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FriCASTimofeev1
FriCASTimofeev37  ← ↑ →  FriCASTimofeev9

FriCASTimofeev5
  
last edited 6 years ago by test1

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

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(1) ->
Timofeev (
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)set break resume

-- Problems 1 - 3 (p. 202-203)
integrate(cos(x)^2, x)
\label{eq1}\frac{{{\cos \left({x}\right)}\ {\sin \left({x}\right)}}+ x}{2}(1)
Type: Union(Expression(Integer),...)
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integrate(cos(x)^3, x)
\label{eq2}\frac{{\left({{\cos \left({x}\right)}^{2}}+ 2 \right)}\ {\sin \left({x}\right)}}{3}(2)
Type: Union(Expression(Integer),...)
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integrate(sin(x)^4, x)
\label{eq3}\frac{{{\left({2 \ {{\cos \left({x}\right)}^{3}}}-{5 \ {\cos \left({x}\right)}}\right)}\ {\sin \left({x}\right)}}+{3 \ x}}{8}(3)
Type: Union(Expression(Integer),...)
fricas
-- Problems 4 - 7 (p. 208)
integrate(cos(x)^6, x)
\label{eq4}\frac{{{\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}}{48}(4)
Type: Union(Expression(Integer),...)
fricas
integrate(sin(x)^8, x)
\label{eq5}\frac{{{\left({{48}\ {{\cos \left({x}\right)}^{7}}}-{{200}\ {{\cos \left({x}\right)}^{5}}}+{{326}\ {{\cos \left({x}\right)}^{3}}}-{{279}\ {\cos \left({x}\right)}}\right)}\ {\sin \left({x}\right)}}+{{105}\ x}}{384}(5)
Type: Union(Expression(Integer),...)
fricas
integrate(cos(1/4*%pi+1/2*x)^4, x)
\label{eq6}\frac{{{\left({4 \ {{\cos \left({\frac{{2 \ x}+ \pi}{4}}\right)}^{3}}}+{6 \ {\cos \left({\frac{{2 \ x}+ \pi}{4}}\right)}}\right)}\ {\sin \left({\frac{{2 \ x}+ \pi}{4}}\right)}}+{3 \ x}}{8}(6)
Type: Union(Expression(Integer),...)
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integrate(sin(-1/12*%pi+3*x)^3, x)
\label{eq7}\frac{{{\cos \left({\frac{{{36}\ x}- \pi}{12}}\right)}^{3}}-{3 \ {\cos \left({\frac{{{36}\ x}- \pi}{12}}\right)}}}{9}(7)
Type: Union(Expression(Integer),...)
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-- Problems 8 - 11 (p. 211)
integrate(1/sin(x)^6, x)
\label{eq8}\frac{-{8 \ {{\cos \left({x}\right)}^{5}}}+{{20}\ {{\cos \left({x}\right)}^{3}}}-{{15}\ {\cos \left({x}\right)}}}{{\left({{15}\ {{\cos \left({x}\right)}^{4}}}-{{30}\ {{\cos \left({x}\right)}^{2}}}+{15}\right)}\ {\sin \left({x}\right)}}(8)
Type: Union(Expression(Integer),...)
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integrate(csc(x)^7, x)
\label{eq9}\frac{{{\left(-{{15}\ {{\cos \left({x}\right)}^{6}}}+{{45}\ {{\cos \left({x}\right)}^{4}}}-{{45}\ {{\cos \left({x}\right)}^{2}}}+{15}\right)}\ {\log \left({\frac{{\cos \left({x}\right)}+ 1}{2}}\right)}}+{{\left({{15}\ {{\cos \left({x}\right)}^{6}}}-{{45}\ {{\cos \left({x}\right)}^{4}}}+{{45}\ {{\cos \left({x}\right)}^{2}}}-{15}\right)}\ {\log \left({\frac{-{\cos \left({x}\right)}+ 1}{2}}\right)}}+{{30}\ {{\cos \left({x}\right)}^{5}}}-{{80}\ {{\cos \left({x}\right)}^{3}}}+{{66}\ {\cos \left({x}\right)}}}{{{96}\ {{\cos \left({x}\right)}^{6}}}-{{288}\ {{\cos \left({x}\right)}^{4}}}+{{288}\ {{\cos \left({x}\right)}^{2}}}-{96}}(9)
Type: Union(Expression(Integer),...)
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integrate(1/cos(x)^12, x)
\label{eq10}\frac{{\left({{256}\ {{\cos \left({x}\right)}^{10}}}+{{128}\ {{\cos \left({x}\right)}^{8}}}+{{96}\ {{\cos \left({x}\right)}^{6}}}+{{80}\ {{\cos \left({x}\right)}^{4}}}+{{70}\ {{\cos \left({x}\right)}^{2}}}+{63}\right)}\ {\sin \left({x}\right)}}{{693}\ {{\cos \left({x}\right)}^{11}}}(10)
Type: Union(Expression(Integer),...)
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integrate(1/cos(1/4*%pi+3*x)^3, x)
\label{eq11}\frac{{{{\cos \left({\frac{{{12}\ x}+ \pi}{4}}\right)}^{2}}\ {\log \left({{\sin \left({\frac{{{12}\ x}+ \pi}{4}}\right)}+ 1}\right)}}-{{{\cos \left({\frac{{{12}\ x}+ \pi}{4}}\right)}^{2}}\ {\log \left({-{\sin \left({\frac{{{12}\ x}+ \pi}{4}}\right)}+ 1}\right)}}+{2 \ {\sin \left({\frac{{{12}\ x}+ \pi}{4}}\right)}}}{{12}\ {{\cos \left({\frac{{{12}\ x}+ \pi}{4}}\right)}^{2}}}(11)
Type: Union(Expression(Integer),...)
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-- Problems 12 - 14 (p. 213)
integrate(tan(x)^6, x)
\label{eq12}\frac{{3 \ {{\tan \left({x}\right)}^{5}}}-{5 \ {{\tan \left({x}\right)}^{3}}}+{{15}\ {\tan \left({x}\right)}}-{{15}\ x}}{15}(12)
Type: Union(Expression(Integer),...)
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integrate(1/tan(x)^5, x)
\label{eq13}\frac{{2 \ {{\tan \left({x}\right)}^{4}}\ {\log \left({\frac{{\tan \left({x}\right)}^{2}}{{{\tan \left({x}\right)}^{2}}+ 1}}\right)}}+{3 \ {{\tan \left({x}\right)}^{4}}}+{2 \ {{\tan \left({x}\right)}^{2}}}- 1}{4 \ {{\tan \left({x}\right)}^{4}}}(13)
Type: Union(Expression(Integer),...)
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integrate(cot(-3/4*%pi+1/3*x)^4, x)
\label{eq14}\frac{{{\left({x \ {\cos \left({\frac{{4 \ x}-{9 \ \pi}}{6}}\right)}}- x \right)}\ {\sin \left({\frac{{4 \ x}-{9 \ \pi}}{6}}\right)}}+{4 \ {{\cos \left({\frac{{4 \ x}-{9 \ \pi}}{6}}\right)}^{2}}}+{2 \ {\cos \left({\frac{{4 \ x}-{9 \ \pi}}{6}}\right)}}- 2}{{\left({\cos \left({\frac{{4 \ x}-{9 \ \pi}}{6}}\right)}- 1 \right)}\ {\sin \left({\frac{{4 \ x}-{9 \ \pi}}{6}}\right)}}(14)
Type: Union(Expression(Integer),...)
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-- Problems 15 - 20 (p. 219-220)
integrate(cos(x)^6*sin(x)^4, x)
\label{eq15}\frac{{{\left({{128}\ {{\cos \left({x}\right)}^{9}}}-{{176}\ {{\cos \left({x}\right)}^{7}}}+{8 \ {{\cos \left({x}\right)}^{5}}}+{{1 0}\ {{\cos \left({x}\right)}^{3}}}+{{15}\ {\cos \left({x}\right)}}\right)}\ {\sin \left({x}\right)}}+{{15}\ x}}{1280}(15)
Type: Union(Expression(Integer),...)
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integrate(cos(x)^6*sin(x)^7, x)
\label{eq16}\frac{{{231}\ {{\cos \left({x}\right)}^{13}}}-{{819}\ {{\cos \left({x}\right)}^{11}}}+{{1001}\ {{\cos \left({x}\right)}^{9}}}-{{429}\ {{\cos \left({x}\right)}^{7}}}}{3003}(16)
Type: Union(Expression(Integer),...)
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integrate(sin(x)^11/cos(x), x)
\label{eq17}\frac{-{{120}\ {\log \left({-{\cos \left({x}\right)}}\right)}}+{{12}\ {{\cos \left({x}\right)}^{10}}}-{{75}\ {{\cos \left({x}\right)}^{8}}}+{{200}\ {{\cos \left({x}\right)}^{6}}}-{{300}\ {{\cos \left({x}\right)}^{4}}}+{{300}\ {{\cos \left({x}\right)}^{2}}}}{120}(17)
Type: Union(Expression(Integer),...)
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integrate(1/(cos(x)^6*sin(x)^6), x)
\label{eq18}\frac{-{{256}\ {{\cos \left({x}\right)}^{10}}}+{{640}\ {{\cos \left({x}\right)}^{8}}}-{{480}\ {{\cos \left({x}\right)}^{6}}}+{{80}\ {{\cos \left({x}\right)}^{4}}}+{{10}\ {{\cos \left({x}\right)}^{2}}}+ 3}{{\left({{15}\ {{\cos \left({x}\right)}^{9}}}-{{30}\ {{\cos \left({x}\right)}^{7}}}+{{15}\ {{\cos \left({x}\right)}^{5}}}\right)}\ {\sin \left({x}\right)}}(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),...)
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integrate(1/(cos(1/4*%pi+2*x)*sin(1/4*%pi+2*x)^3), x)
\label{eq20}\frac{{{\left(-{{\cos \left({\frac{{8 \ x}+ \pi}{4}}\right)}^{2}}+ 1 \right)}\ {\log \left({{\cos \left({\frac{{8 \ x}+ \pi}{4}}\right)}^{2}}\right)}}+{{\left({{\cos \left({\frac{{8 \ x}+ \pi}{4}}\right)}^{2}}- 1 \right)}\ {\log \left({\frac{-{{\cos \left({\frac{{8 \ x}+ \pi}{4}}\right)}^{2}}+ 1}{4}}\right)}}+ 1}{{4 \ {{\cos \left({\frac{{8 \ x}+ \pi}{4}}\right)}^{2}}}- 4}(20)
Type: Union(Expression(Integer),...)
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-- Problems 21 - 29 (p. 223)
integrate(sec(x)^2*tan(x)^2, x)
\label{eq21}\frac{{\left(-{{\cos \left({x}\right)}^{2}}+ 1 \right)}\ {\sin \left({x}\right)}}{3 \ {{\cos \left({x}\right)}^{3}}}(21)
Type: Union(Expression(Integer),...)
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integrate(cot(x)^3*csc(x), x)
\label{eq22}\frac{{3 \ {{\cos \left({x}\right)}^{2}}}- 2}{{\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}\frac{1}{3 \ {{\cos \left({x}\right)}^{3}}}(23)
Type: Union(Expression(Integer),...)
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integrate(cot(x)^2*csc(x)^3, x)
\label{eq24}\frac{{{\left({{\cos \left({x}\right)}^{4}}-{2 \ {{\cos \left({x}\right)}^{2}}}+ 1 \right)}\ {\log \left({\frac{{\cos \left({x}\right)}+ 1}{2}}\right)}}+{{\left(-{{\cos \left({x}\right)}^{4}}+{2 \ {{\cos \left({x}\right)}^{2}}}- 1 \right)}\ {\log \left({\frac{-{\cos \left({x}\right)}+ 1}{2}}\right)}}-{2 \ {{\cos \left({x}\right)}^{3}}}-{2 \ {\cos \left({x}\right)}}}{{{16}\ {{\cos \left({x}\right)}^{4}}}-{{32}\ {{\cos \left({x}\right)}^{2}}}+{16}}(24)
Type: Union(Expression(Integer),...)
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integrate(cos(x)^3/sin(x)^7, x)
\label{eq25}\frac{{3 \ {{\cos \left({x}\right)}^{2}}}- 1}{{{12}\ {{\cos \left({x}\right)}^{6}}}-{{36}\ {{\cos \left({x}\right)}^{4}}}+{{36}\ {{\cos \left({x}\right)}^{2}}}-{12}}(25)
Type: Union(Expression(Integer),...)
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integrate(sqrt(sec(x)^3)*tan(x)^5, x)
\label{eq26}\frac{{\left({{154}\ {{\cos \left({x}\right)}^{4}}}-{{132}\ {{\cos \left({x}\right)}^{2}}}+{42}\right)}\ {\sqrt{\frac{1}{{\cos \left({x}\right)}^{3}}}}}{{231}\ {{\cos \left({x}\right)}^{4}}}(26)
Type: Union(Expression(Integer),...)
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integrate(sec(x)^4*tan(x)^(3/2), x)
\label{eq27}\frac{{\left(-{8 \ {{\cos \left({x}\right)}^{4}}}-{2 \ {{\cos \left({x}\right)}^{2}}}+{10}\right)}\ {\sqrt{\frac{\sin \left({x}\right)}{\cos \left({x}\right)}}}}{{45}\ {{\cos \left({x}\right)}^{4}}}(27)
Type: Union(Expression(Integer),...)
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integrate(cot(x)^4*csc(x)^3, x)
\label{eq28}\frac{{{\left(-{3 \ {{\cos \left({x}\right)}^{6}}}+{9 \ {{\cos \left({x}\right)}^{4}}}-{9 \ {{\cos \left({x}\right)}^{2}}}+ 3 \right)}\ {\log \left({\frac{{\cos \left({x}\right)}+ 1}{2}}\right)}}+{{\left({3 \ {{\cos \left({x}\right)}^{6}}}-{9 \ {{\cos \left({x}\right)}^{4}}}+{9 \ {{\cos \left({x}\right)}^{2}}}- 3 \right)}\ {\log \left({\frac{-{\cos \left({x}\right)}+ 1}{2}}\right)}}+{6 \ {{\cos \left({x}\right)}^{5}}}+{{16}\ {{\cos \left({x}\right)}^{3}}}-{6 \ {\cos \left({x}\right)}}}{{{96}\ {{\cos \left({x}\right)}^{6}}}-{{288}\ {{\cos \left({x}\right)}^{4}}}+{{288}\ {{\cos \left({x}\right)}^{2}}}-{96}}(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)
             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),...)
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)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}\frac{{{\left({2 \ {{\cos \left({x}\right)}^{4}}}+{{\left({2 \ a}- 6 \right)}\ {{\cos \left({x}\right)}^{2}}}-{2 \ a}\right)}\ {\sin \left({x}\right)}}+{2 \ {{\cos \left({x}\right)}^{5}}}+{{\left({6 \ x}- 3 \right)}\ {{\cos \left({x}\right)}^{3}}}+{{\left(-{6 \ x}+ a + 1 \right)}\ {\cos \left({x}\right)}}}{{2 \ {{\cos \left({x}\right)}^{3}}}-{2 \ {\cos \left({x}\right)}}}(29)
Type: Union(Expression(Integer),...)
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integrate((4-3*cos(x))*(1-1/2*sin(x))^4, x)
\label{eq30}\frac{{{\left(-{{18}\ {{\cos \left({x}\right)}^{4}}}+{{30}\ {{\cos \left({x}\right)}^{3}}}+{{756}\ {{\cos \left({x}\right)}^{2}}}-{{1515}\ {\cos \left({x}\right)}}-{2178}\right)}\ {\sin \left({x}\right)}}+{{180}\ {{\cos \left({x}\right)}^{4}}}-{{320}\ {{\cos \left({x}\right)}^{3}}}-{{1800}\ {{\cos \left({x}\right)}^{2}}}+{{4800}\ {\cos \left({x}\right)}}+{{3405}\ x}}{480}(30)
Type: Union(Expression(Integer),...)
fricas
integrate((1/2-3*cot(x))*(3-2*cot(x))^3, x)
\label{eq31}\frac{{{\left({4 \ {\cos \left({2 \ x}\right)}}- 4 \right)}\ {\sin \left({2 \ x}\right)}\ {\log \left({\frac{-{\cos \left({2 \ x}\right)}+ 1}{2}}\right)}}+{{\left(-{{285}\ x \ {\cos \left({2 \ x}\right)}}+{{285}\ x}-{224}\right)}\ {\sin \left({2 \ x}\right)}}-{{296}\ {{\cos \left({2 \ x}\right)}^{2}}}+{{32}\ {\cos \left({2 \ x}\right)}}+{328}}{{\left({2 \ {\cos \left({2 \ x}\right)}}- 2 \right)}\ {\sin \left({2 \ x}\right)}}(31)
Type: Union(Expression(Integer),...)
fricas
-- Problems 33 - 36 (p. 229)
integrate(cos(5*x)/cos(x)^5, x)
\label{eq32}\frac{{{\left(-{{50}\ {{\cos \left({x}\right)}^{2}}}+ 5 \right)}\ {\sin \left({x}\right)}}+{{48}\ x \ {{\cos \left({x}\right)}^{3}}}}{3 \ {{\cos \left({x}\right)}^{3}}}(32)
Type: Union(Expression(Integer),...)
fricas
integrate(cos(4*x)/cos(x), x)
\label{eq33}\frac{{3 \ {\log \left({{\sin \left({x}\right)}+ 1}\right)}}-{3 \ {\log \left({-{\sin \left({x}\right)}+ 1}\right)}}+{{\left({{1 6}\ {{\cos \left({x}\right)}^{2}}}-{16}\right)}\ {\sin \left({x}\right)}}}{6}(33)
Type: Union(Expression(Integer),...)
fricas
integrate(cos(x)*cos(4*x), x)
\label{eq34}\frac{{\left({{24}\ {{\cos \left({x}\right)}^{4}}}-{8 \ {{\cos \left({x}\right)}^{2}}}- 1 \right)}\ {\sin \left({x}\right)}}{1 5}(34)
Type: Union(Expression(Integer),...)
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integrate(cos(4*x)/cos(x)^5, x)
\label{eq35}\frac{{{35}\ {{\cos \left({x}\right)}^{4}}\ {\log \left({{\sin \left({x}\right)}+ 1}\right)}}-{{35}\ {{\cos \left({x}\right)}^{4}}\ {\log \left({-{\sin \left({x}\right)}+ 1}\right)}}+{{\left(-{{58}\ {{\cos \left({x}\right)}^{2}}}+ 4 \right)}\ {\sin \left({x}\right)}}}{{1 6}\ {{\cos \left({x}\right)}^{4}}}(35)
Type: Union(Expression(Integer),...)
fricas
-- Problems 37 - 39 (p. 233)
integrate(cos(x)^4*cos(4*x), x)
\label{eq36}\frac{{{\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}}{48}(36)
Type: Union(Expression(Integer),...)
fricas
integrate(cos(5*x)/sin(x)^5, x)
\label{eq37}\frac{{{\left({{64}\ {{\cos \left({x}\right)}^{4}}}-{{128}\ {{\cos \left({x}\right)}^{2}}}+{64}\right)}\ {\log \left({\frac{\sin \left({x}\right)}{2}}\right)}}-{{24}\ {{\cos \left({x}\right)}^{2}}}+{23}}{{4 \ {{\cos \left({x}\right)}^{4}}}-{8 \ {{\cos \left({x}\right)}^{2}}}+ 4}(37)
Type: Union(Expression(Integer),...)
fricas
integrate(sin(4*x)/sin(x)^4, x)
\label{eq38}\frac{{{\left(-{8 \ {{\cos \left({x}\right)}^{2}}}+ 8 \right)}\ {\log \left({\frac{\sin \left({x}\right)}{2}}\right)}}+ 2}{{{\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}\frac{-{{\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)}}+{2 \ {\sqrt{3}}\ {\log \left({\frac{-{{\cos \left({x}\right)}^{2}}+ 1}{4}}\right)}}-{2 \ {\arctan \left({\frac{{4 \ {\sqrt{3}}\ {\cos \left({x}\right)}\ {\sin \left({x}\right)}}+{\sqrt{3}}}{{6 \ {{\cos \left({x}\right)}^{2}}}- 3}}\right)}}}{8 \ {\sqrt{3}}}(39)
Type: Union(Expression(Integer),...)
fricas
integrate(cos(x)^2/(cos(3*x)*sin(x)), x)
\label{eq40}\frac{{2 \ {\log \left({\frac{\sin \left({x}\right)}{2}}\right)}}-{\log \left({{4 \ {{\cos \left({x}\right)}^{2}}}- 3}\right)}}{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}\frac{\log \left({\frac{{{\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}}{{{\left({{32}\ {\cos \left({x}\right)}}+{24}\right)}\ {\sin \left({x}\right)}}+{{24}\ {\cos \left({x}\right)}}+{25}}}\right)}{2 \ {\sqrt{23}}}(42)
Type: Union(Expression(Integer),...)
fricas
integrate(1/(4-3*cos(x)^2+5*sin(x)^2), x)
\label{eq43}-{\frac{\arctan \left({\frac{{{10}\ {{\cos \left({x}\right)}^{2}}}- 9}{6 \ {\cos \left({x}\right)}\ {\sin \left({x}\right)}}}\right)}{6}}(43)
Type: Union(Expression(Integer),...)
fricas
integrate(1/(4+4*cot(x)+tan(x)), x)
\label{eq44}\frac{{{\left(-{3 \ {\tan \left({x}\right)}}- 6 \right)}\ {\log \left({\frac{{{\tan \left({x}\right)}^{2}}+{4 \ {\tan \left({x}\right)}}+ 4}{{{\tan \left({x}\right)}^{2}}+ 1}}\right)}}+{{\left({8 \ x}- 8 \right)}\ {\tan \left({x}\right)}}+{{16}\ x}+ 4}{{{50}\ {\tan \left({x}\right)}}+{100}}(44)
Type: Union(Expression(Integer),...)
fricas
integrate(1/(2*sec(x)+sin(x))^2, x)
\label{eq45}\frac{{{\left({4 \ {\cos \left({x}\right)}\ {\sin \left({x}\right)}}+ 8 \right)}\ {\arctan \left({\frac{{8 \ {\sqrt{15}}\ {\cos \left({x}\right)}\ {\sin \left({x}\right)}}+{\sqrt{15}}}{{{30}\ {{\cos \left({x}\right)}^{2}}}-{15}}}\right)}}+{{\sqrt{15}}\ {{\cos \left({x}\right)}^{2}}}-{8 \ {\sqrt{15}}}}{{{15}\ {\sqrt{15}}\ {\cos \left({x}\right)}\ {\sin \left({x}\right)}}+{{30}\ {\sqrt{15}}}}(45)
Type: Union(Expression(Integer),...)
fricas
integrate(1/(cos(x)+2*sec(x))^2, x)
\label{eq46}\frac{{{\left(-{{\cos \left({x}\right)}^{2}}- 2 \right)}\ {\arctan \left({\frac{{5 \ {\sqrt{6}}\ {{\cos \left({x}\right)}^{2}}}-{2 \ {\sqrt{6}}}}{{12}\ {\cos \left({x}\right)}\ {\sin \left({x}\right)}}}\right)}}+{2 \ {\sqrt{6}}\ {\cos \left({x}\right)}\ {\sin \left({x}\right)}}}{{{1 2}\ {\sqrt{6}}\ {{\cos \left({x}\right)}^{2}}}+{{24}\ {\sqrt{6}}}}(46)
Type: Union(Expression(Integer),...)
fricas
integrate((5-tan(x)-6*tan(x)^2)/(1+3*tan(x))^3, x)
\label{eq47}\frac{{{\left(-{{504}\ {{\tan \left({x}\right)}^{2}}}-{{336}\ {\tan \left({x}\right)}}-{56}\right)}\ {\log \left({\frac{{9 \ {{\tan \left({x}\right)}^{2}}}+{6 \ {\tan \left({x}\right)}}+ 1}{{{\tan \left({x}\right)}^{2}}+ 1}}\right)}}+{{\left(-{{1206}\ x}+ 9 \right)}\ {{\tan \left({x}\right)}^{2}}}+{{\left(-{{804}\ x}-{864}\right)}\ {\tan \left({x}\right)}}-{{134}\ x}-{639}}{{{4 500}\ {{\tan \left({x}\right)}^{2}}}+{{3000}\ {\tan \left({x}\right)}}+{500}}(47)
Type: Union(Expression(Integer),...)
fricas
-- Problems 50 - 56 (p. 260)
integrate(cos(x)^2/cos(3*x), x)
\label{eq48}\frac{{\log \left({{2 \ {\sin \left({x}\right)}}+ 1}\right)}-{\log \left({-{2 \ {\sin \left({x}\right)}}+ 1}\right)}}{4}(48)
Type: Union(Expression(Integer),...)
fricas
integrate(sin(x)/cos(2*x), x)
\label{eq49}\frac{\log \left({\frac{{2 \ {\sqrt{2}}\ {{\cos \left({x}\right)}^{2}}}+{4 \ {\cos \left({x}\right)}}+{\sqrt{2}}}{{2 \ {{\cos \left({x}\right)}^{2}}}- 1}}\right)}{2 \ {\sqrt{2}}}(49)
Type: Union(Expression(Integer),...)
fricas
integrate(sin(x)^2/cos(2*x), x)
\label{eq50}\frac{{\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}}{8}(50)
Type: Union(Expression(Integer),...)
fricas
integrate(sin(x)^3/cos(3*x), x)
\label{eq51}\frac{-{\log \left({{4 \ {{\cos \left({x}\right)}^{2}}}- 3}\right)}+{8 \ {\log \left({-{\cos \left({x}\right)}}\right)}}}{24}(51)
Type: Union(Expression(Integer),...)
fricas
integrate(cos(x)/sin(3*x), x)
\label{eq52}\frac{{2 \ {\log \left({\frac{\sin \left({x}\right)}{2}}\right)}}-{\log \left({{4 \ {{\cos \left({x}\right)}^{2}}}- 1}\right)}}{6}(52)
Type: Union(Expression(Integer),...)
fricas
integrate(sin(x)/sin(4*x), x)
\label{eq53}\frac{-{{\sqrt{2}}\ {\log \left({{\sin \left({x}\right)}+ 1}\right)}}+{2 \ {\log \left({\frac{{4 \ {\sin \left({x}\right)}}-{2 \ {\sqrt{2}}\ {{\cos \left({x}\right)}^{2}}}+{3 \ {\sqrt{2}}}}{{2 \ {{\cos \left({x}\right)}^{2}}}- 1}}\right)}}+{{\sqrt{2}}\ {\log \left({-{\sin \left({x}\right)}+ 1}\right)}}}{8 \ {\sqrt{2}}}(53)
Type: Union(Expression(Integer),...)
fricas
integrate(sin(x)^3/sin(4*x), x)
\label{eq54}\frac{-{{\sqrt{2}}\ {\log \left({{\sin \left({x}\right)}+ 1}\right)}}+{\log \left({\frac{{4 \ {\sin \left({x}\right)}}-{2 \ {\sqrt{2}}\ {{\cos \left({x}\right)}^{2}}}+{3 \ {\sqrt{2}}}}{{2 \ {{\cos \left({x}\right)}^{2}}}- 1}}\right)}+{{\sqrt{2}}\ {\log \left({-{\sin \left({x}\right)}+ 1}\right)}}}{8 \ {\sqrt{2}}}(54)
Type: Union(Expression(Integer),...)
fricas
-- Problems 57 - 61 (p. 266)
integrate(sqrt(1+sin(2*x)), x)
\label{eq55}\frac{{\left({\sin \left({2 \ x}\right)}-{\cos \left({2 \ x}\right)}- 1 \right)}\ {\sqrt{{\sin \left({2 \ x}\right)}+ 1}}}{{\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}\frac{{\sqrt{2}}\ {\log \left({\frac{{{\left(-{\cos \left({2 \ x}\right)}+ 3 \right)}\ {\sqrt{{\cos \left({2 \ x}\right)}+ 1}}}+{2 \ {\sqrt{2}}\ {\sin \left({2 \ x}\right)}}}{{\left({\cos \left({2 \ x}\right)}+ 1 \right)}\ {\sqrt{{\cos \left({2 \ x}\right)}+ 1}}}}\right)}}{4}(56)
Type: Union(Expression(Integer),...)
fricas
integrate(1/(1-cos(3*x))^(3/2), x)
\label{eq57}\frac{{{\left({{\sqrt{2}}\ {\cos \left({3 \ x}\right)}}-{\sqrt{2}}\right)}\ {\sin \left({3 \ x}\right)}\ {\log \left({\frac{{{\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)}}}{{\left({\cos \left({3 \ x}\right)}- 1 \right)}\ {\sqrt{-{\cos \left({3 \ x}\right)}+ 1}}}}\right)}}+{{\left({4 \ {\cos \left({3 \ x}\right)}}+ 4 \right)}\ {\sqrt{-{\cos \left({3 \ x}\right)}+ 1}}}}{{\left({{2 4}\ {\cos \left({3 \ x}\right)}}-{24}\right)}\ {\sin \left({3 \ x}\right)}}(57)
Type: Union(Expression(Integer),...)
fricas
integrate((1-sin(2/3*x))^(5/2), x)
\label{eq58}\frac{{\left({{\left({3 \ {{\cos \left({\frac{2 \ x}{3}}\right)}^{2}}}+{{14}\ {\cos \left({\frac{2 \ x}{3}}\right)}}-{32}\right)}\ {\sin \left({\frac{2 \ x}{3}}\right)}}+{3 \ {{\cos \left({\frac{2 \ x}{3}}\right)}^{3}}}-{{11}\ {{\cos \left({\frac{2 \ x}{3}}\right)}^{2}}}-{{46}\ {\cos \left({\frac{2 \ x}{3}}\right)}}-{32}\right)}\ {\sqrt{-{\sin \left({\frac{2 \ x}{3}}\right)}+ 1}}}{{5 \ {\sin \left({\frac{2 \ x}{3}}\right)}}-{5 \ {\cos \left({\frac{2 \ x}{3}}\right)}}- 5}(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}\frac{-{{12}\ {{\root{4}\of{{2 \ {\sin \left({x}\right)}}+ 1}}^{3}}}+{{\left(-{2 \ {\sin \left({x}\right)}}-{{\cos \left({x}\right)}^{2}}+ 2 \right)}\ {{\root{4}\of{{2 \ {\sin \left({x}\right)}}+ 1}}^{2}}}}{{6 \ {\sin \left({x}\right)}}+ 3}(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 %%CY1  + (- 2 %%CY0 - 2)%%CY1 - 3 %%CY0  - 2 %%CY0 - 1
         + 
               +------+                           +------+
           (- \|tan(x)  + 1)%%CY1 + (- %%CY0 - 1)\|tan(x)  + %%CY0 + 1
      *
         log
                ((32 %%CY0 + 4)%%CY1 + 4 %%CY0 + 2)
             *
                 +----------------------------------------------------------+
                 |         2                                 2
                \|- 3 %%CY1  + (- 2 %%CY0 - 2)%%CY1 - 3 %%CY0  - 2 %%CY0 - 1
            + 
                                   2              2
              (- 32 %%CY0 - 4)%%CY1  + (- 32 %%CY0  - 32 %%CY0 - 2)%%CY1
            + 
                 +------+          2
              6 \|tan(x)  - 4 %%CY0  - 2 %%CY0 - 3
     + 
               +------+
             (\|tan(x)  - 1)
          *
              +----------------------------------------------------------+
              |         2                                 2
             \|- 3 %%CY1  + (- 2 %%CY0 - 2)%%CY1 - 3 %%CY0  - 2 %%CY0 - 1
         + 
               +------+                           +------+
           (- \|tan(x)  + 1)%%CY1 + (- %%CY0 - 1)\|tan(x)  + %%CY0 + 1
      *
         log
                ((- 32 %%CY0 - 4)%%CY1 - 4 %%CY0 - 2)
             *
                 +----------------------------------------------------------+
                 |         2                                 2
                \|- 3 %%CY1  + (- 2 %%CY0 - 2)%%CY1 - 3 %%CY0  - 2 %%CY0 - 1
            + 
                                   2              2
              (- 32 %%CY0 - 4)%%CY1  + (- 32 %%CY0  - 32 %%CY0 - 2)%%CY1
            + 
                 +------+          2
              6 \|tan(x)  - 4 %%CY0  - 2 %%CY0 - 3
     + 
             +------+
         (2 \|tan(x)  - 2)%%CY1
      *
         log
                                 2            2
              (64 %%CY0 + 8)%%CY1  + (64 %%CY0  + 64 %%CY0 + 4)%%CY1
            + 
                 +------+           3           2
              6 \|tan(x)  + 64 %%CY0  + 64 %%CY0  + 32 %%CY0 - 1
     + 
                  +------+
         (2 %%CY0\|tan(x)  - 2 %%CY0)
      *
                +------+           3           2
         log(6 \|tan(x)  - 64 %%CY0  - 56 %%CY0  - 28 %%CY0 + 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}\frac{{\log \left({{{\left({{\left(-{{16}\ {{\cos \left({x}\right)}^{2}}}- 4 \right)}\ {\sin \left({x}\right)}}+{{16}\ {{\cos \left({x}\right)}^{3}}}-{{20}\ {\cos \left({x}\right)}}\right)}\ {\sqrt{2 \ {\cos \left({x}\right)}\ {\sin \left({x}\right)}}}}+{{16}\ {\cos \left({x}\right)}\ {\sin \left({x}\right)}}-{{32}\ {{\cos \left({x}\right)}^{4}}}+{{32}\ {{\cos \left({x}\right)}^{2}}}+ 1}\right)}-{2 \ {\arctan \left({\frac{{{\sin \left({x}\right)}\ {\sqrt{2 \ {\cos \left({x}\right)}\ {\sin \left({x}\right)}}}}-{{\cos \left({x}\right)}\ {\sin \left({x}\right)}}+{{\cos \left({x}\right)}^{2}}}{{{\cos \left({x}\right)}\ {\sqrt{2 \ {\cos \left({x}\right)}\ {\sin \left({x}\right)}}}}-{{\cos \left({x}\right)}\ {\sin \left({x}\right)}}-{{\cos \left({x}\right)}^{2}}+ 1}}\right)}}+{2 \ {\arctan \left({\frac{{{\left({3 \ {\sin \left({x}\right)}}+{3 \ {\cos \left({x}\right)}}\right)}\ {\sqrt{2 \ {\cos \left({x}\right)}\ {\sin \left({x}\right)}}}}-{6 \ {\cos \left({x}\right)}\ {\sin \left({x}\right)}}- 1}{{{\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)}}}{8}(60)
Type: Union(Expression(Integer),...)
fricas
integrate(cos(x)/sqrt(sin(2*x)), x)
\label{eq61}\frac{-{\log \left({{{\left({{\left(-{{16}\ {{\cos \left({x}\right)}^{2}}}- 4 \right)}\ {\sin \left({x}\right)}}+{{16}\ {{\cos \left({x}\right)}^{3}}}-{{20}\ {\cos \left({x}\right)}}\right)}\ {\sqrt{2 \ {\cos \left({x}\right)}\ {\sin \left({x}\right)}}}}+{{16}\ {\cos \left({x}\right)}\ {\sin \left({x}\right)}}-{{32}\ {{\cos \left({x}\right)}^{4}}}+{{32}\ {{\cos \left({x}\right)}^{2}}}+ 1}\right)}-{2 \ {\arctan \left({\frac{{{\sin \left({x}\right)}\ {\sqrt{2 \ {\cos \left({x}\right)}\ {\sin \left({x}\right)}}}}-{{\cos \left({x}\right)}\ {\sin \left({x}\right)}}+{{\cos \left({x}\right)}^{2}}}{{{\cos \left({x}\right)}\ {\sqrt{2 \ {\cos \left({x}\right)}\ {\sin \left({x}\right)}}}}-{{\cos \left({x}\right)}\ {\sin \left({x}\right)}}-{{\cos \left({x}\right)}^{2}}+ 1}}\right)}}+{2 \ {\arctan \left({\frac{{{\left({3 \ {\sin \left({x}\right)}}+{3 \ {\cos \left({x}\right)}}\right)}\ {\sqrt{2 \ {\cos \left({x}\right)}\ {\sin \left({x}\right)}}}}-{6 \ {\cos \left({x}\right)}\ {\sin \left({x}\right)}}- 1}{{{\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)}}}{8}(61)
Type: Union(Expression(Integer),...)
fricas
integrate(sin(x)*sqrt(sin(2*x)), x)
\label{eq62}\frac{-{\log \left({{{\left({{\left(-{{16}\ {{\cos \left({x}\right)}^{2}}}- 4 \right)}\ {\sin \left({x}\right)}}+{{16}\ {{\cos \left({x}\right)}^{3}}}-{{20}\ {\cos \left({x}\right)}}\right)}\ {\sqrt{2 \ {\cos \left({x}\right)}\ {\sin \left({x}\right)}}}}+{{16}\ {\cos \left({x}\right)}\ {\sin \left({x}\right)}}-{{32}\ {{\cos \left({x}\right)}^{4}}}+{{32}\ {{\cos \left({x}\right)}^{2}}}+ 1}\right)}-{2 \ {\arctan \left({\frac{{{\sin \left({x}\right)}\ {\sqrt{2 \ {\cos \left({x}\right)}\ {\sin \left({x}\right)}}}}-{{\cos \left({x}\right)}\ {\sin \left({x}\right)}}+{{\cos \left({x}\right)}^{2}}}{{{\cos \left({x}\right)}\ {\sqrt{2 \ {\cos \left({x}\right)}\ {\sin \left({x}\right)}}}}-{{\cos \left({x}\right)}\ {\sin \left({x}\right)}}-{{\cos \left({x}\right)}^{2}}+ 1}}\right)}}+{2 \ {\arctan \left({\frac{{{\left({3 \ {\sin \left({x}\right)}}+{3 \ {\cos \left({x}\right)}}\right)}\ {\sqrt{2 \ {\cos \left({x}\right)}\ {\sin \left({x}\right)}}}}-{6 \ {\cos \left({x}\right)}\ {\sin \left({x}\right)}}- 1}{{{\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)}}-{8 \ {\cos \left({x}\right)}\ {\sqrt{2 \ {\cos \left({x}\right)}\ {\sin \left({x}\right)}}}}}{16}(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
     + 
                             +--------------+
       (4 sin(x) + 4 cos(x))\|2 cos(x)sin(x)
  /
     8
Type: Union(Expression(Integer),...)
fricas
integrate(sin(x)^7/sin(2*x)^(7/2), x)

   (71)
       -
                    3
            5 cos(x)
         *
            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
     + 
                                 +--------------+                        2
                  3       sin(x)\|2 cos(x)sin(x)  - cos(x)sin(x) + cos(x)
       - 10 cos(x) atan(----------------------------------------------------)
                               +--------------+                        2
                        cos(x)\|2 cos(x)sin(x)  - cos(x)sin(x) - cos(x)  + 1
     + 
                  3
         10 cos(x)
      *
                                    +--------------+
              (3 sin(x) + 3 cos(x))\|2 cos(x)sin(x)  - 6 cos(x)sin(x) - 1
         atan(-----------------------------------------------------------)
                                     +--------------+           2
                   (sin(x) - cos(x))\|2 cos(x)sin(x)  + 2 cos(x)  - 1
     + 
                   2      +--------------+            3
       (- 48 cos(x)  + 8)\|2 cos(x)sin(x)  - 48 cos(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}\frac{{{\left({5 \ {{\cos \left({x}\right)}^{2}}}- 5 \right)}\ {\sin \left({x}\right)}\ {\log \left({{{\left({{\left(-{{16}\ {{\cos \left({x}\right)}^{2}}}- 4 \right)}\ {\sin \left({x}\right)}}+{{16}\ {{\cos \left({x}\right)}^{3}}}-{{20}\ {\cos \left({x}\right)}}\right)}\ {\sqrt{2 \ {\cos \left({x}\right)}\ {\sin \left({x}\right)}}}}+{{16}\ {\cos \left({x}\right)}\ {\sin \left({x}\right)}}-{{32}\ {{\cos \left({x}\right)}^{4}}}+{{32}\ {{\cos \left({x}\right)}^{2}}}+ 1}\right)}}+{{\left(-{{10}\ {{\cos \left({x}\right)}^{2}}}+{10}\right)}\ {\sin \left({x}\right)}\ {\arctan \left({\frac{{{\sin \left({x}\right)}\ {\sqrt{2 \ {\cos \left({x}\right)}\ {\sin \left({x}\right)}}}}-{{\cos \left({x}\right)}\ {\sin \left({x}\right)}}+{{\cos \left({x}\right)}^{2}}}{{{\cos \left({x}\right)}\ {\sqrt{2 \ {\cos \left({x}\right)}\ {\sin \left({x}\right)}}}}-{{\cos \left({x}\right)}\ {\sin \left({x}\right)}}-{{\cos \left({x}\right)}^{2}}+ 1}}\right)}}+{{\left({{10}\ {{\cos \left({x}\right)}^{2}}}-{10}\right)}\ {\sin \left({x}\right)}\ {\arctan \left({\frac{{{\left({3 \ {\sin \left({x}\right)}}+{3 \ {\cos \left({x}\right)}}\right)}\ {\sqrt{2 \ {\cos \left({x}\right)}\ {\sin \left({x}\right)}}}}-{6 \ {\cos \left({x}\right)}\ {\sin \left({x}\right)}}- 1}{{{\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)}}+{{\left({{48}\ {{\cos \left({x}\right)}^{2}}}-{40}\right)}\ {\sqrt{2 \ {\cos \left({x}\right)}\ {\sin \left({x}\right)}}}}+{{\left({{48}\ {{\cos \left({x}\right)}^{2}}}-{48}\right)}\ {\sin \left({x}\right)}}}{{\left({{320}\ {{\cos \left({x}\right)}^{2}}}-{320}\right)}\ {\sin \left({x}\right)}}(63)
Type: Union(Expression(Integer),...)
fricas
integrate(sin(2*x)^(3/2)/sin(x)^5, x)
\label{eq64}\frac{{4 \ {{\cos \left({x}\right)}^{2}}\ {\sqrt{2 \ {\cos \left({x}\right)}\ {\sin \left({x}\right)}}}}+{{\left({4 \ {{\cos \left({x}\right)}^{2}}}- 4 \right)}\ {\sin \left({x}\right)}}}{{\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}\frac{{{\left({4 \ {{\cos \left({x}\right)}^{2}}}+ 1 \right)}\ {\sqrt{2 \ {\cos \left({x}\right)}\ {\sin \left({x}\right)}}}}+{4 \ {{\cos \left({x}\right)}^{3}}}}{5 \ {{\cos \left({x}\right)}^{3}}}(65)
Type: Union(Expression(Integer),...)
fricas
integrate(1/(sin(x)*sin(2*x)^(3/2)), x)
\label{eq66}\frac{{{\left({4 \ {{\cos \left({x}\right)}^{2}}}- 3 \right)}\ {\sqrt{2 \ {\cos \left({x}\right)}\ {\sin \left({x}\right)}}}}+{4 \ {{\cos \left({x}\right)}^{3}}}-{4 \ {\cos \left({x}\right)}}}{{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}{{\frac{-{3 \ {{\cos \left({\%A}\right)}^{2}}\ {\tan \left({\%A}\right)}}+{{{\cos \left({\%A}\right)}^{2}}\ {\cos \left({2 \ \%A}\right)}}}{{\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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