | 2 2 \|x + a + a +--R \|x + a log(--------------) +--R x +--R (1) - ------------------------------ +--R 2a +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.672~~~~~$\displaystyle +\int{x^m\sinh^{-1}\frac{x}{a}}~dx$} +$$\int{x^m\sinh^{-1}\frac{x}{a}}= +\frac{x^{m+1}}{m+1}\sinh^{-1}\frac{x}{a} +-\frac{1}{m+1}\int{\frac{x^{m+1}}{\sqrt{x^2+a^2}}} +$$ +<<*>>= +)clear all + +--S 27 of 32 +aa:=integrate(x^m*asinh(x/a),x) +--R +--R +--R x +--I ++ %P m +--I (1) | asinh(--)%P d%P +--R ++ a +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.673~~~~~$\displaystyle +\int{x^m\cosh^{-1}\frac{x}{a}}~dx$} +$$\int{x^m\cosh^{-1}\frac{x}{a}}= +\left\{ +\begin{array}{lr} +\displaystyle +\frac{x^{m+1}}{m+1}\cosh^{-1}\frac{x}{a} +-\frac{1}{m+1}\int{\frac{x^{m+1}}{\sqrt{x^2-a^2}}},& +\quad\cosh^{-1}(x/a)>0\\ +\\ +\displaystyle +\frac{x^{m+1}}{m+1}\cosh^{-1}\frac{x}{a} ++\frac{1}{m+1}\int{\frac{x^{m+1}}{\sqrt{x^2-a^2}}},& +\quad\cosh^{-1}(x/a)<0\\ +\end{array} +\right. +$$ +<<*>>= +)clear all + +--S 28 of 32 +aa:=integrate(x^m*acosh(x/a),x) +--R +--R +--R x +--I ++ %P m +--I (1) | acosh(--)%P d%P +--R ++ a +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.674~~~~~$\displaystyle +\int{x^m\tanh^{-1}\frac{x}{a}}~dx$} +$$\int{x^m\tanh^{-1}\frac{x}{a}}= +\frac{x^{m+1}}{m+1}\tanh^{-1}\frac{x}{a} +-\frac{a}{m+1}\int{\frac{x^{m+1}}{a^2-x^2}} +$$ +<<*>>= +)clear all + +--S 29 of 32 +aa:=integrate(x^m*atanh(x/a),x) +--R +--R +--R x +--I ++ %P m +--I (1) | atanh(--)%P d%P +--R ++ a +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.675~~~~~$\displaystyle +\int{x^m\coth^{-1}\frac{x}{a}}~dx$} +$$\int{x^m\coth^{-1}\frac{x}{a}}= +\frac{x^{m+1}}{m+1}\coth^{-1}\frac{x}{a} +-\frac{a}{m+1}\int{\frac{x^{m+1}}{a^2-x^2}} +$$ +<<*>>= +)clear all + +--S 30 of 32 +aa:=integrate(x^m*acoth(x/a),x) +--R +--R +--R x +--I ++ %P m +--I (1) | acoth(--)%P d%P +--R ++ a +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.676~~~~~$\displaystyle +\int{x^m{\rm ~sech}^{-1}\frac{x}{a}}~dx$} +$$\int{x^m{\rm ~sech}^{-1}\frac{x}{a}}= +\left\{ +\begin{array}{lr} +\displaystyle +\frac{x^{m+1}}{m+1}{\rm ~sech}^{-1}\frac{x}{a} ++\frac{a}{m+1}\int{\frac{x^m}{\sqrt{a^2-x^2}}}& +{\rm sech}^{-1}(x/a)>0\\ +\\ +\displaystyle +\frac{x^{m+1}}{m+1}{\rm ~sech}^{-1}\frac{x}{a} +-\frac{a}{m+1}\int{\frac{x^m}{\sqrt{a^2-x^2}}}& +{\rm sech}^{-1}(x/a)<0\\ +\end{array} +\right. +$$ +<<*>>= +)clear all + +--S 31 of 32 +aa:=integrate(x^m*asech(x/a),x) +--R +--R +--R x +--I ++ %P m +--I (1) | asech(--)%P d%P +--R ++ a +--R Type: Union(Expression Integer,...) +--E +@ + +\section{\cite{1}:14.677~~~~~$\displaystyle +\int{x^m{\rm ~csch}^{-1}\frac{x}{a}}~dx$} +$$\int{x^m{\rm ~csch}^{-1}\frac{x}{a}}= +\frac{x^{m+1}}{m+1}{\rm ~csch}^{-1}\frac{x}{a} +\pm\frac{a}{m+1}\int{\frac{x^m}{\sqrt{x^2+a^2}}} +\quad+{\rm if\ }x>0 +~-{\rm if\ }x<0 +$$ +<<*>>= +)clear all + +--S 32 of 32 +aa:=integrate(x^m*acsch(x/a),x) +--R +--R +--R x +--I ++ %P m +--I (1) | acsch(--)%P d%P +--R ++ 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 pp92-93 +\end{thebibliography} +\end{document}