Optimal. Leaf size=42 \[ \frac{1}{8} \sin ^4(2 x)-\sin ^2(2 x)-\frac{1}{8} \csc ^4(2 x)+\csc ^2(2 x)+3 \log (\sin (2 x)) \]
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Rubi [A] time = 0.118309, antiderivative size = 42, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.148, Rules used = {3175, 4360, 266, 43} \[ \frac{1}{8} \sin ^4(2 x)-\sin ^2(2 x)-\frac{1}{8} \csc ^4(2 x)+\csc ^2(2 x)+3 \log (\sin (2 x)) \]
Antiderivative was successfully verified.
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Rule 3175
Rule 4360
Rule 266
Rule 43
Rubi steps
\begin{align*} \int \cot (2 x) \left (-1+\csc ^2(2 x)\right )^2 \left (1-\sin ^2(2 x)\right )^2 \, dx &=\int \cos ^4(2 x) \cot (2 x) \left (-1+\csc ^2(2 x)\right )^2 \, dx\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{\left (1-x^2\right )^4}{x^5} \, dx,x,\sin (2 x)\right )\\ &=\frac{1}{4} \operatorname{Subst}\left (\int \frac{(1-x)^4}{x^3} \, dx,x,\sin ^2(2 x)\right )\\ &=\frac{1}{4} \operatorname{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*}
Mathematica [A] time = 0.0341055, size = 42, normalized size = 1. \[ \frac{1}{8} \sin ^4(2 x)-\sin ^2(2 x)-\frac{1}{8} \csc ^4(2 x)+\csc ^2(2 x)+3 \log (\sin (2 x)) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.05, size = 37, normalized size = 0.9 \begin{align*}{\frac{ \left ( \sin \left ( 2\,x \right ) \right ) ^{4}}{8}}+ \left ( \cos \left ( 2\,x \right ) \right ) ^{2}+3\,\ln \left ( \sin \left ( 2\,x \right ) \right ) + \left ( \sin \left ( 2\,x \right ) \right ) ^{-2}-{\frac{1}{8\, \left ( \sin \left ( 2\,x \right ) \right ) ^{4}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.939247, size = 59, normalized size = 1.4 \begin{align*} \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 ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.51972, size = 220, normalized size = 5.24 \begin{align*} \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 )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.09308, size = 70, normalized size = 1.67 \begin{align*} \frac{1}{8} \, \sin \left (2 \, x\right )^{4} - \sin \left (2 \, x\right )^{2} - \frac{18 \, \sin \left (2 \, x\right )^{4} - 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 ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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