Optimal. Leaf size=79 \[ \frac {\sin (x) \cos ^4(x)}{3 \sin ^{\frac {5}{2}}(2 x)}+\frac {\sin ^2(x) \cos ^3(x)}{2 \sin ^{\frac {5}{2}}(2 x)}-\frac {5 \sin ^5(x) \tanh ^{-1}\left (\frac {\sqrt {\tan (x)}}{\sqrt {2}}\right )}{2 \sqrt {2} \sin ^{\frac {5}{2}}(2 x) \tan ^{\frac {5}{2}}(x)} \]
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Rubi [A] time = 0.57, antiderivative size = 79, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {4390, 898, 1262, 207} \[ \frac {\sin ^2(x) \cos ^3(x)}{2 \sin ^{\frac {5}{2}}(2 x)}+\frac {\sin (x) \cos ^4(x)}{3 \sin ^{\frac {5}{2}}(2 x)}-\frac {5 \sin ^5(x) \tanh ^{-1}\left (\frac {\sqrt {\tan (x)}}{\sqrt {2}}\right )}{2 \sqrt {2} \sin ^{\frac {5}{2}}(2 x) \tan ^{\frac {5}{2}}(x)} \]
Antiderivative was successfully verified.
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Rule 207
Rule 898
Rule 1262
Rule 4390
Rubi steps
\begin {align*} \int \frac {\cos ^2(x) \sin (x)}{\left (\sin ^2(x)-\sin (2 x)\right ) \sin ^{\frac {5}{2}}(2 x)} \, dx &=\frac {\sin ^5(x) \int \frac {\csc ^2(x) \sqrt {\tan (x)}}{\sin ^2(x)-\sin (2 x)} \, dx}{\sin ^{\frac {5}{2}}(2 x) \tan ^{\frac {5}{2}}(x)}\\ &=\frac {\sin ^5(x) \operatorname {Subst}\left (\int \frac {-1-x^2}{(2-x) x^{5/2}} \, dx,x,\tan (x)\right )}{\sin ^{\frac {5}{2}}(2 x) \tan ^{\frac {5}{2}}(x)}\\ &=\frac {\left (2 \sin ^5(x)\right ) \operatorname {Subst}\left (\int \frac {-1-x^4}{x^4 \left (2-x^2\right )} \, dx,x,\sqrt {\tan (x)}\right )}{\sin ^{\frac {5}{2}}(2 x) \tan ^{\frac {5}{2}}(x)}\\ &=\frac {\left (2 \sin ^5(x)\right ) \operatorname {Subst}\left (\int \left (-\frac {1}{2 x^4}-\frac {1}{4 x^2}+\frac {5}{4 \left (-2+x^2\right )}\right ) \, dx,x,\sqrt {\tan (x)}\right )}{\sin ^{\frac {5}{2}}(2 x) \tan ^{\frac {5}{2}}(x)}\\ &=\frac {\cos ^4(x) \sin (x)}{3 \sin ^{\frac {5}{2}}(2 x)}+\frac {\cos ^3(x) \sin ^2(x)}{2 \sin ^{\frac {5}{2}}(2 x)}+\frac {\left (5 \sin ^5(x)\right ) \operatorname {Subst}\left (\int \frac {1}{-2+x^2} \, dx,x,\sqrt {\tan (x)}\right )}{2 \sin ^{\frac {5}{2}}(2 x) \tan ^{\frac {5}{2}}(x)}\\ &=\frac {\cos ^4(x) \sin (x)}{3 \sin ^{\frac {5}{2}}(2 x)}+\frac {\cos ^3(x) \sin ^2(x)}{2 \sin ^{\frac {5}{2}}(2 x)}-\frac {5 \tanh ^{-1}\left (\frac {\sqrt {\tan (x)}}{\sqrt {2}}\right ) \sin ^5(x)}{2 \sqrt {2} \sin ^{\frac {5}{2}}(2 x) \tan ^{\frac {5}{2}}(x)}\\ \end {align*}
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Mathematica [C] time = 4.96, size = 183, normalized size = 2.32 \[ \frac {1}{96} \sqrt {\sin (2 x)} \sec (x) \left (2 \cot ^2(x)+6 \cot (x)+2 \csc ^2(x)+15 \sqrt {2} \sqrt {\frac {\cos (x)}{\cos (x)-1}} \sqrt {\tan \left (\frac {x}{2}\right )} F\left (\left .\sin ^{-1}\left (\frac {1}{\sqrt {\tan \left (\frac {x}{2}\right )}}\right )\right |-1\right )-15 \sqrt {2} \sqrt {\frac {\cos (x)}{\cos (x)-1}} \sqrt {\tan \left (\frac {x}{2}\right )} \Pi \left (-\frac {2}{-1+\sqrt {5}};\left .\sin ^{-1}\left (\frac {1}{\sqrt {\tan \left (\frac {x}{2}\right )}}\right )\right |-1\right )-15 \sqrt {2} \sqrt {\frac {\cos (x)}{\cos (x)-1}} \sqrt {\tan \left (\frac {x}{2}\right )} \Pi \left (\frac {1}{2} \left (-1+\sqrt {5}\right );\left .\sin ^{-1}\left (\frac {1}{\sqrt {\tan \left (\frac {x}{2}\right )}}\right )\right |-1\right )-2\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 1.04, size = 120, normalized size = 1.52 \[ -\frac {4 \, \sqrt {2} \sqrt {\cos \relax (x) \sin \relax (x)} {\left (2 \, \cos \relax (x) + 3 \, \sin \relax (x)\right )} - 4 \, \cos \relax (x)^{2} - 15 \, {\left (\cos \relax (x)^{2} - 1\right )} \log \left (-\frac {1}{2} \, \sqrt {2} \sqrt {\cos \relax (x) \sin \relax (x)} {\left (4 \, \cos \relax (x) + 3 \, \sin \relax (x)\right )} + \frac {1}{2} \, \cos \relax (x)^{2} + \frac {7}{2} \, \cos \relax (x) \sin \relax (x) + \frac {1}{2}\right ) + 15 \, {\left (\cos \relax (x)^{2} - 1\right )} \log \left (\frac {1}{2} \, \cos \relax (x)^{2} + \frac {1}{2} \, \sqrt {2} \sqrt {\cos \relax (x) \sin \relax (x)} \sin \relax (x) - \frac {1}{2} \, \cos \relax (x) \sin \relax (x) + \frac {1}{2}\right ) + 4}{192 \, {\left (\cos \relax (x)^{2} - 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\cos \relax (x)^{2} \sin \relax (x)}{{\left (\sin \relax (x)^{2} - \sin \left (2 \, x\right )\right )} \sin \left (2 \, x\right )^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.28, size = 397, normalized size = 5.03 \[ \frac {\sqrt {-\frac {\tan \left (\frac {x}{2}\right )}{\tan ^{2}\left (\frac {x}{2}\right )-1}}\, \left (140 \sqrt {1+\tan \left (\frac {x}{2}\right )}\, \sqrt {-\tan \left (\frac {x}{2}\right )}\, \sqrt {-2 \tan \left (\frac {x}{2}\right )+2}\, \EllipticF \left (\sqrt {1+\tan \left (\frac {x}{2}\right )}, \frac {\sqrt {2}}{2}\right ) \sqrt {\left (\tan ^{2}\left (\frac {x}{2}\right )-1\right ) \tan \left (\frac {x}{2}\right )}\, \tan \left (\frac {x}{2}\right )-240 \sqrt {1+\tan \left (\frac {x}{2}\right )}\, \sqrt {-\tan \left (\frac {x}{2}\right )}\, \sqrt {-2 \tan \left (\frac {x}{2}\right )+2}\, \EllipticE \left (\sqrt {1+\tan \left (\frac {x}{2}\right )}, \frac {\sqrt {2}}{2}\right ) \sqrt {\left (\tan ^{2}\left (\frac {x}{2}\right )-1\right ) \tan \left (\frac {x}{2}\right )}\, \tan \left (\frac {x}{2}\right )-\sqrt {\tan ^{3}\left (\frac {x}{2}\right )-\tan \left (\frac {x}{2}\right )}\, \sqrt {\left (\tan ^{2}\left (\frac {x}{2}\right )-1\right ) \tan \left (\frac {x}{2}\right )}\, \sqrt {2}\, \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (\textit {\_Z}^{4}+\textit {\_Z}^{3}+2 \textit {\_Z}^{2}-\textit {\_Z} +1\right )}{\sum }\frac {\left (14 \underline {\hspace {1.25 ex}}\alpha ^{3}+3 \underline {\hspace {1.25 ex}}\alpha ^{2}+14 \underline {\hspace {1.25 ex}}\alpha -11\right ) \left (\underline {\hspace {1.25 ex}}\alpha ^{3}+2 \underline {\hspace {1.25 ex}}\alpha -3\right ) \sqrt {1+\tan \left (\frac {x}{2}\right )}\, \sqrt {1-\tan \left (\frac {x}{2}\right )}\, \sqrt {-\tan \left (\frac {x}{2}\right )}\, \EllipticPi \left (\sqrt {1+\tan \left (\frac {x}{2}\right )}, -\frac {1}{4} \underline {\hspace {1.25 ex}}\alpha ^{3}-\frac {1}{2} \underline {\hspace {1.25 ex}}\alpha +\frac {3}{4}, \frac {\sqrt {2}}{2}\right )}{\sqrt {\left (\tan ^{2}\left (\frac {x}{2}\right )-1\right ) \tan \left (\frac {x}{2}\right )}}\right ) \tan \left (\frac {x}{2}\right )-40 \sqrt {\left (\tan ^{2}\left (\frac {x}{2}\right )-1\right ) \tan \left (\frac {x}{2}\right )}\, \left (\tan ^{4}\left (\frac {x}{2}\right )\right )-120 \left (\tan ^{3}\left (\frac {x}{2}\right )\right ) \sqrt {\tan ^{3}\left (\frac {x}{2}\right )-\tan \left (\frac {x}{2}\right )}+120 \sqrt {\tan ^{3}\left (\frac {x}{2}\right )-\tan \left (\frac {x}{2}\right )}\, \tan \left (\frac {x}{2}\right )+40 \sqrt {\left (\tan ^{2}\left (\frac {x}{2}\right )-1\right ) \tan \left (\frac {x}{2}\right )}\right )}{1920 \tan \left (\frac {x}{2}\right )^{2} \sqrt {\tan ^{3}\left (\frac {x}{2}\right )-\tan \left (\frac {x}{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ -\int \frac {{\cos \relax (x)}^2\,\sin \relax (x)}{{\sin \left (2\,x\right )}^{5/2}\,\left (\sin \left (2\,x\right )-{\sin \relax (x)}^2\right )} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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