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Examples from Timofeev (1948) Ch. 5, divided into sections kindly provided by clicliclic vel Martin on sci.math.symbolic

fricas
-- Timofeev (
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)set break resume
-- Problems 1 - 3 (p. 202-203) integrate(cos(x)^2, x)

\label{eq1}{{{\cos \left({x}\right)}\ {\sin \left({x}\right)}}+ x}\over 2(1)
Type: Union(Expression(Integer),...)
fricas
integrate(cos(x)^3, x)

\label{eq2}{{\left({{\cos \left({x}\right)}^{2}}+ 2 \right)}\ {\sin \left({x}\right)}}\over 3(2)
Type: Union(Expression(Integer),...)
fricas
integrate(sin(x)^4, x)

\label{eq3}{{{\left({2 \ {{\cos \left({x}\right)}^{3}}}-{5 \ {\cos \left({x}\right)}}\right)}\ {\sin \left({x}\right)}}+{3 \  x}}\over 8(3)
Type: Union(Expression(Integer),...)
fricas
-- Problems 4 - 7 (p. 208)
integrate(cos(x)^6, x)

\label{eq4}{{{\left({8 \ {{\cos \left({x}\right)}^{5}}}+{{10}\ {{\cos \left({x}\right)}^{3}}}+{{15}\ {\cos \left({x}\right)}}\right)}\ {\sin \left({x}\right)}}+{{15}\  x}}\over{48}(4)
Type: Union(Expression(Integer),...)
fricas
integrate(sin(x)^8, x)

\label{eq5}{\left(
\begin{array}{@{}l}
\displaystyle
{{\left({
\begin{array}{@{}l}
\displaystyle
{{48}\ {{\cos \left({x}\right)}^{7}}}-{{200}\ {{\cos \left({x}\right)}^{5}}}+ 
\
\
\displaystyle
{{326}\ {{\cos \left({x}\right)}^{3}}}-{{279}\ {\cos \left({x}\right)}}
(5)
Type: Union(Expression(Integer),...)
fricas
integrate(cos(1/4*%pi+1/2*x)^4, x)

\label{eq6}{{{\left({4 \ {{\cos \left({{{2 \  x}+ \pi}\over 4}\right)}^{3}}}+{6 \ {\cos \left({{{2 \  x}+ \pi}\over 4}\right)}}\right)}\ {\sin \left({{{2 \  x}+ \pi}\over 4}\right)}}+{3 \  x}}\over 8(6)
Type: Union(Expression(Integer),...)
fricas
integrate(sin(-1/12*%pi+3*x)^3, x)

\label{eq7}{{{\cos \left({{{{36}\  x}- \pi}\over{12}}\right)}^{3}}-{3 \ {\cos \left({{{{36}\  x}- \pi}\over{12}}\right)}}}\over 9(7)
Type: Union(Expression(Integer),...)
fricas
-- Problems 8 - 11 (p. 211)
integrate(1/sin(x)^6, x)

\label{eq8}{-{8 \ {{\cos \left({x}\right)}^{5}}}+{{20}\ {{\cos \left({x}\right)}^{3}}}-{{15}\ {\cos \left({x}\right)}}}\over{{\left({{15}\ {{\cos \left({x}\right)}^{4}}}-{{30}\ {{\cos \left({x}\right)}^{2}}}+{15}\right)}\ {\sin \left({x}\right)}}(8)
Type: Union(Expression(Integer),...)
fricas
integrate(csc(x)^7, x)

\label{eq9}{\left(
\begin{array}{@{}l}
\displaystyle
{{\left({
\begin{array}{@{}l}
\displaystyle
-{{15}\ {{\cos \left({x}\right)}^{6}}}+{{45}\ {{\cos \left({x}\right)}^{4}}}- 
\
\
\displaystyle
{{45}\ {{\cos \left({x}\right)}^{2}}}+{15}
(9)
Type: Union(Expression(Integer),...)
fricas
integrate(1/cos(x)^12, x)

\label{eq10}{\left({\left({
\begin{array}{@{}l}
\displaystyle
{{256}\ {{\cos \left({x}\right)}^{10}}}+{{128}\ {{\cos \left({x}\right)}^{8}}}+ 
\
\
\displaystyle
{{96}\ {{\cos \left({x}\right)}^{6}}}+{{80}\ {{\cos \left({x}\right)}^{4}}}+{{70}\ {{\cos \left({x}\right)}^{2}}}+ 
\
\
\displaystyle
{63}
(10)
Type: Union(Expression(Integer),...)
fricas
integrate(1/cos(1/4*%pi+3*x)^3, x)

\label{eq11}{\left(
\begin{array}{@{}l}
\displaystyle
{{{\cos \left({{{{12}\  x}+ \pi}\over 4}\right)}^{2}}\ {\log \left({{\sin \left({{{{12}\  x}+ \pi}\over 4}\right)}+ 1}\right)}}- 
\
\
\displaystyle
{{{\cos \left({{{{12}\  x}+ \pi}\over 4}\right)}^{2}}\ {\log \left({-{\sin \left({{{{12}\  x}+ \pi}\over 4}\right)}+ 1}\right)}}+ 
\
\
\displaystyle
{2 \ {\sin \left({{{{12}\  x}+ \pi}\over 4}\right)}}
(11)
Type: Union(Expression(Integer),...)
fricas
-- Problems 12 - 14 (p. 213)
integrate(tan(x)^6, x)

\label{eq12}{{3 \ {{\tan \left({x}\right)}^{5}}}-{5 \ {{\tan \left({x}\right)}^{3}}}+{{15}\ {\tan \left({x}\right)}}-{{15}\  x}}\over{15}(12)
Type: Union(Expression(Integer),...)
fricas
integrate(1/tan(x)^5, x)

\label{eq13}{{2 \ {{\tan \left({x}\right)}^{4}}\ {\log \left({{{\tan \left({x}\right)}^{2}}\over{{{\tan \left({x}\right)}^{2}}+ 1}}\right)}}+{3 \ {{\tan \left({x}\right)}^{4}}}+{2 \ {{\tan \left({x}\right)}^{2}}}- 1}\over{4 \ {{\tan \left({x}\right)}^{4}}}(13)
Type: Union(Expression(Integer),...)
fricas
integrate(cot(-3/4*%pi+1/3*x)^4, x)

\label{eq14}{\left(
\begin{array}{@{}l}
\displaystyle
{{\left({x \ {\cos \left({{{4 \  x}-{9 \  \pi}}\over 6}\right)}}- x \right)}\ {\sin \left({{{4 \  x}-{9 \  \pi}}\over 6}\right)}}+ 
\
\
\displaystyle
{4 \ {{\cos \left({{{4 \  x}-{9 \  \pi}}\over 6}\right)}^{2}}}+{2 \ {\cos \left({{{4 \  x}-{9 \  \pi}}\over 6}\right)}}- 2 
(14)
Type: Union(Expression(Integer),...)
fricas
-- Problems 15 - 20 (p. 219-220)
integrate(cos(x)^6*sin(x)^4, x)

\label{eq15}{\left(
\begin{array}{@{}l}
\displaystyle
{{\left({
\begin{array}{@{}l}
\displaystyle
{{128}\ {{\cos \left({x}\right)}^{9}}}-{{176}\ {{\cos \left({x}\right)}^{7}}}+{8 \ {{\cos \left({x}\right)}^{5}}}+ 
\
\
\displaystyle
{{10}\ {{\cos \left({x}\right)}^{3}}}+{{15}\ {\cos \left({x}\right)}}
(15)
Type: Union(Expression(Integer),...)
fricas
integrate(cos(x)^6*sin(x)^7, x)

\label{eq16}{{{231}\ {{\cos \left({x}\right)}^{13}}}-{{819}\ {{\cos \left({x}\right)}^{11}}}+{{1001}\ {{\cos \left({x}\right)}^{9}}}-{{429}\ {{\cos \left({x}\right)}^{7}}}}\over{3003}(16)
Type: Union(Expression(Integer),...)
fricas
integrate(sin(x)^11/cos(x), x)

\label{eq17}{\left(
\begin{array}{@{}l}
\displaystyle
-{{120}\ {\log \left({-{\cos \left({x}\right)}}\right)}}+{{12}\ {{\cos \left({x}\right)}^{10}}}-{{75}\ {{\cos \left({x}\right)}^{8}}}+ 
\
\
\displaystyle
{{200}\ {{\cos \left({x}\right)}^{6}}}-{{300}\ {{\cos \left({x}\right)}^{4}}}+{{300}\ {{\cos \left({x}\right)}^{2}}}
(17)
Type: Union(Expression(Integer),...)
fricas
integrate(1/(cos(x)^6*sin(x)^6), x)

\label{eq18}{\left(
\begin{array}{@{}l}
\displaystyle
-{{256}\ {{\cos \left({x}\right)}^{10}}}+{{640}\ {{\cos \left({x}\right)}^{8}}}-{{480}\ {{\cos \left({x}\right)}^{6}}}+ 
\
\
\displaystyle
{{80}\ {{\cos \left({x}\right)}^{4}}}+{{10}\ {{\cos \left({x}\right)}^{2}}}+ 3 
(18)
Type: Union(Expression(Integer),...)
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integrate(cos(x)^(2*m)*sin(x)^(2*m), x)

\label{eq19}\int^{
\displaystyle
x}{{{\cos \left({\%A}\right)}^{2 \  m}}\ {{\sin \left({\%A}\right)}^{2 \  m}}\ {d \%A}}(19)
Type: Union(Expression(Integer),...)
fricas
integrate(1/(cos(1/4*%pi+2*x)*sin(1/4*%pi+2*x)^3), x)

\label{eq20}{\left(
\begin{array}{@{}l}
\displaystyle
{{\left(-{{\cos \left({{{8 \  x}+ \pi}\over 4}\right)}^{2}}+ 1 \right)}\ {\log \left({{\cos \left({{{8 \  x}+ \pi}\over 4}\right)}^{2}}\right)}}+ 
\
\
\displaystyle
{{\left({{\cos \left({{{8 \  x}+ \pi}\over 4}\right)}^{2}}- 1 \right)}\ {\log \left({{-{{\cos \left({{{8 \  x}+ \pi}\over 4}\right)}^{2}}+ 1}\over 4}\right)}}+ 1 
(20)
Type: Union(Expression(Integer),...)
fricas
-- Problems 21 - 29 (p. 223)
integrate(sec(x)^2*tan(x)^2, x)

\label{eq21}{{\left(-{{\cos \left({x}\right)}^{2}}+ 1 \right)}\ {\sin \left({x}\right)}}\over{3 \ {{\cos \left({x}\right)}^{3}}}(21)
Type: Union(Expression(Integer),...)
fricas
integrate(cot(x)^3*csc(x), x)

\label{eq22}{{3 \ {{\cos \left({x}\right)}^{2}}}- 2}\over{{\left({3 \ {{\cos \left({x}\right)}^{2}}}- 3 \right)}\ {\sin \left({x}\right)}}(22)
Type: Union(Expression(Integer),...)
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integrate(sec(x)^3*tan(x), x)

\label{eq23}1 \over{3 \ {{\cos \left({x}\right)}^{3}}}(23)
Type: Union(Expression(Integer),...)
fricas
integrate(cot(x)^2*csc(x)^3, x)

\label{eq24}{\left(
\begin{array}{@{}l}
\displaystyle
{{\left({{\cos \left({x}\right)}^{4}}-{2 \ {{\cos \left({x}\right)}^{2}}}+ 1 \right)}\ {\log \left({{{\cos \left({x}\right)}+ 1}\over 2}\right)}}+ 
\
\
\displaystyle
{{\left(-{{\cos \left({x}\right)}^{4}}+{2 \ {{\cos \left({x}\right)}^{2}}}- 1 \right)}\ {\log \left({{-{\cos \left({x}\right)}+ 1}\over 2}\right)}}- 
\
\
\displaystyle
{2 \ {{\cos \left({x}\right)}^{3}}}-{2 \ {\cos \left({x}\right)}}
(24)
Type: Union(Expression(Integer),...)
fricas
integrate(cos(x)^3/sin(x)^7, x)

\label{eq25}{{3 \ {{\cos \left({x}\right)}^{2}}}- 1}\over{{{12}\ {{\cos \left({x}\right)}^{6}}}-{{36}\ {{\cos \left({x}\right)}^{4}}}+{{36}\ {{\cos \left({x}\right)}^{2}}}-{12}}(25)
Type: Union(Expression(Integer),...)
fricas
integrate(sqrt(sec(x)^3)*tan(x)^5, x)

\label{eq26}{{\left({{154}\ {{\cos \left({x}\right)}^{4}}}-{{132}\ {{\cos \left({x}\right)}^{2}}}+{42}\right)}\ {\sqrt{1 \over{{\cos \left({x}\right)}^{3}}}}}\over{{231}\ {{\cos \left({x}\right)}^{4}}}(26)
Type: Union(Expression(Integer),...)
fricas
integrate(sec(x)^4*tan(x)^(3/2), x)

\label{eq27}{{\left(-{8 \ {{\cos \left({x}\right)}^{4}}}-{2 \ {{\cos \left({x}\right)}^{2}}}+{10}\right)}\ {\sqrt{{\sin \left({x}\right)}\over{\cos \left({x}\right)}}}}\over{{45}\ {{\cos \left({x}\right)}^{4}}}(27)
Type: Union(Expression(Integer),...)
fricas
integrate(cot(x)^4*csc(x)^3, x)

\label{eq28}{\left(
\begin{array}{@{}l}
\displaystyle
{{\left({
\begin{array}{@{}l}
\displaystyle
-{3 \ {{\cos \left({x}\right)}^{6}}}+{9 \ {{\cos \left({x}\right)}^{4}}}- 
\
\
\displaystyle
{9 \ {{\cos \left({x}\right)}^{2}}}+ 3 
(28)
Type: Union(Expression(Integer),...)
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)set output tex off
 
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)set output algebra on
integrate(sec(1/4*%pi+1/2*x)^3*tan(1/4*%pi+1/2*x)^2, x)
(29) 2 x + %pi 4 2 x + %pi - cos(---------) log(sin(---------) + 1) 4 4 + 2 x + %pi 4 2 x + %pi cos(---------) log(- sin(---------) + 1) 4 4 + 2 x + %pi 2 2 x + %pi (- 2 cos(---------) + 4)sin(---------) 4 4 / 2 x + %pi 4 8 cos(---------) 4
Type: Union(Expression(Integer),...)
fricas
)set output tex on
 
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)set output algebra off
-- Problems 30 - 32 (p. 228) integrate((1+cot(x)^3)*(a*sec(x)^2-sin(2*x)), x)

\label{eq29}{\left(
\begin{array}{@{}l}
\displaystyle
{{\left({2 \ {{\cos \left({x}\right)}^{4}}}+{{\left({2 \  a}- 6 \right)}\ {{\cos \left({x}\right)}^{2}}}-{2 \  a}\right)}\ {\sin \left({x}\right)}}+ 
\
\
\displaystyle
{2 \ {{\cos \left({x}\right)}^{5}}}+{{\left({6 \  x}- 3 \right)}\ {{\cos \left({x}\right)}^{3}}}+ 
\
\
\displaystyle
{{\left(-{6 \  x}+ a + 1 \right)}\ {\cos \left({x}\right)}}
(29)
Type: Union(Expression(Integer),...)
fricas
integrate((4-3*cos(x))*(1-1/2*sin(x))^4, x)

\label{eq30}{\left(
\begin{array}{@{}l}
\displaystyle
{{\left({
\begin{array}{@{}l}
\displaystyle
-{{18}\ {{\cos \left({x}\right)}^{4}}}+{{30}\ {{\cos \left({x}\right)}^{3}}}+ 
\
\
\displaystyle
{{756}\ {{\cos \left({x}\right)}^{2}}}-{{1515}\ {\cos \left({x}\right)}}-{2178}
(30)
Type: Union(Expression(Integer),...)
fricas
integrate((1/2-3*cot(x))*(3-2*cot(x))^3, x)

\label{eq31}{\left(
\begin{array}{@{}l}
\displaystyle
{{\left({4 \ {\cos \left({2 \  x}\right)}}- 4 \right)}\ {\sin \left({2 \  x}\right)}\ {\log \left({{-{\cos \left({2 \  x}\right)}+ 1}\over 2}\right)}}+ 
\
\
\displaystyle
{{\left(-{{285}\  x \ {\cos \left({2 \  x}\right)}}+{{285}\  x}-{224}\right)}\ {\sin \left({2 \  x}\right)}}- 
\
\
\displaystyle
{{296}\ {{\cos \left({2 \  x}\right)}^{2}}}+{{32}\ {\cos \left({2 \  x}\right)}}+{328}
(31)
Type: Union(Expression(Integer),...)
fricas
-- Problems 33 - 36 (p. 229)
integrate(cos(5*x)/cos(x)^5, x)

\label{eq32}{{{\left(-{{50}\ {{\cos \left({x}\right)}^{2}}}+ 5 \right)}\ {\sin \left({x}\right)}}+{{48}\  x \ {{\cos \left({x}\right)}^{3}}}}\over{3 \ {{\cos \left({x}\right)}^{3}}}(32)
Type: Union(Expression(Integer),...)
fricas
integrate(cos(4*x)/cos(x), x)

\label{eq33}{\left(
\begin{array}{@{}l}
\displaystyle
{3 \ {\log \left({{\sin \left({x}\right)}+ 1}\right)}}-{3 \ {\log \left({-{\sin \left({x}\right)}+ 1}\right)}}+ 
\
\
\displaystyle
{{\left({{16}\ {{\cos \left({x}\right)}^{2}}}-{16}\right)}\ {\sin \left({x}\right)}}
(33)
Type: Union(Expression(Integer),...)
fricas
integrate(cos(x)*cos(4*x), x)

\label{eq34}{{\left({{24}\ {{\cos \left({x}\right)}^{4}}}-{8 \ {{\cos \left({x}\right)}^{2}}}- 1 \right)}\ {\sin \left({x}\right)}}\over{15}(34)
Type: Union(Expression(Integer),...)
fricas
integrate(cos(4*x)/cos(x)^5, x)

\label{eq35}{\left(
\begin{array}{@{}l}
\displaystyle
{{35}\ {{\cos \left({x}\right)}^{4}}\ {\log \left({{\sin \left({x}\right)}+ 1}\right)}}- 
\
\
\displaystyle
{{35}\ {{\cos \left({x}\right)}^{4}}\ {\log \left({-{\sin \left({x}\right)}+ 1}\right)}}+ 
\
\
\displaystyle
{{\left(-{{58}\ {{\cos \left({x}\right)}^{2}}}+ 4 \right)}\ {\sin \left({x}\right)}}
(35)
Type: Union(Expression(Integer),...)
fricas
-- Problems 37 - 39 (p. 233)
integrate(cos(x)^4*cos(4*x), x)

\label{eq36}{{{\left({{48}\ {{\cos \left({x}\right)}^{7}}}-{8 \ {{\cos \left({x}\right)}^{5}}}+{2 \ {{\cos \left({x}\right)}^{3}}}+{3 \ {\cos \left({x}\right)}}\right)}\ {\sin \left({x}\right)}}+{3 \  x}}\over{48}(36)
Type: Union(Expression(Integer),...)
fricas
integrate(cos(5*x)/sin(x)^5, x)

\label{eq37}{\left(
\begin{array}{@{}l}
\displaystyle
{{\left({{64}\ {{\cos \left({x}\right)}^{4}}}-{{128}\ {{\cos \left({x}\right)}^{2}}}+{64}\right)}\ {\log \left({{\sin \left({x}\right)}\over 2}\right)}}- 
\
\
\displaystyle
{{24}\ {{\cos \left({x}\right)}^{2}}}+{23}
(37)
Type: Union(Expression(Integer),...)
fricas
integrate(sin(4*x)/sin(x)^4, x)

\label{eq38}{{{\left(-{8 \ {{\cos \left({x}\right)}^{2}}}+ 8 \right)}\ {\log \left({{\sin \left({x}\right)}\over 2}\right)}}+ 2}\over{{{\cos \left({x}\right)}^{2}}- 1}(38)
Type: Union(Expression(Integer),...)
fricas
-- Problems 40 - 49 (p. 254-255)
integrate(cos(x)/(sin(x)*(2+sin(2*x))), x)

\label{eq39}{\left(
\begin{array}{@{}l}
\displaystyle
-{{\sqrt{3}}\ {\log \left({{2 \ {\cos \left({x}\right)}\ {\sin \left({x}\right)}}-{{\cos \left({x}\right)}^{4}}+{{\cos \left({x}\right)}^{2}}+ 1}\right)}}+ 
\
\
\displaystyle
{2 \ {\sqrt{3}}\ {\log \left({{-{{\cos \left({x}\right)}^{2}}+ 1}\over 4}\right)}}- 
\
\
\displaystyle
{2 \ {\arctan \left({{{4 \ {\sqrt{3}}\ {\cos \left({x}\right)}\ {\sin \left({x}\right)}}+{\sqrt{3}}}\over{{6 \ {{\cos \left({x}\right)}^{2}}}- 3}}\right)}}
(39)
Type: Union(Expression(Integer),...)
fricas
integrate(cos(x)^2/(cos(3*x)*sin(x)), x)

\label{eq40}{{2 \ {\log \left({{\sin \left({x}\right)}\over 2}\right)}}-{\log \left({{4 \ {{\cos \left({x}\right)}^{2}}}- 3}\right)}}\over 2(40)
Type: Union(Expression(Integer),...)
fricas
integrate(sin(2*x)/(cos(x)^4+sin(x)^4), x)

\label{eq41}-{\arctan \left({{2 \ {{\cos \left({x}\right)}^{2}}}- 1}\right)}(41)
Type: Union(Expression(Integer),...)
fricas
)set output tex off
 
fricas
)set output algebra on
integrate(1/(4+sin(x)+cos(x)*sqrt(3)), x)
+-+ +-+ 2 \|3 sin(x) + 6 cos(x) + 2 \|3 atan(--------------------------------) +-+ 3 \|3 sin(x) - 3 cos(x) (43) - -------------------------------------- +-+ 2 \|3
Type: Union(Expression(Integer),...)
fricas
)set output tex on
 
fricas
)set output algebra off
integrate(1/(3+4*cos(x)+4*sin(x)), x)

\label{eq42}{\log{\left({{{{\left(-{{14}\ {\sqrt{23}}\ {\cos \left({x}\right)}}+{{24}\ {\sqrt{23}}}+{184}\right)}\ {\sin \left({x}\right)}}-{{1
38}\ {{\cos \left({x}\right)}^{2}}}+{{\left({{24}\ {\sqrt{23}}}-{184}\right)}\ {\cos \left({x}\right)}}+{{48}\ {\sqrt{23}}}+{6
9}}\over{{{\left({{32}\ {\cos \left({x}\right)}}+{24}\right)}\ {\sin \left({x}\right)}}+{{24}\ {\cos \left({x}\right)}}+{25}}}\right)}}\over{2 \ {\sqrt{23}}}(42)
Type: Union(Expression(Integer),...)
fricas
integrate(1/(4-3*cos(x)^2+5*sin(x)^2), x)

\label{eq43}-{{\arctan \left({{{{10}\ {{\cos \left({x}\right)}^{2}}}- 9}\over{6 \ {\cos \left({x}\right)}\ {\sin \left({x}\right)}}}\right)}\over 6}(43)
Type: Union(Expression(Integer),...)
fricas
integrate(1/(4+4*cot(x)+tan(x)), x)

\label{eq44}{\left(
\begin{array}{@{}l}
\displaystyle
{{\left(-{3 \ {\tan \left({x}\right)}}- 6 \right)}\ {\log \left({{{{\tan \left({x}\right)}^{2}}+{4 \ {\tan \left({x}\right)}}+ 4}\over{{{\tan \left({x}\right)}^{2}}+ 1}}\right)}}+ 
\
\
\displaystyle
{{\left({8 \  x}- 8 \right)}\ {\tan \left({x}\right)}}+{{16}\  x}+ 4 
(44)
Type: Union(Expression(Integer),...)
fricas
integrate(1/(2*sec(x)+sin(x))^2, x)

\label{eq45}{\left(
\begin{array}{@{}l}
\displaystyle
{{\left({4 \ {\cos \left({x}\right)}\ {\sin \left({x}\right)}}+ 8 \right)}\ {\arctan{\left({{{8 \ {\sqrt{15}}\ {\cos \left({x}\right)}\ {\sin \left({x}\right)}}+{\sqrt{15}}}\over{{{30}\ {{\cos \left({x}\right)}^{2}}}-{15}}}\right)}}}+ 
\
\
\displaystyle
{{\sqrt{15}}\ {{\cos \left({x}\right)}^{2}}}-{8 \ {\sqrt{15}}}
(45)
Type: Union(Expression(Integer),...)
fricas
integrate(1/(cos(x)+2*sec(x))^2, x)

\label{eq46}{\left(
\begin{array}{@{}l}
\displaystyle
{{\left(-{{\cos \left({x}\right)}^{2}}- 2 \right)}\ {\arctan \left({{{5 \ {\sqrt{6}}\ {{\cos \left({x}\right)}^{2}}}-{2 \ {\sqrt{6}}}}\over{{1
2}\ {\cos \left({x}\right)}\ {\sin \left({x}\right)}}}\right)}}+ 
\
\
\displaystyle
{2 \ {\sqrt{6}}\ {\cos \left({x}\right)}\ {\sin \left({x}\right)}}
(46)
Type: Union(Expression(Integer),...)
fricas
integrate((5-tan(x)-6*tan(x)^2)/(1+3*tan(x))^3, x)

\label{eq47}{\left(
\begin{array}{@{}l}
\displaystyle
{{\left({
\begin{array}{@{}l}
\displaystyle
-{{504}\ {{\tan \left({x}\right)}^{2}}}- 
\
\
\displaystyle
{{336}\ {\tan \left({x}\right)}}-{56}
(47)
Type: Union(Expression(Integer),...)
fricas
-- Problems 50 - 56 (p. 260)
integrate(cos(x)^2/cos(3*x), x)

\label{eq48}{{\log \left({{2 \ {\sin \left({x}\right)}}+ 1}\right)}-{\log \left({-{2 \ {\sin \left({x}\right)}}+ 1}\right)}}\over 4(48)
Type: Union(Expression(Integer),...)
fricas
integrate(sin(x)/cos(2*x), x)

\label{eq49}{\log \left({{{2 \ {\sqrt{2}}\ {{\cos \left({x}\right)}^{2}}}+{4 \ {\cos \left({x}\right)}}+{\sqrt{2}}}\over{{2 \ {{\cos \left({x}\right)}^{2}}}- 1}}\right)}\over{2 \ {\sqrt{2}}}(49)
Type: Union(Expression(Integer),...)
fricas
integrate(sin(x)^2/cos(2*x), x)

\label{eq50}{{\log \left({{2 \ {\cos \left({x}\right)}\ {\sin \left({x}\right)}}+ 1}\right)}-{\log \left({-{2 \ {\cos \left({x}\right)}\ {\sin \left({x}\right)}}+ 1}\right)}-{4 \  x}}\over 8(50)
Type: Union(Expression(Integer),...)
fricas
integrate(sin(x)^3/cos(3*x), x)

\label{eq51}{-{\log \left({{4 \ {{\cos \left({x}\right)}^{2}}}- 3}\right)}+{8 \ {\log \left({-{\cos \left({x}\right)}}\right)}}}\over{24}(51)
Type: Union(Expression(Integer),...)
fricas
integrate(cos(x)/sin(3*x), x)

\label{eq52}{{2 \ {\log \left({{\sin \left({x}\right)}\over 2}\right)}}-{\log \left({{4 \ {{\cos \left({x}\right)}^{2}}}- 1}\right)}}\over 6(52)
Type: Union(Expression(Integer),...)
fricas
integrate(sin(x)/sin(4*x), x)

\label{eq53}{\left(
\begin{array}{@{}l}
\displaystyle
-{{\sqrt{2}}\ {\log \left({{\sin \left({x}\right)}+ 1}\right)}}+ 
\
\
\displaystyle
{2 \ {\log \left({{{4 \ {\sin \left({x}\right)}}-{2 \ {\sqrt{2}}\ {{\cos \left({x}\right)}^{2}}}+{3 \ {\sqrt{2}}}}\over{{2 \ {{\cos \left({x}\right)}^{2}}}- 1}}\right)}}+ 
\
\
\displaystyle
{{\sqrt{2}}\ {\log \left({-{\sin \left({x}\right)}+ 1}\right)}}
(53)
Type: Union(Expression(Integer),...)
fricas
integrate(sin(x)^3/sin(4*x), x)

\label{eq54}{\left(
\begin{array}{@{}l}
\displaystyle
-{{\sqrt{2}}\ {\log \left({{\sin \left({x}\right)}+ 1}\right)}}+ 
\
\
\displaystyle
{\log \left({{{4 \ {\sin \left({x}\right)}}-{2 \ {\sqrt{2}}\ {{\cos \left({x}\right)}^{2}}}+{3 \ {\sqrt{2}}}}\over{{2 \ {{\cos \left({x}\right)}^{2}}}- 1}}\right)}+ 
\
\
\displaystyle
{{\sqrt{2}}\ {\log \left({-{\sin \left({x}\right)}+ 1}\right)}}
(54)
Type: Union(Expression(Integer),...)
fricas
-- Problems 57 - 61 (p. 266)
integrate(sqrt(1+sin(2*x)), x)

\label{eq55}{{\left({\sin \left({2 \  x}\right)}-{\cos \left({2 \  x}\right)}- 1 \right)}\ {\sqrt{{\sin \left({2 \  x}\right)}+ 1}}}\over{{\sin \left({2 \  x}\right)}+{\cos \left({2 \  x}\right)}+ 1}(55)
Type: Union(Expression(Integer),...)
fricas
integrate(1/sqrt(1+cos(2*x)), x)

\label{eq56}{{\sqrt{2}}\ {\log \left({{{{\left(-{\cos \left({2 \  x}\right)}+ 3 \right)}\ {\sqrt{{\cos \left({2 \  x}\right)}+ 1}}}+{2 \ {\sqrt{2}}\ {\sin \left({2 \  x}\right)}}}\over{{\left({\cos \left({2 \  x}\right)}+ 1 \right)}\ {\sqrt{{\cos \left({2 \  x}\right)}+ 1}}}}\right)}}\over 4(56)
Type: Union(Expression(Integer),...)
fricas
integrate(1/(1-cos(3*x))^(3/2), x)

\label{eq57}{\left(
\begin{array}{@{}l}
\displaystyle
{
\begin{array}{@{}l}
\displaystyle
{\left({{\sqrt{2}}\ {\cos \left({3 \  x}\right)}}-{\sqrt{2}}\right)}\ {\sin \left({3 \  x}\right)}\  \cdot 
\
\
\displaystyle
{\log{\left({{{{\left(-{{\sqrt{2}}\ {\cos \left({3 \  x}\right)}}-{3 \ {\sqrt{2}}}\right)}\ {\sqrt{-{\cos \left({3 \  x}\right)}+ 1}}}+{4 \ {\sin \left({3 \  x}\right)}}}\over{{\left({\cos \left({3 \  x}\right)}- 1 \right)}\ {\sqrt{-{\cos \left({3 \  x}\right)}+ 1}}}}\right)}}
(57)
Type: Union(Expression(Integer),...)
fricas
integrate((1-sin(2/3*x))^(5/2), x)

\label{eq58}{\left({\left({
\begin{array}{@{}l}
\displaystyle
{{\left({
\begin{array}{@{}l}
\displaystyle
{3 \ {{\cos \left({{2 \  x}\over 3}\right)}^{2}}}+ 
\
\
\displaystyle
{{14}\ {\cos{\left({{2 \  x}\over 3}\right)}}}- 
\
\
\displaystyle
{32}
(58)
Type: Union(Expression(Integer),...)
fricas
integrate(cos(x)*(-cos(x)^2+2*(1+2*sin(x))^(1/4))/(1+2*sin(x))^(3/2), x)

\label{eq59}{\left(
\begin{array}{@{}l}
\displaystyle
-{{12}\ {{\root{4}\of{{2 \ {\sin \left({x}\right)}}+ 1}}^{3}}}+ 
\
\
\displaystyle
{{\left(-{2 \ {\sin \left({x}\right)}}-{{\cos \left({x}\right)}^{2}}+ 2 \right)}\ {{\root{4}\of{{2 \ {\sin \left({x}\right)}}+ 1}}^{2}}}
(59)
Type: Union(Expression(Integer),...)
fricas
-- Problems 62 - 66 (p. 268)
fricas
)set output tex off
 
fricas
)set output algebra on
integrate(sqrt(tan(x)), x)
(62) - +-+ 4 \|2 * atan 1 / +---------------------------------------------+ | +------+ | +-+ |sin(x) |- 2 \|2 cos(x) |------ + 2 sin(x) + 2 cos(x) | \|cos(x) |--------------------------------------------- \| cos(x) + +------+ +-+ |sin(x) \|2 |------ - 1 \|cos(x) + - +-+ 4 \|2 * atan 1 / +-------------------------------------------+ | +------+ | +-+ |sin(x) |2 \|2 cos(x) |------ + 2 sin(x) + 2 cos(x) | \|cos(x) |------------------------------------------- \| cos(x) + +------+ +-+ |sin(x) \|2 |------ + 1 \|cos(x) + +------+ +-+ |sin(x) 2 \|2 cos(x) |------ + 2 sin(x) + 2 cos(x) +-+ \|cos(x) - \|2 log(-------------------------------------------) cos(x) + +------+ +-+ |sin(x) - 2 \|2 cos(x) |------ + 2 sin(x) + 2 cos(x) +-+ \|cos(x) \|2 log(---------------------------------------------) cos(x) / 4
Type: Union(Expression(Integer),...)
fricas
integrate(1/tan(5*x)^(1/3), x)
(63) 3+--------+2 3+--------+2 3+--------+ 2 log(\|tan(5 x) + 1) - log(- \|tan(5 x) + tan(5 x)\|tan(5 x) + 1) + 3+--------+2 +-+ 2 \|tan(5 x) - 1 2 \|3 atan(------------------) +-+ \|3 / 20
Type: Union(Expression(Integer),...)
fricas
integrate(1/(4+3*tan(2*x))^(3/2), x)
(64) - +-----------------------+ +-+ |3 sin(2 x) + 4 cos(2 x) 36 \|5 |----------------------- \| cos(2 x) * atan +-+ 125 \|5 / +--+ \|10 * ROOT +-----------------------+ +-+ +--+ |3 sin(2 x) + 4 cos(2 x) - 9375 \|5 \|10 cos(2 x) |----------------------- \| cos(2 x) + 46875 sin(2 x) + 140625 cos(2 x) / cos(2 x) + +-----------------------+ +--+ |3 sin(2 x) + 4 cos(2 x) +-+ 125 \|10 |----------------------- - 375 \|5 \| cos(2 x) + - +-----------------------+ +-+ |3 sin(2 x) + 4 cos(2 x) 36 \|5 |----------------------- \| cos(2 x) * atan +-+ 125 \|5 / +--+ \|10 * ROOT +-----------------------+ +-+ +--+ |3 sin(2 x) + 4 cos(2 x) 9375 \|5 \|10 cos(2 x) |----------------------- \| cos(2 x) + 46875 sin(2 x) + 140625 cos(2 x) / cos(2 x) + +-----------------------+ +--+ |3 sin(2 x) + 4 cos(2 x) +-+ 125 \|10 |----------------------- + 375 \|5 \| cos(2 x) + +-----------------------+ +-+ |3 sin(2 x) + 4 cos(2 x) 13 \|5 |----------------------- \| cos(2 x) * log +-----------------------+ +-+ +--+ |3 sin(2 x) + 4 cos(2 x) 9375 \|5 \|10 cos(2 x) |----------------------- \| cos(2 x) + 46875 sin(2 x) + 140625 cos(2 x) / cos(2 x) + - +-----------------------+ +-+ |3 sin(2 x) + 4 cos(2 x) 13 \|5 |----------------------- \| cos(2 x) * log +-----------------------+ +-+ +--+ |3 sin(2 x) + 4 cos(2 x) - 9375 \|5 \|10 cos(2 x) |----------------------- \| cos(2 x) + 46875 sin(2 x) + 140625 cos(2 x) / cos(2 x) + +--+ - 60 \|10 / +-----------------------+ +--+ |3 sin(2 x) + 4 cos(2 x) 500 \|10 |----------------------- \| cos(2 x)
Type: Union(Expression(Integer),...)
fricas
integrate((-sqrt(4-3*tan(x))+3*tan(x))/(cos(x)^2*(4-3*tan(x))^(3/2)), x)
(65) +---------------------+ |- 3 sin(x) + 4 cos(x) (12 sin(x) - 32 cos(x)) |--------------------- \| cos(x) + 2 (- 3 sin(x) + 4 cos(x))log(cos(x) ) + 2 - 24 cos(x)sin(x) + 7 cos(x) + 9 (3 sin(x) - 4 cos(x))log(---------------------------------) 4 / 18 sin(x) - 24 cos(x)
Type: Union(Expression(Integer),...)
fricas
integrate(tan(x)/(-1+sqrt(tan(x)))^2, x)
(66) +------+ (- \|tan(x) + 1) * +----------------------------------------------------------+ | 2 2 \|- 3 %%CZ1 + (- 2 %%CZ0 - 2)%%CZ1 - 3 %%CZ0 - 2 %%CZ0 - 1 + +------+ +------+ (- \|tan(x) + 1)%%CZ1 + (- %%CZ0 - 1)\|tan(x) + %%CZ0 + 1 * log ((32 %%CZ0 + 4)%%CZ1 + 4 %%CZ0 + 2) * +----------------------------------------------------------+ | 2 2 \|- 3 %%CZ1 + (- 2 %%CZ0 - 2)%%CZ1 - 3 %%CZ0 - 2 %%CZ0 - 1 + 2 2 (- 32 %%CZ0 - 4)%%CZ1 + (- 32 %%CZ0 - 32 %%CZ0 - 2)%%CZ1 + +------+ 2 6 \|tan(x) - 4 %%CZ0 - 2 %%CZ0 - 3 + +------+ (\|tan(x) - 1) * +----------------------------------------------------------+ | 2 2 \|- 3 %%CZ1 + (- 2 %%CZ0 - 2)%%CZ1 - 3 %%CZ0 - 2 %%CZ0 - 1 + +------+ +------+ (- \|tan(x) + 1)%%CZ1 + (- %%CZ0 - 1)\|tan(x) + %%CZ0 + 1 * log ((- 32 %%CZ0 - 4)%%CZ1 - 4 %%CZ0 - 2) * +----------------------------------------------------------+ | 2 2 \|- 3 %%CZ1 + (- 2 %%CZ0 - 2)%%CZ1 - 3 %%CZ0 - 2 %%CZ0 - 1 + 2 2 (- 32 %%CZ0 - 4)%%CZ1 + (- 32 %%CZ0 - 32 %%CZ0 - 2)%%CZ1 + +------+ 2 6 \|tan(x) - 4 %%CZ0 - 2 %%CZ0 - 3 + +------+ (2 \|tan(x) - 2)%%CZ1 * log 2 2 (64 %%CZ0 + 8)%%CZ1 + (64 %%CZ0 + 64 %%CZ0 + 4)%%CZ1 + +------+ 3 2 6 \|tan(x) + 64 %%CZ0 + 64 %%CZ0 + 32 %%CZ0 - 1 + +------+ (2 %%CZ0\|tan(x) - 2 %%CZ0) * +------+ 3 2 log(6 \|tan(x) - 64 %%CZ0 - 56 %%CZ0 - 28 %%CZ0 + 7) + +------+ +------+ (2 \|tan(x) - 2)log(\|tan(x) - 1) - 2 / +------+ 2 \|tan(x) - 2
Type: Union(Expression(Integer),...)
fricas
)set output tex on
 
fricas
)set output algebra off
-- Problems 67 - 75 (p. 272-273) integrate(sin(x)/sqrt(sin(2*x)), x)

\label{eq60}{\left(
\begin{array}{@{}l}
\displaystyle
- 
\
\
\displaystyle
{\log{{\left({{{\left({2 \ {\sin \left({x}\right)}}+{2 \ {\cos \left({x}\right)}}\right)}\ {\sqrt{2 \ {\cos \left({x}\right)}\ {\sin \left({x}\right)}}}}+{4 \ {\cos \left({x}\right)}\ {\sin \left({x}\right)}}+ 1}\right)}}}+ 
\
\
\displaystyle
{\log{\left({{{\left(-{2 \ {\sin \left({x}\right)}}-{2 \ {\cos \left({x}\right)}}\right)}\ {\sqrt{2 \ {\cos \left({x}\right)}\ {\sin \left({x}\right)}}}}+{4 \ {\cos \left({x}\right)}\ {\sin \left({x}\right)}}+ 1}\right)}}- 
\
\
\displaystyle
{\arctan{\left({{{{\left({\sin \left({x}\right)}+{\cos \left({x}\right)}\right)}\ {\sqrt{2 \ {\cos \left({x}\right)}\ {\sin \left({x}\right)}}}}+{2 \ {\cos \left({x}\right)}\ {\sin \left({x}\right)}}}\over{{{\left({\sin \left({x}\right)}-{\cos \left({x}\right)}\right)}\ {\sqrt{2 \ {\cos \left({x}\right)}\ {\sin \left({x}\right)}}}}-{2 \ {{\cos \left({x}\right)}^{2}}}+ 1}}\right)}}+ 
\
\
\displaystyle
{\arctan \left({{{{\left({\sin \left({x}\right)}+{\cos \left({x}\right)}\right)}\ {\sqrt{2 \ {\cos \left({x}\right)}\ {\sin \left({x}\right)}}}}+ 1}\over{{\left({\sin \left({x}\right)}-{\cos \left({x}\right)}\right)}\ {\sqrt{2 \ {\cos \left({x}\right)}\ {\sin \left({x}\right)}}}}}\right)}- 
\
\
\displaystyle
{\arctan \left({{{{\left({\sin \left({x}\right)}+{\cos \left({x}\right)}\right)}\ {\sqrt{2 \ {\cos \left({x}\right)}\ {\sin \left({x}\right)}}}}- 1}\over{{\left({\sin \left({x}\right)}-{\cos \left({x}\right)}\right)}\ {\sqrt{2 \ {\cos \left({x}\right)}\ {\sin \left({x}\right)}}}}}\right)}+ 
\
\
\displaystyle
{\arctan{\left({{{{\left({\sin \left({x}\right)}+{\cos \left({x}\right)}\right)}\ {\sqrt{2 \ {\cos \left({x}\right)}\ {\sin \left({x}\right)}}}}-{2 \ {\cos \left({x}\right)}\ {\sin \left({x}\right)}}}\over{{{\left({\sin \left({x}\right)}-{\cos \left({x}\right)}\right)}\ {\sqrt{2 \ {\cos \left({x}\right)}\ {\sin \left({x}\right)}}}}+{2 \ {{\cos \left({x}\right)}^{2}}}- 1}}\right)}}
(60)
Type: Union(Expression(Integer),...)
fricas
integrate(cos(x)/sqrt(sin(2*x)), x)

\label{eq61}{\left(
\begin{array}{@{}l}
\displaystyle
{\log{\left({{{\left({2 \ {\sin \left({x}\right)}}+{2 \ {\cos \left({x}\right)}}\right)}\ {\sqrt{2 \ {\cos \left({x}\right)}\ {\sin \left({x}\right)}}}}+{4 \ {\cos \left({x}\right)}\ {\sin \left({x}\right)}}+ 1}\right)}}- 
\
\
\displaystyle
{\log{\left({{{\left(-{2 \ {\sin \left({x}\right)}}-{2 \ {\cos \left({x}\right)}}\right)}\ {\sqrt{2 \ {\cos \left({x}\right)}\ {\sin \left({x}\right)}}}}+{4 \ {\cos \left({x}\right)}\ {\sin \left({x}\right)}}+ 1}\right)}}+ 
\
\
\displaystyle
{\arctan \left({{{{\left({\sin \left({x}\right)}+{\cos \left({x}\right)}\right)}\ {\sqrt{2 \ {\cos \left({x}\right)}\ {\sin \left({x}\right)}}}}+ 1}\over{{\left({\sin \left({x}\right)}-{\cos \left({x}\right)}\right)}\ {\sqrt{2 \ {\cos \left({x}\right)}\ {\sin \left({x}\right)}}}}}\right)}- 
\
\
\displaystyle
{\arctan \left({{{{\left({\sin \left({x}\right)}+{\cos \left({x}\right)}\right)}\ {\sqrt{2 \ {\cos \left({x}\right)}\ {\sin \left({x}\right)}}}}- 1}\over{{\left({\sin \left({x}\right)}-{\cos \left({x}\right)}\right)}\ {\sqrt{2 \ {\cos \left({x}\right)}\ {\sin \left({x}\right)}}}}}\right)}- 
\
\
\displaystyle
{\arctan{\left({{{{\left({\sin \left({x}\right)}-{\cos \left({x}\right)}\right)}\ {\sqrt{2 \ {\cos \left({x}\right)}\ {\sin \left({x}\right)}}}}+{2 \ {{\cos \left({x}\right)}^{2}}}- 1}\over{{{\left({\sin \left({x}\right)}+{\cos \left({x}\right)}\right)}\ {\sqrt{2 \ {\cos \left({x}\right)}\ {\sin \left({x}\right)}}}}-{2 \ {\cos \left({x}\right)}\ {\sin \left({x}\right)}}}}\right)}}+ 
\
\
\displaystyle
{\arctan{\left({{{{\left({\sin \left({x}\right)}-{\cos \left({x}\right)}\right)}\ {\sqrt{2 \ {\cos \left({x}\right)}\ {\sin \left({x}\right)}}}}-{2 \ {{\cos \left({x}\right)}^{2}}}+ 1}\over{{{\left({\sin \left({x}\right)}+{\cos \left({x}\right)}\right)}\ {\sqrt{2 \ {\cos \left({x}\right)}\ {\sin \left({x}\right)}}}}+{2 \ {\cos \left({x}\right)}\ {\sin \left({x}\right)}}}}\right)}}
(61)
Type: Union(Expression(Integer),...)
fricas
integrate(sin(x)*sqrt(sin(2*x)), x)

\label{eq62}{\left(
\begin{array}{@{}l}
\displaystyle
{\log{\left({{{\left({2 \ {\sin \left({x}\right)}}+{2 \ {\cos \left({x}\right)}}\right)}\ {\sqrt{2 \ {\cos \left({x}\right)}\ {\sin \left({x}\right)}}}}+{4 \ {\cos \left({x}\right)}\ {\sin \left({x}\right)}}+ 1}\right)}}- 
\
\
\displaystyle
{\log{\left({{{\left(-{2 \ {\sin \left({x}\right)}}-{2 \ {\cos \left({x}\right)}}\right)}\ {\sqrt{2 \ {\cos \left({x}\right)}\ {\sin \left({x}\right)}}}}+{4 \ {\cos \left({x}\right)}\ {\sin \left({x}\right)}}+ 1}\right)}}+ 
\
\
\displaystyle
{\arctan \left({{{{\left({\sin \left({x}\right)}+{\cos \left({x}\right)}\right)}\ {\sqrt{2 \ {\cos \left({x}\right)}\ {\sin \left({x}\right)}}}}+ 1}\over{{\left({\sin \left({x}\right)}-{\cos \left({x}\right)}\right)}\ {\sqrt{2 \ {\cos \left({x}\right)}\ {\sin \left({x}\right)}}}}}\right)}- 
\
\
\displaystyle
{\arctan \left({{{{\left({\sin \left({x}\right)}+{\cos \left({x}\right)}\right)}\ {\sqrt{2 \ {\cos \left({x}\right)}\ {\sin \left({x}\right)}}}}- 1}\over{{\left({\sin \left({x}\right)}-{\cos \left({x}\right)}\right)}\ {\sqrt{2 \ {\cos \left({x}\right)}\ {\sin \left({x}\right)}}}}}\right)}- 
\
\
\displaystyle
{\arctan{\left({{{{\left({\sin \left({x}\right)}-{\cos \left({x}\right)}\right)}\ {\sqrt{2 \ {\cos \left({x}\right)}\ {\sin \left({x}\right)}}}}+{2 \ {{\cos \left({x}\right)}^{2}}}- 1}\over{{{\left({\sin \left({x}\right)}+{\cos \left({x}\right)}\right)}\ {\sqrt{2 \ {\cos \left({x}\right)}\ {\sin \left({x}\right)}}}}-{2 \ {\cos \left({x}\right)}\ {\sin \left({x}\right)}}}}\right)}}+ 
\
\
\displaystyle
{\arctan{\left({{{{\left({\sin \left({x}\right)}-{\cos \left({x}\right)}\right)}\ {\sqrt{2 \ {\cos \left({x}\right)}\ {\sin \left({x}\right)}}}}-{2 \ {{\cos \left({x}\right)}^{2}}}+ 1}\over{{{\left({\sin \left({x}\right)}+{\cos \left({x}\right)}\right)}\ {\sqrt{2 \ {\cos \left({x}\right)}\ {\sin \left({x}\right)}}}}+{2 \ {\cos \left({x}\right)}\ {\sin \left({x}\right)}}}}\right)}}- 
\
\
\displaystyle
{8 \ {\cos \left({x}\right)}\ {\sqrt{2 \ {\cos \left({x}\right)}\ {\sin \left({x}\right)}}}}
(62)
Type: Union(Expression(Integer),...)
fricas
)set output tex off
 
fricas
)set output algebra on
integrate((cos(x)-sin(x))*sqrt(sin(2*x)), x)
(70) - log 2 3 ((16 cos(x) + 4)sin(x) - 16 cos(x) + 20 cos(x)) * +--------------+ \|2 cos(x)sin(x) + 4 2 16 cos(x)sin(x) - 32 cos(x) + 32 cos(x) + 1 + log 2 3 +--------------+ ((- 16 cos(x) - 4)sin(x) + 16 cos(x) - 20 cos(x))\|2 cos(x)sin(x) + 4 2 16 cos(x)sin(x) - 32 cos(x) + 32 cos(x) + 1 + +--------------+ (8 sin(x) + 8 cos(x))\|2 cos(x)sin(x) / 16
Type: Union(Expression(Integer),...)
fricas
integrate(sin(x)^7/sin(2*x)^(7/2), x)
(71) 3 +--------------+ 5 cos(x) log((2 sin(x) + 2 cos(x))\|2 cos(x)sin(x) + 4 cos(x)sin(x) + 1) + - 3 5 cos(x) * +--------------+ log((- 2 sin(x) - 2 cos(x))\|2 cos(x)sin(x) + 4 cos(x)sin(x) + 1) + +--------------+ 3 (sin(x) + cos(x))\|2 cos(x)sin(x) + 1 5 cos(x) atan(--------------------------------------) +--------------+ (sin(x) - cos(x))\|2 cos(x)sin(x) + +--------------+ 3 (sin(x) + cos(x))\|2 cos(x)sin(x) - 1 - 5 cos(x) atan(--------------------------------------) +--------------+ (sin(x) - cos(x))\|2 cos(x)sin(x) + +--------------+ 2 3 (sin(x) - cos(x))\|2 cos(x)sin(x) + 2 cos(x) - 1 - 5 cos(x) atan(---------------------------------------------------) +--------------+ (sin(x) + cos(x))\|2 cos(x)sin(x) - 2 cos(x)sin(x) + +--------------+ 2 3 (sin(x) - cos(x))\|2 cos(x)sin(x) - 2 cos(x) + 1 5 cos(x) atan(---------------------------------------------------) +--------------+ (sin(x) + cos(x))\|2 cos(x)sin(x) + 2 cos(x)sin(x) + 2 +--------------+ (- 48 cos(x) + 8)\|2 cos(x)sin(x) / 3 320 cos(x)
Type: Union(Expression(Integer),...)
fricas
)set output algebra off
 
fricas
)set output tex on
integrate(cos(x)^7/sin(2*x)^(7/2), x)

\label{eq63}{\left(
\begin{array}{@{}l}
\displaystyle
{
\begin{array}{@{}l}
\displaystyle
{\left(-{5 \ {{\cos \left({x}\right)}^{2}}}+ 5 \right)}\ {\sin \left({x}\right)}\  \cdot 
\
\
\displaystyle
{\log{\left({{{\left({2 \ {\sin \left({x}\right)}}+{2 \ {\cos \left({x}\right)}}\right)}\ {\sqrt{2 \ {\cos \left({x}\right)}\ {\sin \left({x}\right)}}}}+{4 \ {\cos \left({x}\right)}\ {\sin \left({x}\right)}}+ 1}\right)}}
(63)
Type: Union(Expression(Integer),...)
fricas
integrate(sin(2*x)^(3/2)/sin(x)^5, x)

\label{eq64}{4 \ {{\cos \left({x}\right)}^{2}}\ {\sqrt{2 \ {\cos \left({x}\right)}\ {\sin \left({x}\right)}}}}\over{{\left({5 \ {{\cos \left({x}\right)}^{2}}}- 5 \right)}\ {\sin \left({x}\right)}}(64)
Type: Union(Expression(Integer),...)
fricas
integrate(1/(cos(x)^3*sqrt(sin(2*x))), x)

\label{eq65}{{\left({4 \ {{\cos \left({x}\right)}^{2}}}+ 1 \right)}\ {\sqrt{2 \ {\cos \left({x}\right)}\ {\sin \left({x}\right)}}}}\over{5 \ {{\cos \left({x}\right)}^{3}}}(65)
Type: Union(Expression(Integer),...)
fricas
integrate(1/(sin(x)*sin(2*x)^(3/2)), x)

\label{eq66}{{\left({4 \ {{\cos \left({x}\right)}^{2}}}- 3 \right)}\ {\sqrt{2 \ {\cos \left({x}\right)}\ {\sin \left({x}\right)}}}}\over{{6 \ {{\cos \left({x}\right)}^{3}}}-{6 \ {\cos \left({x}\right)}}}(66)
Type: Union(Expression(Integer),...)
fricas
-- Fails, nonelementary
integrate(cos(x)^2*(cos(2*x)-3*tan(x))/((sin(x)^2-sin(2*x))*sin(2*x)^(5/2)), x)

\label{eq67}\int^{
\displaystyle
x}{{{-{3 \ {{\cos \left({\%A}\right)}^{2}}\ {\tan \left({\%A}\right)}}+{{{\cos \left({\%A}\right)}^{2}}\ {\cos \left({2 \  \%A}\right)}}}\over{{\left({{\left({{\cos \left({2 \  \%A}\right)}^{2}}- 1 \right)}\ {\sin \left({2 \  \%A}\right)}}+{{\left({{\cos \left({\%A}\right)}^{2}}- 1 \right)}\ {{\cos \left({2 \  \%A}\right)}^{2}}}-{{\cos \left({\%A}\right)}^{2}}+ 1 \right)}\ {\sqrt{\sin \left({2 \  \%A}\right)}}}}\ {d \%A}}(67)
Type: Union(Expression(Integer),...)

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