Optimal. Leaf size=95 \[ \frac {\cos ^5(x)}{5 \sin ^{\frac {5}{2}}(2 x)}+\frac {\sin (x) \cos ^4(x)}{6 \sin ^{\frac {5}{2}}(2 x)}-\frac {3 \sin ^2(x) \cos ^3(x)}{4 \sin ^{\frac {5}{2}}(2 x)}+\frac {3 \sin ^5(x) \tanh ^{-1}\left (\frac {\sqrt {\tan (x)}}{\sqrt {2}}\right )}{4 \sqrt {2} \sin ^{\frac {5}{2}}(2 x) \tan ^{\frac {5}{2}}(x)} \]
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Rubi [A] time = 0.58, antiderivative size = 95, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {4390, 898, 1262, 207} \[ \frac {\cos ^5(x)}{5 \sin ^{\frac {5}{2}}(2 x)}+\frac {\sin (x) \cos ^4(x)}{6 \sin ^{\frac {5}{2}}(2 x)}-\frac {3 \sin ^2(x) \cos ^3(x)}{4 \sin ^{\frac {5}{2}}(2 x)}+\frac {3 \sin ^5(x) \tanh ^{-1}\left (\frac {\sqrt {\tan (x)}}{\sqrt {2}}\right )}{4 \sqrt {2} \sin ^{\frac {5}{2}}(2 x) \tan ^{\frac {5}{2}}(x)} \]
Antiderivative was successfully verified.
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Rule 207
Rule 898
Rule 1262
Rule 4390
Rubi steps
\begin {align*} \int \frac {\cos ^3(x) \cos (2 x)}{\left (\sin ^2(x)-\sin (2 x)\right ) \sin ^{\frac {5}{2}}(2 x)} \, dx &=\frac {\sin ^5(x) \int \frac {\cos (2 x) \csc ^2(x)}{\left (\sin ^2(x)-\sin (2 x)\right ) \sqrt {\tan (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^{7/2}} \, 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^6 \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^6}-\frac {1}{4 x^4}+\frac {3}{8 x^2}-\frac {3}{8 \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 ^5(x)}{5 \sin ^{\frac {5}{2}}(2 x)}+\frac {\cos ^4(x) \sin (x)}{6 \sin ^{\frac {5}{2}}(2 x)}-\frac {3 \cos ^3(x) \sin ^2(x)}{4 \sin ^{\frac {5}{2}}(2 x)}-\frac {\left (3 \sin ^5(x)\right ) \operatorname {Subst}\left (\int \frac {1}{-2+x^2} \, dx,x,\sqrt {\tan (x)}\right )}{4 \sin ^{\frac {5}{2}}(2 x) \tan ^{\frac {5}{2}}(x)}\\ &=\frac {\cos ^5(x)}{5 \sin ^{\frac {5}{2}}(2 x)}+\frac {\cos ^4(x) \sin (x)}{6 \sin ^{\frac {5}{2}}(2 x)}-\frac {3 \cos ^3(x) \sin ^2(x)}{4 \sin ^{\frac {5}{2}}(2 x)}+\frac {3 \tanh ^{-1}\left (\frac {\sqrt {\tan (x)}}{\sqrt {2}}\right ) \sin ^5(x)}{4 \sqrt {2} \sin ^{\frac {5}{2}}(2 x) \tan ^{\frac {5}{2}}(x)}\\ \end {align*}
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Mathematica [C] time = 15.72, size = 184, normalized size = 1.94 \[ \frac {1}{960} \sqrt {\sin (2 x)} \sec (x) \left (20 \cot ^2(x)-114 \cot (x)+24 \cot (x) \csc ^2(x)-45 \sqrt {2} \sqrt {\frac {\cos (x)}{\cos (x)-1}} \sqrt {\tan \left (\frac {x}{2}\right )} F\left (\left .\sin ^{-1}\left (\frac {1}{\sqrt {\tan \left (\frac {x}{2}\right )}}\right )\right |-1\right )+45 \sqrt {2} \sqrt {\frac {\cos (x)}{\cos (x)-1}} \sqrt {\tan \left (\frac {x}{2}\right )} \Pi \left (-\frac {2}{-1+\sqrt {5}};\left .\sin ^{-1}\left (\frac {1}{\sqrt {\tan \left (\frac {x}{2}\right )}}\right )\right |-1\right )+45 \sqrt {2} \sqrt {\frac {\cos (x)}{\cos (x)-1}} \sqrt {\tan \left (\frac {x}{2}\right )} \Pi \left (\frac {1}{2} \left (-1+\sqrt {5}\right );\left .\sin ^{-1}\left (\frac {1}{\sqrt {\tan \left (\frac {x}{2}\right )}}\right )\right |-1\right )\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.47, size = 136, normalized size = 1.43 \[ -\frac {45 \, {\left (\cos \relax (x)^{2} - 1\right )} \log \left (-\frac {1}{2} \, \sqrt {2} \sqrt {\cos \relax (x) \sin \relax (x)} {\left (4 \, \cos \relax (x) + 3 \, \sin \relax (x)\right )} + \frac {1}{2} \, \cos \relax (x)^{2} + \frac {7}{2} \, \cos \relax (x) \sin \relax (x) + \frac {1}{2}\right ) \sin \relax (x) - 45 \, {\left (\cos \relax (x)^{2} - 1\right )} \log \left (\frac {1}{2} \, \cos \relax (x)^{2} + \frac {1}{2} \, \sqrt {2} \sqrt {\cos \relax (x) \sin \relax (x)} \sin \relax (x) - \frac {1}{2} \, \cos \relax (x) \sin \relax (x) + \frac {1}{2}\right ) \sin \relax (x) + 4 \, \sqrt {2} {\left (57 \, \cos \relax (x)^{2} + 10 \, \cos \relax (x) \sin \relax (x) - 45\right )} \sqrt {\cos \relax (x) \sin \relax (x)} + 268 \, {\left (\cos \relax (x)^{2} - 1\right )} \sin \relax (x)}{1920 \, {\left (\cos \relax (x)^{2} - 1\right )} \sin \relax (x)} \]
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 \[ \int \frac {\cos \left (2 \, x\right ) \cos \relax (x)^{3}}{{\left (\sin \relax (x)^{2} - \sin \left (2 \, x\right )\right )} \sin \left (2 \, x\right )^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.38, size = 761, normalized size = 8.01 \[ \text {result too large to display} \]
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 \[ \int -\frac {\cos \left (2\,x\right )\,{\cos \relax (x)}^3}{{\sin \left (2\,x\right )}^{5/2}\,\left (\sin \left (2\,x\right )-{\sin \relax (x)}^2\right )} \,d x \]
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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