Integrand size = 13, antiderivative size = 42 \[ \int \cos ^4(2 x) \cot ^5(2 x) \, dx=\csc ^2(2 x)-\frac {1}{8} \csc ^4(2 x)+3 \log (\sin (2 x))-\sin ^2(2 x)+\frac {1}{8} \sin ^4(2 x) \]
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Time = 0.05 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {2670, 272, 45} \[ \int \cos ^4(2 x) \cot ^5(2 x) \, dx=\frac {1}{8} \sin ^4(2 x)-\sin ^2(2 x)-\frac {1}{8} \csc ^4(2 x)+\csc ^2(2 x)+3 \log (\sin (2 x)) \]
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Rule 45
Rule 272
Rule 2670
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \text {Subst}\left (\int \frac {\left (1-x^2\right )^4}{x^5} \, dx,x,-\sin (2 x)\right ) \\ & = \frac {1}{4} \text {Subst}\left (\int \frac {(1-x)^4}{x^3} \, dx,x,\sin ^2(2 x)\right ) \\ & = \frac {1}{4} \text {Subst}\left (\int \left (-4+\frac {1}{x^3}-\frac {4}{x^2}+\frac {6}{x}+x\right ) \, dx,x,\sin ^2(2 x)\right ) \\ & = \csc ^2(2 x)-\frac {1}{8} \csc ^4(2 x)+3 \log (\sin (2 x))-\sin ^2(2 x)+\frac {1}{8} \sin ^4(2 x) \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.00 \[ \int \cos ^4(2 x) \cot ^5(2 x) \, dx=\csc ^2(2 x)-\frac {1}{8} \csc ^4(2 x)+3 \log (\sin (2 x))-\sin ^2(2 x)+\frac {1}{8} \sin ^4(2 x) \]
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Time = 0.29 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.64
\[-\frac {\cos \left (2 x \right )^{10}}{8 \sin \left (2 x \right )^{4}}+\frac {3 \cos \left (2 x \right )^{10}}{8 \sin \left (2 x \right )^{2}}+\frac {3 \cos \left (2 x \right )^{8}}{8}+\frac {\cos \left (2 x \right )^{6}}{2}+\frac {3 \cos \left (2 x \right )^{4}}{4}+\frac {3 \cos \left (2 x \right )^{2}}{2}+3 \ln \left (\sin \left (2 x \right )\right )\]
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Leaf count of result is larger than twice the leaf count of optimal. 79 vs. \(2 (38) = 76\).
Time = 0.27 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.88 \[ \int \cos ^4(2 x) \cot ^5(2 x) \, dx=\frac {8 \, \cos \left (2 \, x\right )^{8} + 32 \, \cos \left (2 \, x\right )^{6} - 115 \, \cos \left (2 \, x\right )^{4} + 38 \, \cos \left (2 \, x\right )^{2} + 192 \, {\left (\cos \left (2 \, x\right )^{4} - 2 \, \cos \left (2 \, x\right )^{2} + 1\right )} \log \left (\frac {1}{2} \, \sin \left (2 \, x\right )\right ) + 29}{64 \, {\left (\cos \left (2 \, x\right )^{4} - 2 \, \cos \left (2 \, x\right )^{2} + 1\right )}} \]
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Time = 0.05 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.98 \[ \int \cos ^4(2 x) \cot ^5(2 x) \, dx=\frac {8 \sin ^{2}{\left (2 x \right )} - 1}{8 \sin ^{4}{\left (2 x \right )}} + 3 \log {\left (\sin {\left (2 x \right )} \right )} + \frac {\sin ^{4}{\left (2 x \right )}}{8} - \sin ^{2}{\left (2 x \right )} \]
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Time = 0.20 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.05 \[ \int \cos ^4(2 x) \cot ^5(2 x) \, dx=\frac {1}{8} \, \sin \left (2 \, x\right )^{4} - \sin \left (2 \, x\right )^{2} + \frac {8 \, \sin \left (2 \, x\right )^{2} - 1}{8 \, \sin \left (2 \, x\right )^{4}} + \frac {3}{2} \, \log \left (\sin \left (2 \, x\right )^{2}\right ) \]
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Time = 0.28 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.24 \[ \int \cos ^4(2 x) \cot ^5(2 x) \, dx=\frac {1}{8} \, \cos \left (2 \, x\right )^{4} + \frac {3}{4} \, \cos \left (2 \, x\right )^{2} - \frac {8 \, \cos \left (2 \, x\right )^{2} - 7}{8 \, {\left (\cos \left (2 \, x\right )^{2} - 1\right )}^{2}} + \frac {3}{2} \, \log \left (-\cos \left (2 \, x\right )^{2} + 1\right ) \]
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Time = 27.18 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.69 \[ \int \cos ^4(2 x) \cot ^5(2 x) \, dx=3\,\ln \left (\mathrm {tan}\left (2\,x\right )\right )-\frac {3\,\ln \left ({\mathrm {tan}\left (2\,x\right )}^2+1\right )}{2}+\frac {3\,{\mathrm {tan}\left (2\,x\right )}^6+\frac {9\,{\mathrm {tan}\left (2\,x\right )}^4}{2}+{\mathrm {tan}\left (2\,x\right )}^2-\frac {1}{4}}{2\,\left ({\mathrm {tan}\left (2\,x\right )}^8+2\,{\mathrm {tan}\left (2\,x\right )}^6+{\mathrm {tan}\left (2\,x\right )}^4\right )} \]
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