Optimal. Leaf size=9 \[ \frac{1}{4} \log ^2(\tan (x)) \]
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Rubi [A] time = 0.0195538, antiderivative size = 9, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 2, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, 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*}
Mathematica [A] time = 0.0109094, size = 9, normalized size = 1. \[ \frac{1}{4} \log ^2(\tan (x)) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.016, size = 8, normalized size = 0.9 \begin{align*}{\frac{ \left ( \ln \left ( \tan \left ( x \right ) \right ) \right ) ^{2}}{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.54833, size = 358, normalized size = 39.78 \begin{align*} \frac{1}{4} \,{\left (\pi - 2 \, \arctan \left (\sin \left (x\right ), \cos \left (x\right ) + 1\right ) - 2 \, \arctan \left (\sin \left (x\right ), \cos \left (x\right ) - 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 \left (x\right ), \cos \left (x\right ) - 1\right )\right )} \arctan \left (\sin \left (x\right ), \cos \left (x\right ) + 1\right ) + \frac{1}{4} \, \arctan \left (\sin \left (x\right ), \cos \left (x\right ) + 1\right )^{2} - \frac{1}{4} \, \pi \arctan \left (\sin \left (x\right ), \cos \left (x\right ) - 1\right ) + \frac{1}{4} \, \arctan \left (\sin \left (x\right ), \cos \left (x\right ) - 1\right )^{2} + \frac{1}{8} \,{\left (\log \left (\cos \left (x\right )^{2} + \sin \left (x\right )^{2} + 2 \, \cos \left (x\right ) + 1\right ) + \log \left (\cos \left (x\right )^{2} + \sin \left (x\right )^{2} - 2 \, \cos \left (x\right ) + 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 \left (x\right )^{2} + \sin \left (x\right )^{2} + 2 \, \cos \left (x\right ) + 1\right )^{2} - \frac{1}{8} \, \log \left (\cos \left (x\right )^{2} + \sin \left (x\right )^{2} + 2 \, \cos \left (x\right ) + 1\right ) \log \left (\cos \left (x\right )^{2} + \sin \left (x\right )^{2} - 2 \, \cos \left (x\right ) + 1\right ) - \frac{1}{16} \, \log \left (\cos \left (x\right )^{2} + \sin \left (x\right )^{2} - 2 \, \cos \left (x\right ) + 1\right )^{2} - \frac{1}{2} \, \log \left (\cot \left (2 \, x\right ) + \csc \left (2 \, x\right )\right ) \log \left (\tan \left (x\right )\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.01379, size = 26, normalized size = 2.89 \begin{align*} \frac{1}{4} \, \log \left (\tan \left (x\right )\right )^{2} \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 [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \csc \left (2 \, x\right ) \log \left (\tan \left (x\right )\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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