Optimal. Leaf size=15 \[ \tanh (x)-\log (\tanh (x))-\log (\coth (x)+1) \]
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Rubi [A] time = 0.0411416, antiderivative size = 15, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {3516, 44} \[ \tanh (x)-\log (\tanh (x))-\log (\coth (x)+1) \]
Antiderivative was successfully verified.
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Rule 3516
Rule 44
Rubi steps
\begin{align*} \int \frac{\text{sech}^2(x)}{1+\coth (x)} \, dx &=-\operatorname{Subst}\left (\int \frac{1}{x^2 (1+x)} \, dx,x,\coth (x)\right )\\ &=-\operatorname{Subst}\left (\int \left (\frac{1}{x^2}-\frac{1}{x}+\frac{1}{1+x}\right ) \, dx,x,\coth (x)\right )\\ &=-\log (1+\coth (x))-\log (\tanh (x))+\tanh (x)\\ \end{align*}
Mathematica [A] time = 0.0296206, size = 9, normalized size = 0.6 \[ -x+\tanh (x)+\log (\cosh (x)) \]
Antiderivative was successfully verified.
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Maple [B] time = 0.028, size = 36, normalized size = 2.4 \begin{align*} -2\,\ln \left ( \tanh \left ( x/2 \right ) +1 \right ) +2\,{\frac{\tanh \left ( x/2 \right ) }{ \left ( \tanh \left ( x/2 \right ) \right ) ^{2}+1}}+\ln \left ( \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{2}+1 \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.56431, size = 24, normalized size = 1.6 \begin{align*} \frac{2}{e^{\left (-2 \, x\right )} + 1} + \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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Fricas [B] time = 2.6983, size = 269, normalized size = 17.93 \begin{align*} -\frac{2 \, x \cosh \left (x\right )^{2} + 4 \, x \cosh \left (x\right ) \sinh \left (x\right ) + 2 \, x \sinh \left (x\right )^{2} -{\left (\cosh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2} + 1\right )} \log \left (\frac{2 \, \cosh \left (x\right )}{\cosh \left (x\right ) - \sinh \left (x\right )}\right ) + 2 \, x + 2}{\cosh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2} + 1} \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 )}}{\coth{\left (x \right )} + 1}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.18812, size = 36, normalized size = 2.4 \begin{align*} -2 \, x - \frac{e^{\left (2 \, x\right )} + 3}{e^{\left (2 \, x\right )} + 1} + \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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