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 = 0.04, antiderivative size = 42, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {2590, 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 43
Rule 266
Rule 2590
Rubi steps
\begin {align*} \int \cos ^4(2 x) \cot ^5(2 x) \, 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*}
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Mathematica [A] time = 0.03, size = 42, normalized size = 1.00 \[ \frac {1}{8} \sin ^4(2 x)-\sin ^2(2 x)-\frac {1}{8} \csc ^4(2 x)+\csc ^2(2 x)+3 \log (\sin (2 x)) \]
Antiderivative was successfully verified.
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fricas [B] time = 1.96, size = 79, normalized size = 1.88 \[ \frac {8 \, \cos \left (2 \, x\right )^{8} + 32 \, \cos \left (2 \, x\right )^{6} - 115 \, \cos \left (2 \, x\right )^{4} + 38 \, \cos \left (2 \, x\right )^{2} + 192 \, {\left (\cos \left (2 \, x\right )^{4} - 2 \, \cos \left (2 \, x\right )^{2} + 1\right )} \log \left (\frac {1}{2} \, \sin \left (2 \, x\right )\right ) + 29}{64 \, {\left (\cos \left (2 \, x\right )^{4} - 2 \, \cos \left (2 \, x\right )^{2} + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 52, normalized size = 1.24 \[ \frac {1}{8} \, \cos \left (2 \, x\right )^{4} + \frac {3}{4} \, \cos \left (2 \, x\right )^{2} - \frac {8 \, \cos \left (2 \, x\right )^{2} - 7}{8 \, {\left (\cos \left (2 \, x\right )^{2} - 1\right )}^{2}} + \frac {3}{2} \, \log \left (-\cos \left (2 \, x\right )^{2} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.11, size = 69, normalized size = 1.64 \[ -\frac {\cos ^{10}\left (2 x \right )}{8 \sin \left (2 x \right )^{4}}+\frac {3 \left (\cos ^{10}\left (2 x \right )\right )}{8 \sin \left (2 x \right )^{2}}+\frac {3 \left (\cos ^{8}\left (2 x \right )\right )}{8}+\frac {\left (\cos ^{6}\left (2 x \right )\right )}{2}+\frac {3 \left (\cos ^{4}\left (2 x \right )\right )}{4}+\frac {3 \left (\cos ^{2}\left (2 x \right )\right )}{2}+3 \ln \left (\sin \left (2 x \right )\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.34, size = 44, normalized size = 1.05 \[ \frac {1}{8} \, \sin \left (2 \, x\right )^{4} - \sin \left (2 \, x\right )^{2} + \frac {8 \, \sin \left (2 \, x\right )^{2} - 1}{8 \, \sin \left (2 \, x\right )^{4}} + \frac {3}{2} \, \log \left (\sin \left (2 \, x\right )^{2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.05, size = 71, normalized size = 1.69 \[ 3\,\ln \left (\mathrm {tan}\left (2\,x\right )\right )-\frac {3\,\ln \left ({\mathrm {tan}\left (2\,x\right )}^2+1\right )}{2}+\frac {3\,{\mathrm {tan}\left (2\,x\right )}^6+\frac {9\,{\mathrm {tan}\left (2\,x\right )}^4}{2}+{\mathrm {tan}\left (2\,x\right )}^2-\frac {1}{4}}{2\,\left ({\mathrm {tan}\left (2\,x\right )}^8+2\,{\mathrm {tan}\left (2\,x\right )}^6+{\mathrm {tan}\left (2\,x\right )}^4\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.10, size = 41, normalized size = 0.98 \[ \frac {8 \sin ^{2}{\left (2 x \right )} - 1}{8 \sin ^{4}{\left (2 x \right )}} + 3 \log {\left (\sin {\left (2 x \right )} \right )} + \frac {\sin ^{4}{\left (2 x \right )}}{8} - \sin ^{2}{\left (2 x \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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