Optimal. Leaf size=7 \[ -\log (\coth (x)+1) \]
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Rubi [A] time = 0.0357077, antiderivative size = 7, 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} \[ -\log (\coth (x)+1) \]
Antiderivative was successfully verified.
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Rule 3487
Rule 31
Rubi steps
\begin{align*} \int \frac{\text{csch}^2(x)}{1+\coth (x)} \, dx &=-\operatorname{Subst}\left (\int \frac{1}{1+x} \, dx,x,\coth (x)\right )\\ &=-\log (1+\coth (x))\\ \end{align*}
Mathematica [A] time = 0.0029622, size = 7, normalized size = 1. \[ \log (\sinh (x))-x \]
Antiderivative was successfully verified.
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Maple [A] time = 0.017, size = 8, normalized size = 1.1 \begin{align*} -\ln \left ( 1+{\rm coth} \left (x\right ) \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.05112, size = 9, normalized size = 1.29 \begin{align*} -\log \left (\coth \left (x\right ) + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.59185, size = 58, normalized size = 8.29 \begin{align*} -2 \, x + \log \left (\frac{2 \, \sinh \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{csch}^{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 = 1.15829, size = 16, normalized size = 2.29 \begin{align*} -2 \, x + \log \left ({\left | e^{\left (2 \, x\right )} - 1 \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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