Optimal. Leaf size=9 \[ \frac {1}{2} \log ^2(\tanh (x)) \]
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Rubi [A] time = 0.03, antiderivative size = 9, normalized size of antiderivative = 1.00, 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 29
Rule 2620
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*}
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Mathematica [A] time = 0.01, size = 9, normalized size = 1.00 \[ \frac {1}{2} \log ^2(\tanh (x)) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 12, normalized size = 1.33 \[ \frac {1}{2} \, \log \left (\frac {\sinh \relax (x)}{\cosh \relax (x)}\right )^{2} \]
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 \[ \int \operatorname {csch}\relax (x) \log \left (\tanh \relax (x)\right ) \operatorname {sech}\relax (x)\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.17, size = 8, normalized size = 0.89 \[ \frac {\ln \left (\tanh \relax (x )\right )^{2}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.48, size = 95, normalized size = 10.56 \[ {\left (\log \left (e^{x} + 1\right ) + \log \left (-e^{x} + 1\right )\right )} \log \left (e^{\left (2 \, x\right )} + 1\right ) - \frac {1}{2} \, \log \left (e^{\left (2 \, x\right )} + 1\right )^{2} - \frac {1}{2} \, \log \left (e^{x} + 1\right )^{2} - \log \left (e^{x} + 1\right ) \log \left (-e^{x} + 1\right ) - \frac {1}{2} \, \log \left (-e^{x} + 1\right )^{2} + {\left (\log \left (e^{\left (-x\right )} + 1\right ) + \log \left (e^{\left (-x\right )} - 1\right ) - \log \left (e^{\left (-2 \, x\right )} + 1\right )\right )} \log \left (\tanh \relax (x)\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.76, size = 21, normalized size = 2.33 \[ \frac {{\left (\ln \left ({\mathrm {e}}^{2\,x}-1\right )-\ln \left ({\mathrm {e}}^{2\,x}+1\right )\right )}^2}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \log {\left (\tanh {\relax (x )} \right )} \operatorname {csch}{\relax (x )} \operatorname {sech}{\relax (x )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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