Optimal. Leaf size=5 \[ \log (\tanh (x)+1) \]
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Rubi [A] time = 0.0334008, antiderivative size = 5, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {3487, 31} \[ \log (\tanh (x)+1) \]
Antiderivative was successfully verified.
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Rule 3487
Rule 31
Rubi steps
\begin{align*} \int \frac{\text{sech}^2(x)}{1+\tanh (x)} \, dx &=\operatorname{Subst}\left (\int \frac{1}{1+x} \, dx,x,\tanh (x)\right )\\ &=\log (1+\tanh (x))\\ \end{align*}
Mathematica [A] time = 0.0038613, size = 7, normalized size = 1.4 \[ x-\log (\cosh (x)) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.018, size = 6, normalized size = 1.2 \begin{align*} \ln \left ( 1+\tanh \left ( x \right ) \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.16682, size = 7, normalized size = 1.4 \begin{align*} \log \left (\tanh \left (x\right ) + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.31621, size = 57, normalized size = 11.4 \begin{align*} 2 \, x - \log \left (\frac{2 \, \cosh \left (x\right )}{\cosh \left (x\right ) - \sinh \left (x\right )}\right ) \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 \frac{\operatorname{sech}^{2}{\left (x \right )}}{\tanh{\left (x \right )} + 1}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.26671, size = 18, normalized size = 3.6 \begin{align*} 2 \, x - \log \left (e^{\left (2 \, x\right )} + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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