Integrand size = 13, antiderivative size = 43 \[ \int \cos ^5(2 x) \cot ^4(2 x) \, dx=2 \csc (2 x)-\frac {1}{6} \csc ^3(2 x)+3 \sin (2 x)-\frac {2}{3} \sin ^3(2 x)+\frac {1}{10} \sin ^5(2 x) \]
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Time = 0.05 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {2670, 276} \[ \int \cos ^5(2 x) \cot ^4(2 x) \, dx=\frac {1}{10} \sin ^5(2 x)-\frac {2}{3} \sin ^3(2 x)+3 \sin (2 x)-\frac {1}{6} \csc ^3(2 x)+2 \csc (2 x) \]
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Rule 276
Rule 2670
Rubi steps \begin{align*} \text {integral}& = -\left (\frac {1}{2} \text {Subst}\left (\int \frac {\left (1-x^2\right )^4}{x^4} \, dx,x,-\sin (2 x)\right )\right ) \\ & = -\left (\frac {1}{2} \text {Subst}\left (\int \left (6+\frac {1}{x^4}-\frac {4}{x^2}-4 x^2+x^4\right ) \, dx,x,-\sin (2 x)\right )\right ) \\ & = 2 \csc (2 x)-\frac {1}{6} \csc ^3(2 x)+3 \sin (2 x)-\frac {2}{3} \sin ^3(2 x)+\frac {1}{10} \sin ^5(2 x) \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.00 \[ \int \cos ^5(2 x) \cot ^4(2 x) \, dx=2 \csc (2 x)-\frac {1}{6} \csc ^3(2 x)+3 \sin (2 x)-\frac {2}{3} \sin ^3(2 x)+\frac {1}{10} \sin ^5(2 x) \]
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Time = 4.93 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.58
\[-\frac {\cos \left (2 x \right )^{10}}{6 \sin \left (2 x \right )^{3}}+\frac {7 \cos \left (2 x \right )^{10}}{6 \sin \left (2 x \right )}+\frac {7 \left (\frac {128}{35}+\cos \left (2 x \right )^{8}+\frac {8 \cos \left (2 x \right )^{6}}{7}+\frac {48 \cos \left (2 x \right )^{4}}{35}+\frac {64 \cos \left (2 x \right )^{2}}{35}\right ) \sin \left (2 x \right )}{6}\]
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Time = 0.25 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.21 \[ \int \cos ^5(2 x) \cot ^4(2 x) \, dx=-\frac {3 \, \cos \left (2 \, x\right )^{8} + 8 \, \cos \left (2 \, x\right )^{6} + 48 \, \cos \left (2 \, x\right )^{4} - 192 \, \cos \left (2 \, x\right )^{2} + 128}{30 \, {\left (\cos \left (2 \, x\right )^{2} - 1\right )} \sin \left (2 \, x\right )} \]
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Time = 0.05 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.98 \[ \int \cos ^5(2 x) \cot ^4(2 x) \, dx=\frac {12 \sin ^{2}{\left (2 x \right )} - 1}{6 \sin ^{3}{\left (2 x \right )}} + \frac {\sin ^{5}{\left (2 x \right )}}{10} - \frac {2 \sin ^{3}{\left (2 x \right )}}{3} + 3 \sin {\left (2 x \right )} \]
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Time = 0.19 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.95 \[ \int \cos ^5(2 x) \cot ^4(2 x) \, dx=\frac {1}{10} \, \sin \left (2 \, x\right )^{5} - \frac {2}{3} \, \sin \left (2 \, x\right )^{3} + \frac {12 \, \sin \left (2 \, x\right )^{2} - 1}{6 \, \sin \left (2 \, x\right )^{3}} + 3 \, \sin \left (2 \, x\right ) \]
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Time = 0.29 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.95 \[ \int \cos ^5(2 x) \cot ^4(2 x) \, dx=\frac {1}{10} \, \sin \left (2 \, x\right )^{5} - \frac {2}{3} \, \sin \left (2 \, x\right )^{3} + \frac {12 \, \sin \left (2 \, x\right )^{2} - 1}{6 \, \sin \left (2 \, x\right )^{3}} + 3 \, \sin \left (2 \, x\right ) \]
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Time = 26.54 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.98 \[ \int \cos ^5(2 x) \cot ^4(2 x) \, dx=\frac {3\,{\sin \left (2\,x\right )}^8-20\,{\sin \left (2\,x\right )}^6+90\,{\sin \left (2\,x\right )}^4+60\,{\sin \left (2\,x\right )}^2-5}{30\,{\sin \left (2\,x\right )}^3} \]
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