Optimal. Leaf size=9 \[ \frac {1}{4} \log ^2(\tan (x)) \]
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Rubi [A] time = 0.02, antiderivative size = 9, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 2, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3770, 6686} \[ \frac {1}{4} \log ^2(\tan (x)) \]
Antiderivative was successfully verified.
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Rule 3770
Rule 6686
Rubi steps
\begin {align*} \int \csc (2 x) \log (\tan (x)) \, dx &=\frac {1}{4} \log ^2(\tan (x))\\ \end {align*}
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Mathematica [A] time = 0.01, size = 9, normalized size = 1.00 \[ \frac {1}{4} \log ^2(\tan (x)) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.51, size = 7, normalized size = 0.78 \[ \frac {1}{4} \, \log \left (\tan \relax (x)\right )^{2} \]
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 \[ \int \csc \left (2 \, x\right ) \log \left (\tan \relax (x)\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 8, normalized size = 0.89 \[ \frac {\ln \left (\tan \relax (x )\right )^{2}}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.46, size = 265, normalized size = 29.44 \[ \frac {1}{4} \, {\left (\pi - 2 \, \arctan \left (\sin \relax (x), \cos \relax (x) + 1\right ) - 2 \, \arctan \left (\sin \relax (x), \cos \relax (x) - 1\right )\right )} \arctan \left (\sin \left (2 \, x\right ), \cos \left (2 \, x\right ) + 1\right ) + \frac {1}{4} \, \arctan \left (\sin \left (2 \, x\right ), \cos \left (2 \, x\right ) + 1\right )^{2} - \frac {1}{4} \, {\left (\pi - 2 \, \arctan \left (\sin \relax (x), \cos \relax (x) - 1\right )\right )} \arctan \left (\sin \relax (x), \cos \relax (x) + 1\right ) + \frac {1}{4} \, \arctan \left (\sin \relax (x), \cos \relax (x) + 1\right )^{2} - \frac {1}{4} \, \pi \arctan \left (\sin \relax (x), \cos \relax (x) - 1\right ) + \frac {1}{4} \, \arctan \left (\sin \relax (x), \cos \relax (x) - 1\right )^{2} + \frac {1}{8} \, {\left (\log \left (\cos \relax (x)^{2} + \sin \relax (x)^{2} + 2 \, \cos \relax (x) + 1\right ) + \log \left (\cos \relax (x)^{2} + \sin \relax (x)^{2} - 2 \, \cos \relax (x) + 1\right )\right )} \log \left (\cos \left (2 \, x\right )^{2} + \sin \left (2 \, x\right )^{2} + 2 \, \cos \left (2 \, x\right ) + 1\right ) - \frac {1}{16} \, \log \left (\cos \left (2 \, x\right )^{2} + \sin \left (2 \, x\right )^{2} + 2 \, \cos \left (2 \, x\right ) + 1\right )^{2} - \frac {1}{16} \, \log \left (\cos \relax (x)^{2} + \sin \relax (x)^{2} + 2 \, \cos \relax (x) + 1\right )^{2} - \frac {1}{8} \, \log \left (\cos \relax (x)^{2} + \sin \relax (x)^{2} + 2 \, \cos \relax (x) + 1\right ) \log \left (\cos \relax (x)^{2} + \sin \relax (x)^{2} - 2 \, \cos \relax (x) + 1\right ) - \frac {1}{16} \, \log \left (\cos \relax (x)^{2} + \sin \relax (x)^{2} - 2 \, \cos \relax (x) + 1\right )^{2} - \frac {1}{2} \, \log \left (\cot \left (2 \, x\right ) + \csc \left (2 \, x\right )\right ) \log \left (\tan \relax (x)\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.53, size = 27, normalized size = 3.00 \[ \frac {{\ln \left (-\frac {{\mathrm {e}}^{x\,2{}\mathrm {i}}\,1{}\mathrm {i}-\mathrm {i}}{{\mathrm {e}}^{x\,2{}\mathrm {i}}+1}\right )}^2}{4} \]
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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