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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),...)
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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)}+{\cos \left({{{{12}\  x}+ \pi}\over 4}\right)}+ 1}\over{{\cos \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)}-{\cos \left({{{{12}\  x}+ \pi}\over 4}\right)}- 1}\over{{\cos \left({{{{12}\  x}+ \pi}\over 4}\right)}+ 1}}\right)}}}+ 
\
\
\displaystyle
{\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}{\left(
\begin{array}{@{}l}
\displaystyle
-{2 \ {{\tan \left({x}\right)}^{4}}\ {\log \left({{{\tan \left({x}\right)}^{2}}+ 1}\right)}}+ 
\
\
\displaystyle
{4 \ {{\tan \left({x}\right)}^{4}}\ {\log \left({\tan \left({x}\right)}\right)}}+{2 \ {{\tan \left({x}\right)}^{2}}}- 1 
(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({2 \over{{\cos \left({x}\right)}+ 1}}\right)}}-{{120}\ {\log \left({-{{2 \ {\cos \left({x}\right)}}\over{{\cos \left({x}\right)}+ 1}}}\right)}}+ 
\
\
\displaystyle
{{12}\ {{\cos \left({x}\right)}^{10}}}-{{75}\ {{\cos \left({x}\right)}^{8}}}+{{200}\ {{\cos \left({x}\right)}^{6}}}- 
\
\
\displaystyle
{{300}\ {{\cos \left({x}\right)}^{4}}}+{{300}\ {{\cos \left({x}\right)}^{2}}}
(17)
Type: Union(Expression(Integer),...)
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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),...)
fricas
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({4 \ {{\cos \left({{{8 \  x}+ \pi}\over 4}\right)}^{2}}}- 4 \right)}\ {\log \left({{\sin \left({{{8 \  x}+ \pi}\over 4}\right)}\over{{\cos \left({{{8 \  x}+ \pi}\over 4}\right)}+ 1}}\right)}}+ 
\
\
\displaystyle
{{\left(-{4 \ {{\cos \left({{{8 \  x}+ \pi}\over 4}\right)}^{2}}}+ 4 \right)}\ {\log \left({-{{2 \ {\cos \left({{{8 \  x}+ \pi}\over 4}\right)}}\over{{\cos \left({{{8 \  x}+ \pi}\over 4}\right)}+ 1}}}\right)}}+ 
\
\
\displaystyle
{{\cos \left({{{8 \  x}+ \pi}\over 4}\right)}^{2}}+ 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),...)
fricas
integrate(sec(x)^3*tan(x), x)

\label{eq23}{{{\cos \left({x}\right)}^{3}}+ 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({{\sin \left({x}\right)}\over{{\cos \left({x}\right)}+ 1}}\right)}}- 
\
\
\displaystyle
{{\cos \left({x}\right)}^{3}}-{\cos \left({x}\right)}
(24)
Type: Union(Expression(Integer),...)
fricas
integrate(cos(x)^3/sin(x)^7, x)

\label{eq25}{-{{\cos \left({x}\right)}^{6}}+{3 \ {{\cos \left({x}\right)}^{4}}}+{3 \ {{\cos \left({x}\right)}^{2}}}- 1}\over{{{24}\ {{\cos \left({x}\right)}^{6}}}-{{72}\ {{\cos \left({x}\right)}^{4}}}+{{72}\ {{\cos \left({x}\right)}^{2}}}-{24}}(25)
Type: Union(Expression(Integer),...)
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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),...)
fricas
)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) 2x + %pi 2x + %pi sin(--------) + cos(--------) + 1 2x + %pi 4 4 4 - cos(--------) log(---------------------------------) 4 2x + %pi cos(--------) + 1 4 + 2x + %pi 2x + %pi sin(--------) - cos(--------) - 1 2x + %pi 4 4 4 cos(--------) log(---------------------------------) 4 2x + %pi cos(--------) + 1 4 + 2x + %pi 2 2x + %pi (- cos(--------) + 2)sin(--------) 4 4 / 2x + %pi 4 4cos(--------) 4
Type: Union(Expression(Integer),...)
fricas
)set output tex on
 
fricas
)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({4 \ {{\cos \left({x}\right)}^{4}}}+{{\left({4 \  a}-{12}\right)}\ {{\cos \left({x}\right)}^{2}}}-{4 \  a}\right)}\ {\sin \left({x}\right)}}+ 
\
\
\displaystyle
{4 \ {{\cos \left({x}\right)}^{5}}}+{{\left({{12}\  x}+ a - 8 \right)}\ {{\cos \left({x}\right)}^{3}}}+ 
\
\
\displaystyle
{{\left(-{{12}\  x}+ a + 4 \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({8 \ {\cos \left({2 \  x}\right)}}- 8 \right)}\ {\sin \left({2 \  x}\right)}\ {\log \left({{\sin \left({2 \  x}\right)}\over{{\cos \left({2 \  x}\right)}+ 1}}\right)}}+ 
\
\
\displaystyle
{{\left(-{4 \ {\cos \left({2 \  x}\right)}}+ 4 \right)}\ {\sin \left({2 \  x}\right)}\ {\log \left({2 \over{{\cos \left({2 \  x}\right)}+ 1}}\right)}}+ 
\
\
\displaystyle
{{\left({{\left(-{{285}\  x}-{112}\right)}\ {\cos \left({2 \  x}\right)}}+{{285}\  x}-{112}\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)}+{\cos \left({x}\right)}+ 1}\over{{\cos \left({x}\right)}+ 1}}\right)}}- 
\
\
\displaystyle
{3 \ {\log \left({{{\sin \left({x}\right)}-{\cos \left({x}\right)}- 1}\over{{\cos \left({x}\right)}+ 1}}\right)}}+{{\left({8 \ {{\cos \left({x}\right)}^{2}}}- 8 \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)}+{\cos \left({x}\right)}+ 1}\over{{\cos \left({x}\right)}+ 1}}\right)}}- 
\
\
\displaystyle
{{35}\ {{\cos \left({x}\right)}^{4}}\ {\log \left({{{\sin \left({x}\right)}-{\cos \left({x}\right)}- 1}\over{{\cos \left({x}\right)}+ 1}}\right)}}+ 
\
\
\displaystyle
{{\left(-{{29}\ {{\cos \left({x}\right)}^{2}}}+ 2 \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({{512}\ {{\cos \left({x}\right)}^{4}}}-{{1024}\ {{\cos \left({x}\right)}^{2}}}+{512}\right)}\ {\log \left({{\sin \left({x}\right)}\over{{\cos \left({x}\right)}+ 1}}\right)}}+ 
\
\
\displaystyle
{{\left({
\begin{array}{@{}l}
\displaystyle
-{{512}\ {{\cos \left({x}\right)}^{4}}}+ 
\
\
\displaystyle
{{1024}\ {{\cos \left({x}\right)}^{2}}}-{512}
(37)
Type: Union(Expression(Integer),...)
fricas
integrate(sin(4*x)/sin(x)^4, x)

\label{eq38}{\left(
\begin{array}{@{}l}
\displaystyle
{{\left(-{8 \ {{\cos \left({x}\right)}^{2}}}+ 8 \right)}\ {\log \left({{\sin \left({x}\right)}\over{{\cos \left({x}\right)}+ 1}}\right)}}+ 
\
\
\displaystyle
{{\left({8 \ {{\cos \left({x}\right)}^{2}}}- 8 \right)}\ {\log \left({2 \over{{\cos \left({x}\right)}+ 1}}\right)}}+{{\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({{{4 \ {\cos \left({x}\right)}\ {\sin \left({x}\right)}}+ 4}\over{{{\cos \left({x}\right)}^{2}}+{2 \ {\cos \left({x}\right)}}+ 1}}\right)}}+ 
\
\
\displaystyle
{2 \ {\sqrt{3}}\ {\log \left({{\sin \left({x}\right)}\over{{\cos \left({x}\right)}+ 1}}\right)}}- 
\
\
\displaystyle
{2 \ {\arctan \left({{{{\sqrt{3}}\ {\sin \left({x}\right)}}-{{\sqrt{3}}\ {\cos \left({x}\right)}}-{\sqrt{3}}}\over{{3 \ {\cos \left({x}\right)}}+ 3}}\right)}}- 
\
\
\displaystyle
{2 \ {\arctan{\left({{{{\left({2 \ {\cos \left({x}\right)}}+ 4 \right)}\ {\sin \left({x}\right)}}+{3 \ {{\cos \left({x}\right)}^{2}}}+{2 \ {\cos \left({x}\right)}}- 1}\over{{{\sqrt{3}}\ {{\cos \left({x}\right)}^{2}}}+{2 \ {\sqrt{3}}\ {\cos \left({x}\right)}}+{\sqrt{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{{\cos \left({x}\right)}+ 1}}\right)}}-{\log \left({{{{16}\ {{\cos \left({x}\right)}^{2}}}-{12}}\over{{{\cos \left({x}\right)}^{2}}+{2 \ {\cos \left({x}\right)}}+ 1}}\right)}}\over 2(40)
Type: Union(Expression(Integer),...)
fricas
integrate(sin(2*x)/(cos(x)^4+sin(x)^4), x)

\label{eq41}\begin{array}{@{}l}
\displaystyle
-{\arctan \left({{{7 \ {{\cos \left({x}\right)}^{3}}}+{5 \ {{\cos \left({x}\right)}^{2}}}-{4 \ {\cos \left({x}\right)}}- 4}\over{{{\cos \left({x}\right)}^{3}}+{3 \ {{\cos \left({x}\right)}^{2}}}+{3 \ {\cos \left({x}\right)}}+ 1}}\right)}+ 
\
\
\displaystyle
{\arctan \left({{{3 \ {\cos \left({x}\right)}}+ 1}\over{{4 \ {\cos \left({x}\right)}}+ 4}}\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)
(43) - atan +-+ (104765547927309152411486\|3 - 181459251892870078413729)sin(x) + +-+ (18277149216643579325555\|3 - 31656951060734835109266)cos(x) + +-+ 18277149216643579325555\|3 - 31656951060734835109266 / +-+ (63313902121469670218532\|3 - 109662895299861475953330)cos(x) + +-+ 63313902121469670218532\|3 - 109662895299861475953330 / +-+ \|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({4 \ {\sqrt{23}}}-{23}\right)}\ {\sin \left({x}\right)}}+{{\left(-{{19}\ {\sqrt{23}}}+{92}\right)}\ {\cos \left({x}\right)}}-{{20}\ {\sqrt{23}}}+{92}}\over{{4 \ {\sin \left({x}\right)}}+{4 \ {\cos \left({x}\right)}}+ 3}}\right)}}\over{\sqrt{23}}(42)
Type: Union(Expression(Integer),...)
fricas
integrate(1/(4-3*cos(x)^2+5*sin(x)^2), x)

\label{eq43}{{\arctan \left({{{\left({{17}\ {\cos \left({x}\right)}}+{18}\right)}\ {\sin \left({x}\right)}}\over{{3 \ {{\cos \left({x}\right)}^{2}}}+{6 \ {\cos \left({x}\right)}}+ 3}}\right)}+{\arctan \left({{\sin \left({x}\right)}\over{{6 \ {\cos \left({x}\right)}}+ 6}}\right)}}\over 3(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}}+ 1}\right)}}+ 
\
\
\displaystyle
{{\left(-{6 \ {\tan \left({x}\right)}}-{12}\right)}\ {\log \left({{\tan \left({x}\right)}+ 2}\right)}}+{8 \  x \ {\tan \left({x}\right)}}+ 
\
\
\displaystyle
{{16}\  x}+{20}
(44)
Type: Union(Expression(Integer),...)
fricas
integrate(1/(2*sec(x)+sin(x))^2, x)

\label{eq45}{\left(
\begin{array}{@{}l}
\displaystyle
{{\left({{16}\ {\cos \left({x}\right)}\ {\sin \left({x}\right)}}+{32}\right)}\ {\arctan{\left({{{2 \ {\sqrt{15}}\ {\sin \left({x}\right)}}-{{\sqrt{15}}\ {\cos \left({x}\right)}}-{\sqrt{15}}}\over{{{15}\ {\cos \left({x}\right)}}+{15}}}\right)}}}+ 
\
\
\displaystyle
{{\left({{16}\ {\cos \left({x}\right)}\ {\sin \left({x}\right)}}+{32}\right)}\ {\arctan{\left({{{{\left({4 \ {\cos \left({x}\right)}}+ 8 \right)}\ {\sin \left({x}\right)}}+{3 \ {{\cos \left({x}\right)}^{2}}}+{2 \ {\cos \left({x}\right)}}- 1}\over{{{\sqrt{15}}\ {{\cos \left({x}\right)}^{2}}}+{2 \ {\sqrt{15}}\ {\cos \left({x}\right)}}+{\sqrt{15}}}}\right)}}}+ 
\
\
\displaystyle
{7 \ {\sqrt{15}}\ {\cos \left({x}\right)}\ {\sin \left({x}\right)}}+{2 \ {\sqrt{15}}\ {{\cos \left({x}\right)}^{2}}}-{2 \ {\sqrt{1
5}}}
(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({{{\sqrt{6}}\ {\sin \left({x}\right)}}\over{{4 \ {\cos \left({x}\right)}}+ 4}}\right)}}+ 
\
\
\displaystyle
{{\left({{\cos \left({x}\right)}^{2}}+ 2 \right)}\ {\arctan{\left({{{\left({\cos \left({x}\right)}+ 4 \right)}\ {\sin \left({x}\right)}}\over{{{\sqrt{6}}\ {{\cos \left({x}\right)}^{2}}}+{2 \ {\sqrt{6}}\ {\cos \left({x}\right)}}+{\sqrt{6}}}}\right)}}}+ 
\
\
\displaystyle
{{\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({{252}\ {{\tan \left({x}\right)}^{2}}}+{{168}\ {\tan \left({x}\right)}}+{28}\right)}\ {\log \left({{{\tan \left({x}\right)}^{2}}+ 1}\right)}}+ 
\
\
\displaystyle
{{\left(-{{504}\ {{\tan \left({x}\right)}^{2}}}-{{336}\ {\tan \left({x}\right)}}-{56}\right)}\ {\log \left({{3 \ {\tan \left({x}\right)}}+ 1}\right)}}- 
\
\
\displaystyle
{{603}\  x \ {{\tan \left({x}\right)}^{2}}}+{{\left(-{{402}\  x}-{435}\right)}\ {\tan \left({x}\right)}}-{{67}\  x}- 
\
\
\displaystyle
{320}
(47)
Type: Union(Expression(Integer),...)
fricas
-- Problems 50 - 56 (p. 260)
integrate(cos(x)^2/cos(3*x), x)

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

\label{eq49}{\log \left({{{{\left({6 \ {\sqrt{2}}}+ 8 \right)}\ {{\cos \left({x}\right)}^{2}}}+{{\left({8 \ {\sqrt{2}}}+{12}\right)}\ {\cos \left({x}\right)}}+{3 \ {\sqrt{2}}}+ 4}\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}{\left(
\begin{array}{@{}l}
\displaystyle
-{\log \left({{{2 \ {\sin \left({x}\right)}}-{2 \ {\cos \left({x}\right)}}}\over{{\cos \left({x}\right)}+ 1}}\right)}+ 
\
\
\displaystyle
{\log \left({{-{2 \ {\sin \left({x}\right)}}-{2 \ {\cos \left({x}\right)}}}\over{{\cos \left({x}\right)}+ 1}}\right)}-{2 \  x}
(50)
Type: Union(Expression(Integer),...)
fricas
integrate(sin(x)^3/cos(3*x), x)

\label{eq51}{\left(
\begin{array}{@{}l}
\displaystyle
-{\log \left({{{{16}\ {{\cos \left({x}\right)}^{2}}}-{12}}\over{{{\cos \left({x}\right)}^{2}}+{2 \ {\cos \left({x}\right)}}+ 1}}\right)}-{6 \ {\log \left({2 \over{{\cos \left({x}\right)}+ 1}}\right)}}+ 
\
\
\displaystyle
{8 \ {\log \left({-{{2 \ {\cos \left({x}\right)}}\over{{\cos \left({x}\right)}+ 1}}}\right)}}
(51)
Type: Union(Expression(Integer),...)
fricas
integrate(cos(x)/sin(3*x), x)

\label{eq52}{{2 \ {\log \left({{\sin \left({x}\right)}\over{{\cos \left({x}\right)}+ 1}}\right)}}-{\log \left({{{{16}\ {{\cos \left({x}\right)}^{2}}}- 4}\over{{{\cos \left({x}\right)}^{2}}+{2 \ {\cos \left({x}\right)}}+ 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)}+{\cos \left({x}\right)}+ 1}\over{{\cos \left({x}\right)}+ 1}}\right)}}+ 
\
\
\displaystyle
{{\sqrt{2}}\ {\log \left({{{\sin \left({x}\right)}-{\cos \left({x}\right)}- 1}\over{{\cos \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)}
(53)
Type: Union(Expression(Integer),...)
fricas
integrate(sin(x)^3/sin(4*x), x)

\label{eq54}{\left(
\begin{array}{@{}l}
\displaystyle
-{2 \ {\sqrt{2}}\ {\log \left({{{\sin \left({x}\right)}+{\cos \left({x}\right)}+ 1}\over{{\cos \left({x}\right)}+ 1}}\right)}}+ 
\
\
\displaystyle
{2 \ {\sqrt{2}}\ {\log \left({{{\sin \left({x}\right)}-{\cos \left({x}\right)}- 1}\over{{\cos \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)}
(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) |------ + 2sin(x) + 2cos(x) +------+ | \|cos(x) +-+ |sin(x) |------------------------------------------ + \|2 |------ - 1 \| cos(x) \|cos(x) + - +-+ 4\|2 * 1 atan(---------------------------------------------------------------) +----------------------------------------+ | +------+ | +-+ |sin(x) |2\|2 cos(x) |------ + 2sin(x) + 2cos(x) +------+ | \|cos(x) +-+ |sin(x) |---------------------------------------- + \|2 |------ + 1 \| cos(x) \|cos(x) + +------+ +-+ |sin(x) 2\|2 cos(x) |------ + 2sin(x) + 2cos(x) +-+ \|cos(x) - \|2 log(----------------------------------------) cos(x) + +------+ +-+ |sin(x) - 2\|2 cos(x) |------ + 2sin(x) + 2cos(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+-------+ 2log(\|tan(5x) + 1) - log(- \|tan(5x) + tan(5x)\|tan(5x) + 1) + 3+-------+2 +-+ 2\|tan(5x) - 1 2\|3 atan(----------------) +-+ \|3 / 20
Type: Union(Expression(Integer),...)
fricas
integrate(1/(4+3*tan(2*x))^(3/2), x)
(64) - +-------------------+ +-+ |3sin(2x) + 4cos(2x) 36\|5 |------------------- \| cos(2x) * atan +-+ 125\|5 / +--+ \|10 * ROOT +-------------------+ +-+ +--+ |3sin(2x) + 4cos(2x) - 9375\|5 \|10 cos(2x) |------------------- \| cos(2x) + 46875sin(2x) + 140625cos(2x) / cos(2x) + +-------------------+ +--+ |3sin(2x) + 4cos(2x) +-+ 125\|10 |------------------- - 375\|5 \| cos(2x) + - +-------------------+ +-+ |3sin(2x) + 4cos(2x) 36\|5 |------------------- \| cos(2x) * atan +-+ 125\|5 / +--+ \|10 * ROOT +-------------------+ +-+ +--+ |3sin(2x) + 4cos(2x) 9375\|5 \|10 cos(2x) |------------------- \| cos(2x) + 46875sin(2x) + 140625cos(2x) / cos(2x) + +-------------------+ +--+ |3sin(2x) + 4cos(2x) +-+ 125\|10 |------------------- + 375\|5 \| cos(2x) + +-------------------+ +-+ |3sin(2x) + 4cos(2x) 13\|5 |------------------- \| cos(2x) * log +-------------------+ +-+ +--+ |3sin(2x) + 4cos(2x) 9375\|5 \|10 cos(2x) |------------------- + 46875sin(2x) \| cos(2x) + 140625cos(2x) / cos(2x) + - +-------------------+ +-+ |3sin(2x) + 4cos(2x) 13\|5 |------------------- \| cos(2x) * log +-------------------+ +-+ +--+ |3sin(2x) + 4cos(2x) - 9375\|5 \|10 cos(2x) |------------------- + 46875sin(2x) \| cos(2x) + 140625cos(2x) / cos(2x) + +--+ - 60\|10 / +-------------------+ +--+ |3sin(2x) + 4cos(2x) 500\|10 |------------------- \| cos(2x)
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) +-------------------+ |- 3sin(x) + 4cos(x) (12sin(x) - 32cos(x)) |------------------- \| cos(x) + 2 (- 3sin(x) + 4cos(x))log(cos(x) ) + 2 - 24cos(x)sin(x) + 7cos(x) + 9 (3sin(x) - 4cos(x))log(-------------------------------) 4 / 18sin(x) - 24cos(x)
Type: Union(Expression(Integer),...)
fricas
integrate(tan(x)/(-1+sqrt(tan(x)))^2, x)
(66) +------+ (- \|tan(x) + 1) * +------------------------------------------------------+ | 2 2 \|- 3%%BS1 + (- 2%%BS0 - 2)%%BS1 - 3%%BS0 - 2%%BS0 - 1 + +------+ +------+ (- \|tan(x) + 1)%%BS1 + (- %%BS0 - 1)\|tan(x) + %%BS0 + 1 * log ((32%%BS0 + 4)%%BS1 + 4%%BS0 + 2) * +------------------------------------------------------+ | 2 2 \|- 3%%BS1 + (- 2%%BS0 - 2)%%BS1 - 3%%BS0 - 2%%BS0 - 1 + 2 2 (- 32%%BS0 - 4)%%BS1 + (- 32%%BS0 - 32%%BS0 - 2)%%BS1 + +------+ 2 6\|tan(x) - 4%%BS0 - 2%%BS0 - 3 + +------+ (\|tan(x) - 1) * +------------------------------------------------------+ | 2 2 \|- 3%%BS1 + (- 2%%BS0 - 2)%%BS1 - 3%%BS0 - 2%%BS0 - 1 + +------+ +------+ (- \|tan(x) + 1)%%BS1 + (- %%BS0 - 1)\|tan(x) + %%BS0 + 1 * log ((- 32%%BS0 - 4)%%BS1 - 4%%BS0 - 2) * +------------------------------------------------------+ | 2 2 \|- 3%%BS1 + (- 2%%BS0 - 2)%%BS1 - 3%%BS0 - 2%%BS0 - 1 + 2 2 (- 32%%BS0 - 4)%%BS1 + (- 32%%BS0 - 32%%BS0 - 2)%%BS1 + +------+ 2 6\|tan(x) - 4%%BS0 - 2%%BS0 - 3 + +------+ (2\|tan(x) - 2)%%BS1 * log 2 2 +------+ (64%%BS0 + 8)%%BS1 + (64%%BS0 + 64%%BS0 + 4)%%BS1 + 6\|tan(x) + 3 2 64%%BS0 + 64%%BS0 + 32%%BS0 - 1 + +------+ (2%%BS0\|tan(x) - 2%%BS0) * +------+ 3 2 log(6\|tan(x) - 64%%BS0 - 56%%BS0 - 28%%BS0 + 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
{2 \ {\arctan \left({{{2 \ {\sqrt{2 \ {\cos \left({x}\right)}\ {\sin \left({x}\right)}}}}+{\sin \left({x}\right)}+{\cos \left({x}\right)}}\over{{\sin \left({x}\right)}-{\cos \left({x}\right)}}}\right)}}- 
\
\
\displaystyle
{2 \ {\arctan \left({{{2 \ {\sqrt{2 \ {\cos \left({x}\right)}\ {\sin \left({x}\right)}}}}-{\sin \left({x}\right)}-{\cos \left({x}\right)}}\over{{\sin \left({x}\right)}-{\cos \left({x}\right)}}}\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
{2 \ {\arctan \left({{{2 \ {\sqrt{2 \ {\cos \left({x}\right)}\ {\sin \left({x}\right)}}}}+{\sin \left({x}\right)}+{\cos \left({x}\right)}}\over{{\sin \left({x}\right)}-{\cos \left({x}\right)}}}\right)}}- 
\
\
\displaystyle
{2 \ {\arctan \left({{{2 \ {\sqrt{2 \ {\cos \left({x}\right)}\ {\sin \left({x}\right)}}}}-{\sin \left({x}\right)}-{\cos \left({x}\right)}}\over{{\sin \left({x}\right)}-{\cos \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
{2 \ {\arctan \left({{{2 \ {\sqrt{2 \ {\cos \left({x}\right)}\ {\sin \left({x}\right)}}}}+{\sin \left({x}\right)}+{\cos \left({x}\right)}}\over{{\sin \left({x}\right)}-{\cos \left({x}\right)}}}\right)}}- 
\
\
\displaystyle
{2 \ {\arctan \left({{{2 \ {\sqrt{2 \ {\cos \left({x}\right)}\ {\sin \left({x}\right)}}}}-{\sin \left({x}\right)}-{\cos \left({x}\right)}}\over{{\sin \left({x}\right)}-{\cos \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 +-------------+ ((16cos(x) + 4)sin(x) - 16cos(x) + 20cos(x))\|2cos(x)sin(x) + 4 2 16cos(x)sin(x) - 32cos(x) + 32cos(x) + 1 + log 2 3 +-------------+ ((- 16cos(x) - 4)sin(x) + 16cos(x) - 20cos(x))\|2cos(x)sin(x) + 4 2 16cos(x)sin(x) - 32cos(x) + 32cos(x) + 1 + +-------------+ (8sin(x) + 8cos(x))\|2cos(x)sin(x) / 16
Type: Union(Expression(Integer),...)
fricas
integrate(sin(x)^7/sin(2*x)^(7/2), x)
(71) 3 +-------------+ 5cos(x) log((2sin(x) + 2cos(x))\|2cos(x)sin(x) + 4cos(x)sin(x) + 1) + 3 +-------------+ - 5cos(x) log((- 2sin(x) - 2cos(x))\|2cos(x)sin(x) + 4cos(x)sin(x) + 1) + +-------------+ 3 2\|2cos(x)sin(x) + sin(x) + cos(x) - 10cos(x) atan(-----------------------------------) sin(x) - cos(x) + +-------------+ 3 2\|2cos(x)sin(x) - sin(x) - cos(x) - 10cos(x) atan(-----------------------------------) sin(x) - cos(x) + 2 +-------------+ (- 48cos(x) + 8)\|2cos(x)sin(x) / 3 320cos(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
integrate(cos(x)^2*(cos(2*x)-3*tan(x))/((sin(x)^2-sin(2*x))*sin(2*x)^(5/2)), x)
>> Error detected within library code: Imaginary part is nonzero. Cannot retract.
Continuing to read the file...

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