Optimal. Leaf size=9 \[ \frac{1}{2} \log ^2(\tanh (x)) \]
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Rubi [A] time = 0.0260572, antiderivative size = 9, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 3, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375, Rules used = {2620, 29, 6686} \[ \frac{1}{2} \log ^2(\tanh (x)) \]
Antiderivative was successfully verified.
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Rule 2620
Rule 29
Rule 6686
Rubi steps
\begin{align*} \int \text{csch}(x) \log (\tanh (x)) \text{sech}(x) \, dx &=\frac{1}{2} \log ^2(\tanh (x))\\ \end{align*}
Mathematica [A] time = 0.0058762, size = 9, normalized size = 1. \[ \frac{1}{2} \log ^2(\tanh (x)) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.017, size = 8, normalized size = 0.9 \begin{align*}{\frac{ \left ( \ln \left ( \tanh \left ( x \right ) \right ) \right ) ^{2}}{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 4.90216, size = 128, normalized size = 14.22 \begin{align*}{\left (\log \left (e^{x} + 1\right ) + \log \left (-e^{x} + 1\right )\right )} \log \left (e^{\left (2 \, x\right )} + 1\right ) - \frac{1}{2} \, \log \left (e^{\left (2 \, x\right )} + 1\right )^{2} - \frac{1}{2} \, \log \left (e^{x} + 1\right )^{2} - \log \left (e^{x} + 1\right ) \log \left (-e^{x} + 1\right ) - \frac{1}{2} \, \log \left (-e^{x} + 1\right )^{2} +{\left (\log \left (e^{\left (-x\right )} + 1\right ) + \log \left (e^{\left (-x\right )} - 1\right ) - \log \left (e^{\left (-2 \, x\right )} + 1\right )\right )} \log \left (\tanh \left (x\right )\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.98701, size = 38, normalized size = 4.22 \begin{align*} \frac{1}{2} \, \log \left (\frac{\sinh \left (x\right )}{\cosh \left (x\right )}\right )^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \log{\left (\tanh{\left (x \right )} \right )} \operatorname{csch}{\left (x \right )} \operatorname{sech}{\left (x \right )}\, dx \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 \operatorname{csch}\left (x\right ) \log \left (\tanh \left (x\right )\right ) \operatorname{sech}\left (x\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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