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[Axiom-developer] 20080403.01.tpd.patch (CATS integration regression tes
|
From: |
daly |
|
Subject: |
[Axiom-developer] 20080403.01.tpd.patch (CATS integration regression testing) |
|
Date: |
Thu, 3 Apr 2008 23:26:49 -0600 |
More integrals
========================================================================
diff --git a/changelog b/changelog
index 102ba8b..13f5f2e 100644
--- a/changelog
+++ b/changelog
@@ -1,3 +1,10 @@
+20080403 tpd src/input/Makefile add integration regression testing
+20080403 tpd src/input/schaum18.input integrals of cos(ax)
+20080403 tpd src/input/schaum17.input ignore gensyms
+20080403 tpd src/input/schaum16.input ignore gensyms
+20080403 tpd src/input/schaum14.input ignore gensyms
+20080403 tpd src/input/schaum13.input ignore gensyms
+20080403 tpd src/input/schaum12.input ignore gensyms
20080402 tpd src/input/Makefile add integration regression testing
20080402 tpd src/input/schaum17.input integrals of sin(ax)
20080402 tpd src/input/schaum16.input integrals of x^n \pm a^n
diff --git a/src/input/Makefile.pamphlet b/src/input/Makefile.pamphlet
index 6de47f3..dbeb3d7 100644
--- a/src/input/Makefile.pamphlet
+++ b/src/input/Makefile.pamphlet
@@ -359,7 +359,7 @@ REGRES= algaggr.regress algbrbf.regress algfacob.regress
alist.regress \
schaum5.regress schaum6.regress schaum7.regress schaum8.regress \
schaum9.regress schaum10.regress schaum11.regress schaum12.regress \
schaum13.regress schaum14.regress schaum15.regress schaum16.regress \
- schaum17.regress \
+ schaum17.regress schaum18.regress \
scherk.regress scope.regress seccsc.regress \
segbind.regress seg.regress \
series2.regress series.regress sersolve.regress set.regress \
@@ -637,7 +637,7 @@ FILES= ${OUT}/algaggr.input ${OUT}/algbrbf.input
${OUT}/algfacob.input \
${OUT}/schaum8.input ${OUT}/schaum9.input ${OUT}/schaum10.input \
${OUT}/schaum11.input ${OUT}/schaum12.input ${OUT}/schaum13.input \
${OUT}/schaum14.input ${OUT}/schaum15.input ${OUT}/schaum16.input \
- ${OUT}/schaum17.input \
+ ${OUT}/schaum17.input ${OUT}/schaum18.input \
${OUT}/saddle.input \
${OUT}/scherk.input ${OUT}/scope.input ${OUT}/seccsc.input \
${OUT}/segbind.input ${OUT}/seg.input ${OUT}/series2.input \
@@ -944,7 +944,7 @@ DOCFILES= \
${DOC}/schaum11.input.dvi ${DOC}/schaum12.input.dvi \
${DOC}/schaum13.input.dvi ${DOC}/schaum14.input.dvi \
${DOC}/schaum15.input.dvi ${DOC}/schaum16.input.dvi \
- ${DOC}/schaum17.input.dvi \
+ ${DOC}/schaum17.input.dvi ${DOC}/schaum18.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/schaum12.input.pamphlet
b/src/input/schaum12.input.pamphlet
index 6669ade..644ebed 100644
--- a/src/input/schaum12.input.pamphlet
+++ b/src/input/schaum12.input.pamphlet
@@ -189,10 +189,10 @@ aa:=integrate(x^m/(a*x^2+b*x+c),x)
--R
--R
--R x m
---R ++ %N
---R (1) | --------------- d%N
+--I ++ %N
+--I (1) | --------------- d%N
--R ++ 2
---R c + %N b + %N a
+--I c + %N b + %N a
--R Type: Union(Expression
Integer,...)
--E
@
@@ -326,9 +326,9 @@ aa:=integrate(1/(x^n*(a*x^2+b*x+c)),x)
--R
--R x
--R ++ 1
---R (1) | -------------------- d%N
+--I (1) | -------------------- d%N
--R ++ 2 n
---R (c + %N b + %N a)%N
+--I (c + %N b + %N a)%N
--R Type: Union(Expression
Integer,...)
--E
@
@@ -519,10 +519,10 @@ aa:=integrate(x^m/(a*x^2+b*x+c)^n,x)
--R
--R
--R x m
---R ++ %N
---R (1) | ------------------ d%N
+--I ++ %N
+--I (1) | ------------------ d%N
--R ++ 2 n
---R (c + %N b + %N a)
+--I (c + %N b + %N a)
--R Type: Union(Expression
Integer,...)
--E
@
@@ -546,10 +546,10 @@ aa:=integrate(x^(2*n-1)/(a*x^2+b*x+c)^n,x)
--R
--R
--R x 2n - 1
---R ++ %N
---R (1) | ------------------ d%N
+--I ++ %N
+--I (1) | ------------------ d%N
--R ++ 2 n
---R (c + %N b + %N a)
+--I (c + %N b + %N a)
--R Type: Union(Expression
Integer,...)
--E
@
@@ -762,9 +762,9 @@ aa:=integrate(1/(x^m*(a*x^2+b*x+c)^n),x)
--R
--R x
--R ++ 1
---R (1) | --------------------- d%N
+--I (1) | --------------------- d%N
--R ++ m 2 n
---R %N (c + %N b + %N a)
+--I %N (c + %N b + %N a)
--R Type: Union(Expression
Integer,...)
--E
diff --git a/src/input/schaum13.input.pamphlet
b/src/input/schaum13.input.pamphlet
index 04551e6..eb1a6eb 100644
--- a/src/input/schaum13.input.pamphlet
+++ b/src/input/schaum13.input.pamphlet
@@ -1167,7 +1167,7 @@ aa:=integrate((a*x^2+b*x+c)^(n+1/2),x)
--R 2n + 1
--R x ------
--R ++ 2 2
---R (1) | (c + %N b + %N a) d%N
+--I (1) | (c + %N b + %N a) d%N
--R ++
--R Type: Union(Expression
Integer,...)
--E
@@ -1189,7 +1189,7 @@ aa:=integrate(x*(a*x^2+b*x+c)^(n+1/2),x)
--R 2n + 1
--R x ------
--R ++ 2 2
---R (1) | %N (c + %N b + %N a) d%N
+--I (1) | %N (c + %N b + %N a) d%N
--R ++
--R Type: Union(Expression
Integer,...)
--E
@@ -1214,11 +1214,11 @@ aa:=integrate(1/(a*x^2+b*x+c)^(n+1/2),x)
--R
--R x
--R ++ 1
---R (1) | ----------------------- d%N
+--I (1) | ----------------------- d%N
--R ++ 2n + 1
--R ------
--R 2 2
---R (c + %N b + %N a)
+--I (c + %N b + %N a)
--R Type: Union(Expression
Integer,...)
--E
@
@@ -1245,11 +1245,11 @@ aa:=integrate(1/(x*(a*x^2+b*x+c)^(n+1/2)),x)
--R
--R x
--R ++ 1
---R (1) | -------------------------- d%N
+--I (1) | -------------------------- d%N
--R ++ 2n + 1
--R ------
--R 2 2
---R %N (c + %N b + %N a)
+--I %N (c + %N b + %N a)
--R Type: Union(Expression
Integer,...)
--E
diff --git a/src/input/schaum14.input.pamphlet
b/src/input/schaum14.input.pamphlet
index 560ab37..532a38a 100644
--- a/src/input/schaum14.input.pamphlet
+++ b/src/input/schaum14.input.pamphlet
@@ -264,10 +264,10 @@ aa:=integrate(x^m/(x^3+a^3),x)
--R
--R
--R x m
---R ++ %L
---R (1) | -------- d%L
+--I ++ %L
+--I (1) | -------- d%L
--R ++ 3 3
---R a + %L
+--I a + %L
--R Type: Union(Expression
Integer,...)
--E
@
@@ -286,9 +286,9 @@ aa:=integrate(1/(x^n*(x^3+a^3)),x)
--R
--R x
--R ++ 1
---R (1) | ------------- d%L
+--I (1) | ------------- d%L
--R ++ 3 3 n
---R (a + %L )%L
+--I (a + %L )%L
--R Type: Union(Expression
Integer,...)
--E
diff --git a/src/input/schaum16.input.pamphlet
b/src/input/schaum16.input.pamphlet
index 2246f9a..f6b849c 100644
--- a/src/input/schaum16.input.pamphlet
+++ b/src/input/schaum16.input.pamphlet
@@ -64,10 +64,10 @@ aa:=integrate(x^m/(x^n+a^n)^r,x)
--R
--R
--R x m
---R ++ %J
---R (1) | ----------- d%J
+--I ++ %J
+--I (1) | ----------- d%J
--R ++ n n r
---R (a + %J )
+--I (a + %J )
--R Type: Union(Expression
Integer,...)
--E
@
@@ -87,9 +87,9 @@ aa:=integrate(1/(x^m*(x^n+a^n)^r),x)
--R
--R x
--R ++ 1
---R (1) | -------------- d%J
+--I (1) | -------------- d%J
--R ++ m n n r
---R %J (a + %J )
+--I %J (a + %J )
--R Type: Union(Expression
Integer,...)
--E
@
@@ -187,10 +187,10 @@ aa:=integrate(x^m/(x^n-a^n)^r,x)
--R
--R
--R x m
---R ++ %J
---R (1) | ------------- d%J
+--I ++ %J
+--I (1) | ------------- d%J
--R ++ n n r
---R (- a + %J )
+--I (- a + %J )
--R Type: Union(Expression
Integer,...)
--E
@
@@ -210,9 +210,9 @@ aa:=integrate(1/(x^m*(x^n-a^n)^r),x)
--R
--R x
--R ++ 1
---R (1) | ---------------- d%J
+--I (1) | ---------------- d%J
--R ++ m n n r
---R %J (- a + %J )
+--I %J (- a + %J )
--R Type: Union(Expression
Integer,...)
--E
@
@@ -274,10 +274,10 @@ aa:=integrate(x^(p-1)/(x^(2*m)+a^(2*m)),x)
--R
--R
--R x p - 1
---R ++ %J
---R (1) | ---------- d%J
+--I ++ %J
+--I (1) | ---------- d%J
--R ++ 2m 2m
---R a + %J
+--I a + %J
--R Type: Union(Expression
Integer,...)
--E
@
@@ -304,10 +304,10 @@ aa:=integrate(x^(p-1)/(x^(2*m)-a^(2*m)),x)
--R
--R
--R x p - 1
---R ++ %J
---R (1) | - ---------- d%J
+--I ++ %J
+--I (1) | - ---------- d%J
--R ++ 2m 2m
---R a - %J
+--I a - %J
--R Type: Union(Expression
Integer,...)
--E
@
@@ -338,10 +338,10 @@ aa:=integrate(x^(p-1)/(x^(2*m+1)+a^(2*m+1)),x)
--R
--R
--R x p - 1
---R ++ %J
---R (1) | ------------------ d%J
+--I ++ %J
+--I (1) | ------------------ d%J
--R ++ 2m + 1 2m + 1
---R a + %J
+--I a + %J
--R Type: Union(Expression
Integer,...)
--E
@
@@ -374,10 +374,10 @@ aa:=integrate(x^(p-1)/(x^(2*m+1)-a^(2*m+1)),x)
--R
--R
--R x p - 1
---R ++ %J
---R (1) | - ------------------ d%J
+--I ++ %J
+--I (1) | - ------------------ d%J
--R ++ 2m + 1 2m + 1
---R a - %J
+--I a - %J
--R Type: Union(Expression
Integer,...)
--E
diff --git a/src/input/schaum17.input.pamphlet
b/src/input/schaum17.input.pamphlet
index 7ab11d7..fd40132 100644
--- a/src/input/schaum17.input.pamphlet
+++ b/src/input/schaum17.input.pamphlet
@@ -32,7 +32,7 @@ aa:=integrate(sin(a*x),x)
\section{\cite{1}:14.340~~~~~$\displaystyle
\int{x\sin{ax}~dx}$}
$$\int{x\sin{ax}}=
-\frac{sin{ax}}{a^2}-\frac{x\cos{ax}}{a}
+\frac{\sin{ax}}{a^2}-\frac{x\cos{ax}}{a}
$$
<<*>>=
)clear all
@@ -122,10 +122,10 @@ aa:=integrate(sin(a*x)/x^2,x)
--R
--R
--R x
---R ++ sin(%I a)
---R (1) | --------- d%I
+--I ++ sin(%I a)
+--I (1) | --------- d%I
--R ++ 2
---R %I
+--I %I
--R Type: Union(Expression
Integer,...)
--E
@
@@ -167,9 +167,9 @@ aa:=integrate(x/sin(a*x),x)
--R
--R
--R x
---R ++ %I
---R (1) | --------- d%I
---R ++ sin(%I a)
+--I ++ %I
+--I (1) | --------- d%I
+--I ++ sin(%I a)
--R Type: Union(Expression
Integer,...)
--E
@
@@ -675,7 +675,7 @@ aa:=integrate(x^m*sin(a*x),x)
--R
--R x
--R ++ m
---R (1) | sin(%I a)%I d%I
+--I (1) | sin(%I a)%I d%I
--R ++
--R Type: Union(Expression
Integer,...)
--E
@@ -694,10 +694,10 @@ aa:=integrate(sin(a*x)/x^n,x)
--R
--R
--R x
---R ++ sin(%I a)
---R (1) | --------- d%I
+--I ++ sin(%I a)
+--I (1) | --------- d%I
--R ++ n
---R %I
+--I %I
--R Type: Union(Expression
Integer,...)
--E
@
@@ -716,7 +716,7 @@ aa:=integrate(sin(a*x)^n,x)
--R
--R x
--R ++ n
---R (1) | sin(%I a) d%I
+--I (1) | sin(%I a) d%I
--R ++
--R Type: Union(Expression
Integer,...)
--E
@@ -737,9 +737,9 @@ aa:=integrate(1/(sin(a*x))^n,x)
--R
--R x
--R ++ 1
---R (1) | ---------- d%I
+--I (1) | ---------- d%I
--R ++ n
---R sin(%I a)
+--i sin(%I a)
--R Type: Union(Expression
Integer,...)
--E
@
@@ -759,10 +759,10 @@ aa:=integrate(x/sin(a*x)^n,x)
--R
--R
--R x
---R ++ %I
---R (1) | ---------- d%I
+--I ++ %I
+--I (1) | ---------- d%I
--R ++ n
---R sin(%I a)
+--I sin(%I a)
--R Type: Union(Expression
Integer,...)
--E
diff --git a/src/input/schaum18.input.pamphlet
b/src/input/schaum18.input.pamphlet
new file mode 100644
index 0000000..3cb4766
--- /dev/null
+++ b/src/input/schaum18.input.pamphlet
@@ -0,0 +1,754 @@
+\documentclass{article}
+\usepackage{axiom}
+\begin{document}
+\title{\$SPAD/input schaum18.input}
+\author{Timothy Daly}
+\maketitle
+\eject
+\tableofcontents
+\eject
+\section{\cite{1}:14.369~~~~~$\displaystyle
+\int{\cos ax ~dx}$}
+$$\int{\cos ax}=
+\frac{\sin{ax}}{a}
+$$
+<<*>>=
+)spool schaum18.output
+)set message test on
+)set message auto off
+)clear all
+
+--S 1 of 30
+aa:=integrate(cos(a*x),x)
+--R
+--R
+--R sin(a x)
+--R (1) --------
+--R a
+--R Type: Union(Expression
Integer,...)
+--E
+@
+
+\section{\cite{1}:14.370~~~~~$\displaystyle
+\int{x\cos{ax}~dx}$}
+$$\int{x\cos{ax}}=
+\frac{\cos{ax}}{a^2}+\frac{x\sin{ax}}{a}
+$$
+<<*>>=
+)clear all
+
+--S 2 of 30
+aa:=integrate(x*cos(a*x),x)
+--R
+--R
+--R a x sin(a x) + cos(a x)
+--R (1) -----------------------
+--R 2
+--R a
+--R Type: Union(Expression
Integer,...)
+--E
+@
+
+\section{\cite{1}:14.371~~~~~$\displaystyle
+\int{x^2\cos{ax}~dx}$}
+$$\int{x^2\cos{ax}}=
+\frac{2x}{a^2}\cos{ax}+\left(\frac{x^2}{a}-\frac{2}{a^3}\right)\sin{ax}
+$$
+<<*>>=
+)clear all
+
+--S 3 of 30
+aa:=integrate(x^2*cos(a*x),x)
+--R
+--R
+--R 2 2
+--R (a x - 2)sin(a x) + 2a x cos(a x)
+--R (1) ----------------------------------
+--R 3
+--R a
+--R Type: Union(Expression
Integer,...)
+--E
+@
+
+\section{\cite{1}:14.372~~~~~$\displaystyle
+\int{x^3\cos{ax}~dx}$}
+$$\int{x^3\cos{ax}}=
+\left(\frac{3x^2}{a^2}-\frac{6}{a^4}\right)\cos{ax}
++\left(\frac{x^3}{a}-\frac{6x}{a^3}\right)\sin{ax}
+$$
+<<*>>=
+)clear all
+
+--S 4 of 30
+aa:=integrate(x^3*cos(a*x),x)
+--R
+--R
+--R 3 3 2 2
+--R (a x - 6a x)sin(a x) + (3a x - 6)cos(a x)
+--R (1) -------------------------------------------
+--R 4
+--R a
+--R Type: Union(Expression
Integer,...)
+--E
+@
+
+\section{\cite{1}:14.373~~~~~$\displaystyle
+\int{\frac{\cos{ax}}{x}}~dx$}
+$$\int{\frac{\cos{ax}}{x}}=
+\ln{x}-\frac{(ax)^2}{2\cdot 2!}+\frac{(ax)^4}{4\cdot 4!}
+-\frac{(ax)^6}{6\cdot 6!}+\cdots
+$$
+<<*>>=
+)clear all
+
+--S 5 of 30
+aa:=integrate(cos(x)/x,x)
+--R
+--R
+--R (1) Ci(x)
+--R Type: Union(Expression
Integer,...)
+--E
+@
+
+\section{\cite{1}:14.374~~~~~$\displaystyle
+\int{\frac{\cos{ax}}{x^2}}~dx$}
+$$\int{\frac{\cos{ax}}{x^2}}=
+-\frac{\cos{ax}}{x}-a\int{\frac{\sin{ax}}{x}}
+$$
+<<*>>=
+)clear all
+
+--S 6 of 30
+aa:=integrate(cos(a*x)/x^2,x)
+--R
+--R
+--R x
+--I ++ cos(%I a)
+--I (1) | --------- d%I
+--R ++ 2
+--I %I
+--R Type: Union(Expression
Integer,...)
+--E
+@
+
+\section{\cite{1}:14.375~~~~~$\displaystyle
+\int{\frac{dx}{\cos{ax}}}$}
+$$\int{\frac{1}{\cos{ax}}}=
+\frac{1}{a}\ln(\sec{ax}-\tan{ax})=
+\frac{1}{a}\ln\tan\left(\frac{\pi}{4}+\frac{ax}{2}\right)
+$$
+<<*>>=
+)clear all
+
+--S 7 of 30
+aa:=integrate(1/cos(a*x),x)
+--R
+--R
+--R sin(a x) + cos(a x) + 1 sin(a x) - cos(a x) - 1
+--R log(-----------------------) - log(-----------------------)
+--R cos(a x) + 1 cos(a x) + 1
+--R (1) -----------------------------------------------------------
+--R a
+--R Type: Union(Expression
Integer,...)
+--E
+@
+
+\section{\cite{1}:14.376~~~~~$\displaystyle
+\int{\frac{x~dx}{\cos{ax}}}$}
+$$\int{\frac{x}{\cos{ax}}}=
+\frac{1}{a^2}\left\{
+\frac{(ax)^2}{2}+\frac{(ax)^4}{8}+\frac{5(ax)^6}{144}+\cdots+
+\frac{E_n(ax)^{2n+2}}{(2n+2)(2n)!}+\cdots
+\right\}
+$$
+<<*>>=
+)clear all
+
+--S 8 of 30
+aa:=integrate(x/cos(a*x),x)
+--R
+--R
+--R x
+--I ++ %I
+--I (1) | --------- d%I
+--I ++ cos(%I a)
+--R Type: Union(Expression
Integer,...)
+--E
+@
+
+\section{\cite{1}:14.377~~~~~$\displaystyle
+\int{\cos^2{ax}}~dx$}
+$$\int{\cos^2{ax}}=
+\frac{x}{2}+\frac{\sin{2ax}}{4a}
+$$
+<<*>>=
+)clear all
+
+--S 9 of 30
+aa:=integrate(cos(a*x)^2,x)
+--R
+--R
+--R cos(a x)sin(a x) + a x
+--R (1) ----------------------
+--R 2a
+--R Type: Union(Expression
Integer,...)
+--E
+@
+
+\section{\cite{1}:14.378~~~~~$\displaystyle
+\int{x\cos^2{ax}}~dx$}
+$$\int{x\cos^2{ax}}=
+\frac{x^2}{4}+\frac{x\sin{2ax}}{4a}+\frac{\cos{2ax}}{8a^2}
+$$
+<<*>>=
+)clear all
+
+--S 10 of 30
+aa:=integrate(x*cos(a*x)^2,x)
+--R
+--R
+--R 2 2 2
+--R 2a x cos(a x)sin(a x) + cos(a x) + a x
+--R (1) ----------------------------------------
+--R 2
+--R 4a
+--R Type: Union(Expression
Integer,...)
+--E
+@
+
+\section{\cite{1}:14.379~~~~~$\displaystyle
+\int{\cos^3{ax}}~dx$}
+$$\int{\cos^3{ax}}=
+\frac{\sin{ax}}{a}-\frac{\sin^3{ax}}{3a}
+$$
+<<*>>=
+)clear all
+
+--S 11 of 30
+aa:=integrate(cos(a*x)^3,x)
+--R
+--R
+--R 2
+--R (cos(a x) + 2)sin(a x)
+--R (1) -----------------------
+--R 3a
+--R Type: Union(Expression
Integer,...)
+--E
+@
+
+\section{\cite{1}:14.380~~~~~$\displaystyle
+\int{\cos^4{ax}}~dx$}
+$$\int{\cos^4{ax}}=
+\frac{3x}{8}+\frac{\sin{2ax}}{4a}+\frac{\sin{4ax}}{32a}
+$$
+<<*>>=
+)clear all
+
+--S 12 of 30
+aa:=integrate(cos(a*x)^4,x)
+--R
+--R
+--R 3
+--R (2cos(a x) + 3cos(a x))sin(a x) + 3a x
+--R (1) ---------------------------------------
+--R 8a
+--R Type: Union(Expression
Integer,...)
+--E
+@
+
+\section{\cite{1}:14.381~~~~~$\displaystyle
+\int{\frac{dx}{\cos^2{ax}}}$}
+$$\int{\frac{1}{\cos^2{ax}}}=
+\frac{\tan{ax}}{a}
+$$
+<<*>>=
+)clear all
+
+--S 13 of 30
+aa:=integrate(1/cos(a*x)^2,x)
+--R
+--R
+--R sin(a x)
+--R (1) ----------
+--R a cos(a x)
+--R Type: Union(Expression
Integer,...)
+--E
+@
+
+\section{\cite{1}:14.382~~~~~$\displaystyle
+\int{\frac{dx}{\cos^3{ax}}}$}
+$$\int{\frac{1}{\cos^3{ax}}}=
+\frac{\sin{ax}}{2a\cos^2{ax}}
++\frac{1}{2a}\ln\tan\left(\frac{\pi}{4}+\frac{ax}{2}\right)
+$$
+<<*>>=
+)clear all
+
+--S 14 of 30
+aa:=integrate(1/cos(a*x)^3,x)
+--R
+--R
+--R (1)
+--R 2 sin(a x) + cos(a x) + 1
+--R cos(a x) log(-----------------------)
+--R cos(a x) + 1
+--R +
+--R 2 sin(a x) - cos(a x) - 1
+--R - cos(a x) log(-----------------------) + sin(a x)
+--R cos(a x) + 1
+--R /
+--R 2
+--R 2a cos(a x)
+--R Type: Union(Expression
Integer,...)
+--E
+@
+
+\section{\cite{1}:14.383~~~~~$\displaystyle
+\int{\cos{px}\cos{qx}}~dx$}
+$$\int{\cos{ax}\cos{px}}=
+\frac{\sin{(a-p)x}}{2(a-p)}+\frac{\sin{(a+p)x}}{2(a+p)}
+$$
+<<*>>=
+)clear all
+
+--S 15 of 30
+aa:=integrate(cos(p*x)*cos(q*x),x)
+--R
+--R
+--R q cos(p x)sin(q x) - p cos(q x)sin(p x)
+--R (1) ---------------------------------------
+--R 2 2
+--R q - p
+--R Type: Union(Expression
Integer,...)
+--E
+@
+
+\section{\cite{1}:14.384~~~~~$\displaystyle
+\int{\frac{dx}{1-\cos{ax}}}$}
+$$\int{\frac{1}{1-\cos{ax}}}=
+-\frac{1}{a}\cot\frac{ax}{2}
+$$
+<<*>>=
+)clear all
+
+--S 16 of 30
+aa:=integrate(1/(1-cos(a*x)),x)
+--R
+--R
+--R - cos(a x) - 1
+--R (1) --------------
+--R a sin(a x)
+--R Type: Union(Expression
Integer,...)
+--E
+@
+
+\section{\cite{1}:14.385~~~~~$\displaystyle
+\int{\frac{x~dx}{1-\cos{ax}}}$}
+$$\int{\frac{x}{1-\cos{ax}}}=
+-\frac{x}{a}\cot\frac{ax}{2}
++\frac{2}{a^2}\ln~\sin\frac{ax}{2}
+$$
+<<*>>=
+)clear all
+
+--S 17 of 30
+aa:=integrate(x/(1-cos(ax)),x)
+--R
+--R
+--R 2
+--R x
+--R (1) - ------------
+--R 2cos(ax) - 2
+--R Type: Union(Expression
Integer,...)
+--E
+@
+
+\section{\cite{1}:14.386~~~~~$\displaystyle
+\int{\frac{dx}{1+\cos{ax}}}$}
+$$\int{\frac{1}{1+\cos{ax}}}=
+\frac{1}{a}\tan\frac{ax}{2}
+$$
+<<*>>=
+)clear all
+
+--S 18 of 30
+aa:=integrate(1/(1+cos(ax)),x)
+--R
+--R
+--R x
+--R (1) -----------
+--R cos(ax) + 1
+--R Type: Union(Expression
Integer,...)
+--E
+@
+
+\section{\cite{1}:14.387~~~~~$\displaystyle
+\int{\frac{x~dx}{1+\cos{ax}}}$}
+$$\int{\frac{x}{1+\cos{ax}}}=
+\frac{x}{a}\tan\frac{ax}{2}
++\frac{2}{a^2}\ln~\cos\frac{ax}{2}
+$$
+<<*>>=
+)clear all
+
+--S 19 of 30
+aa:=integrate(x/(1+cos(a*x)),x)
+--R
+--R
+--R 2
+--R (- cos(a x) - 1)log(------------) + a x sin(a x)
+--R cos(a x) + 1
+--R (1) ------------------------------------------------
+--R 2 2
+--R a cos(a x) + a
+--R Type: Union(Expression
Integer,...)
+--E
+@
+
+\section{\cite{1}:14.388~~~~~$\displaystyle
+\int{\frac{dx}{(1-\cos{ax})^2}}$}
+$$\int{\frac{1}{(1-\cos{ax})^2}}=
+-\frac{1}{2a}\cot\frac{ax}{2}
+-\frac{1}{6a}\cot^3\frac{ax}{2}
+$$
+<<*>>=
+)clear all
+
+--S 20 of 30
+aa:=integrate(1/(1-cos(a*x))^2,x)
+--R
+--R
+--R 2
+--R - cos(a x) + cos(a x) + 2
+--R (1) --------------------------
+--R (3a cos(a x) - 3a)sin(a x)
+--R Type: Union(Expression
Integer,...)
+--E
+@
+
+\section{\cite{1}:14.389~~~~~$\displaystyle
+\int{\frac{dx}{(1+\cos{ax})^2}}$}
+$$\int{\frac{1}{(1+\cos{ax})^2}}=
+\frac{1}{2a}\tan\frac{ax}{2}
++\frac{1}{6a}\tan^3\frac{ax}{2}
+$$
+<<*>>=
+)clear all
+
+--S 21 of 30
+aa:=integrate(1/(1+cos(a*x))^2,x)
+--R
+--R
+--R (cos(a x) + 2)sin(a x)
+--R (1) -------------------------------
+--R 2
+--R 3a cos(a x) + 6a cos(a x) + 3a
+--R Type: Union(Expression
Integer,...)
+--E
+@
+
+\section{\cite{1}:14.390~~~~~$\displaystyle
+\int{\frac{dx}{p+q\cos{ax}}}$}
+$$\int{\frac{1}{p+q\cos{ax}}}=
+\left\{
+\begin{array}{l}
+\displaystyle
+\frac{2}{a\sqrt{p^2-q^q}}
+\tan^{-1}\sqrt{(p-q)/(p+q)}\tan\frac{1}{2}ax
+\\
+\displaystyle
+\frac{1}{a\sqrt{q^2-p^2}}\ln\left(
+\frac{\tan\frac{1}{2}ax+\sqrt{(q+p)/(q-p)}}
+{\tan\frac{1}{2}ax-\sqrt{(q+p)(q-p)}}\right)
+\end{array}
+\right.
+$$
+<<*>>=
+)clear all
+
+--S 22 of 30
+aa:=integrate(1/(p+q*cos(a*x)),x)
+--R
+--R
+--R (1)
+--R +-------+
+--R | 2 2 2 2
+--R (- p cos(a x) - q)\|q - p + (- q + p )sin(a x)
+--R log(--------------------------------------------------)
+--R q cos(a x) + p
+--R [-------------------------------------------------------,
+--R +-------+
+--R | 2 2
+--R a\|q - p
+--R +---------+
+--R | 2 2
+--R sin(a x)\|- q + p
+--R 2atan(-----------------------)
+--R (q + p)cos(a x) + q + p
+--R ------------------------------]
+--R +---------+
+--R | 2 2
+--R a\|- q + p
+--R Type: Union(List Expression
Integer,...)
+--E
+@
+
+\section{\cite{1}:14.391~~~~~$\displaystyle
+\int{\frac{dx}{(p+q\cos{ax})^2}}$}
+$$\int{\frac{1}{(p+q\cos{ax})^2}}=
+\frac{q\sin{ax}}{a(q^2-p^2)(p+q\cos{ax})}
+-\frac{p}{q^2-p^2}\int{\frac{1}{p+q\cos{ax}}}
+$$
+<<*>>=
+)clear all
+
+--S 23 of 30
+aa:=integrate(1/(p+q*cos(a*x))^2,x)
+--R
+--R
+--R (1)
+--R [
+--R 2
+--R (p q cos(a x) + p )
+--R *
+--R +-------+
+--R | 2 2 2 2
+--R (- p cos(a x) - q)\|q - p + (q - p )sin(a x)
+--R log(------------------------------------------------)
+--R q cos(a x) + p
+--R +
+--R +-------+
+--R | 2 2
+--R q sin(a x)\|q - p
+--R /
+--R +-------+
+--R 3 2 2 3 | 2 2
+--R ((a q - a p q)cos(a x) + a p q - a p )\|q - p
+--R ,
+--R
+--R +---------+
+--R | 2 2
+--R 2 sin(a x)\|- q + p
+--R (- 2p q cos(a x) - 2p )atan(-----------------------)
+--R (q + p)cos(a x) + q + p
+--R +
+--R +---------+
+--R | 2 2
+--R q sin(a x)\|- q + p
+--R /
+--R +---------+
+--R 3 2 2 3 | 2 2
+--R ((a q - a p q)cos(a x) + a p q - a p )\|- q + p
+--R ]
+--R Type: Union(List Expression
Integer,...)
+--E
+@
+
+\section{\cite{1}:14.392~~~~~$\displaystyle
+\int{\frac{dx}{p^2+q^2\cos^2{ax}}}$}
+$$\int{\frac{1}{p^2+q^2\cos^2{ax}}}=
+\frac{1}{ap\sqrt{p^2+q^2}}\tan^{-1}\frac{p\tan{ax}}{\sqrt{p^2+q^2}}
+$$
+<<*>>=
+)clear all
+
+--S 24 of 30
+aa:=integrate(1/(p^2+a^2*cos(a*x)),x)
+--R
+--R
+--R (1)
+--R +---------+
+--R 2 2 | 4 4 4 4
+--R (- p cos(a x) - a )\|- p + a + (p - a )sin(a x)
+--R log(---------------------------------------------------)
+--R 2 2
+--R a cos(a x) + p
+--R [--------------------------------------------------------,
+--R +---------+
+--R | 4 4
+--R a\|- p + a
+--R +-------+
+--R | 4 4
+--R sin(a x)\|p - a
+--R 2atan(---------------------------)
+--R 2 2 2 2
+--R (p + a )cos(a x) + p + a
+--R ----------------------------------]
+--R +-------+
+--R | 4 4
+--R a\|p - a
+--R Type: Union(List Expression
Integer,...)
+--E
+@
+
+\section{\cite{1}:14.393~~~~~$\displaystyle
+\int{\frac{dx}{p^2-q^2\cos^2{ax}}}$}
+$$\int{\frac{1}{p^2-q^2\cos^2{ax}}}=
+\left\{
+\begin{array}{l}
+\displaystyle
+\frac{1}{ap\sqrt{p^2-q^2}}\tan^{-1}\frac{p\tan{ax}}{\sqrt{p^2-q^2}}\\
+\\
+\displaystyle
+\frac{1}{2ap\sqrt{q^2-p^2}}\ln\left(\frac{p\tan{ax}-\sqrt{q^2-p^2}}
+{p\tan{ax}+\sqrt{q^2-p^2}}\right)
+\end{array}
+\right.
+$$
+<<*>>=
+)clear all
+
+--S 25 of 30
+aa:=integrate(1/(p^2-q^2*cos(a*x)^2),x)
+--R
+--R
+--R (1)
+--R +-------+
+--R 2 2 2 2 | 2 2 2 3
+--R ((q - 2p )cos(a x) + p )\|q - p + (- 2p q + 2p )cos(a x)sin(a
x)
+--R
log(----------------------------------------------------------------------)
+--R 2 2 2
+--R q cos(a x) - p
+--R
[---------------------------------------------------------------------------,
+--R +-------+
+--R | 2 2
+--R 2a p\|q - p
+--R
+--R +---------+
+--R | 2 2
+--R sin(a x)\|- q + p
+--R atan(--------------------)
+--R 2p cos(a x) + 2p
+--R +
+--R 2 2 2
+--R ((q + p )cos(a x) + 2p )sin(a x)
+--R atan(-------------------------------------------)
+--R +---------+
+--R 2 | 2 2
+--R (p cos(a x) + 2p cos(a x) + p)\|- q + p
+--R /
+--R +---------+
+--R | 2 2
+--R a p\|- q + p
+--R ]
+--R Type: Union(List Expression
Integer,...)
+--E
+@
+
+\section{\cite{1}:14.394~~~~~$\displaystyle
+\int{x^m\cos{ax}}~dx$}
+$$\int{x^m\cos{ax}}=
+\frac{x^m\sin{ax}}{a}+\frac{mx^{m-1}}{a^2}\cos{ax}
+-\frac{m(m-1)}{a^2}\int{x^{m-2}\cos{ax}}
+$$
+<<*>>=
+)clear all
+
+--S 26 of 30
+aa:=integrate(x^m*cos(a*x),x)
+--R
+--R
+--R x
+--R ++ m
+--I (1) | cos(%I a)%I d%I
+--R ++
+--R Type: Union(Expression
Integer,...)
+--E
+@
+
+\section{\cite{1}:14.395~~~~~$\displaystyle
+\int{\frac{\cos{ax}}{x^n}}~dx$}
+$$\int{\frac{\cos{ax}}{x^n}}=
+-\frac{\cos{ax}}{(n-1)x^{n-1}}-\frac{a}{n-1}\int{\frac{\sin{ax}}{x^{n-1}}}
+$$
+<<*>>=
+)clear all
+
+--S 26 of 30
+aa:=integrate(cos(a*x)/x^n,x)
+--R
+--R
+--R x
+--I ++ cos(%I a)
+--I (1) | --------- d%I
+--R ++ n
+--I %I
+--R Type: Union(Expression
Integer,...)
+--E
+@
+
+\section{\cite{1}:14.396~~~~~$\displaystyle
+\int{\cos^n{ax}}~dx$}
+$$\int{\cos^n{ax}}=
+\frac{\sin{ax}\cos^{n-1}{ax}}{an}+\frac{n-1}{n}\int{\cos^{n-2}{ax}}
+$$
+<<*>>=
+)clear all
+
+--S 28 of 30
+aa:=integrate(cos(a*x)^n,x)
+--R
+--R
+--R x
+--R ++ n
+--I (1) | cos(%I a) d%I
+--R ++
+--R Type: Union(Expression
Integer,...)
+--E
+@
+
+\section{\cite{1}:14.397~~~~~$\displaystyle
+\int{\frac{1}{\cos^n{ax}}}~dx$}
+$$\int{\frac{1}{\cos^n{ax}}}=
+\frac{\sin{ax}}{a(n-1)\cos^{n-1}{ax}}
++\frac{n-2}{n-1}\int{\frac{1}{\cos^{n-2}{ax}}}
+$$
+<<*>>=
+)clear all
+
+--S 29 of 30
+aa:=integrate(1/(cos(a*x))^n,x)
+--R
+--R
+--R x
+--R ++ 1
+--I (1) | ---------- d%I
+--R ++ n
+--I cos(%I a)
+--R Type: Union(Expression
Integer,...)
+--E
+@
+
+\section{\cite{1}:14.398~~~~~$\displaystyle
+\int{\frac{x~dx}{cos^n{ax}}}$}
+$$\int{\frac{x}{cos^n{ax}}}=
+\frac{x\sin{ax}}{a(n-1)\cos^{n-1}{ax}}
+-\frac{1}{a^2(n-1)(n-2)\cos^{n-2}{ax}}
++\frac{n-2}{n-1}\int{\frac{x}{\cos^{n-2}{ax}}}
+$$
+<<*>>=
+)clear all
+
+--S 30 of 30
+aa:=integrate(x/cos(a*x)^n,x)
+--R
+--R
+--R x
+--I ++ %I
+--I (1) | ---------- d%I
+--R ++ n
+--I cos(%I 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 pp77-78
+\end{thebibliography}
+\end{document}
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