Optimal. Leaf size=5 \[ \log (\tanh (x)+1) \]
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Rubi [A] time = 0.03, antiderivative size = 5, normalized size of antiderivative = 1.00, 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 31
Rule 3487
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*}
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Mathematica [A] time = 0.00, size = 7, normalized size = 1.40 \[ x-\log (\cosh (x)) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.57, size = 20, normalized size = 4.00 \[ 2 \, x - \log \left (\frac {2 \, \cosh \relax (x)}{\cosh \relax (x) - \sinh \relax (x)}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.11, size = 13, normalized size = 2.60 \[ 2 \, x - \log \left (e^{\left (2 \, x\right )} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.08, size = 6, normalized size = 1.20 \[ \ln \left (1+\tanh \relax (x )\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 5, normalized size = 1.00 \[ \log \left (\tanh \relax (x) + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.05, size = 13, normalized size = 2.60 \[ 2\,x-\ln \left ({\mathrm {e}}^{2\,x}+1\right ) \]
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 \[ \int \frac {\operatorname {sech}^{2}{\relax (x )}}{\tanh {\relax (x )} + 1}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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