Optimal. Leaf size=22 \[ -\frac{2}{1-e^x}-2 \log \left (1-e^x\right ) \]
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Rubi [A] time = 0.0242695, antiderivative size = 22, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {2282, 12, 43} \[ -\frac{2}{1-e^x}-2 \log \left (1-e^x\right ) \]
Antiderivative was successfully verified.
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Rule 2282
Rule 12
Rule 43
Rubi steps
\begin{align*} \int \frac{e^x}{1-\cosh (x)} \, dx &=\operatorname{Subst}\left (\int -\frac{2 x}{(1-x)^2} \, dx,x,e^x\right )\\ &=-\left (2 \operatorname{Subst}\left (\int \frac{x}{(1-x)^2} \, dx,x,e^x\right )\right )\\ &=-\left (2 \operatorname{Subst}\left (\int \left (\frac{1}{(-1+x)^2}+\frac{1}{-1+x}\right ) \, dx,x,e^x\right )\right )\\ &=-\frac{2}{1-e^x}-2 \log \left (1-e^x\right )\\ \end{align*}
Mathematica [A] time = 0.0318872, size = 36, normalized size = 1.64 \[ \frac{4 \left (\frac{1}{1-e^x}+\log \left (1-e^x\right )\right ) \sinh ^2\left (\frac{x}{2}\right )}{1-\cosh (x)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.01, size = 24, normalized size = 1.1 \begin{align*} \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{-1}-2\,\ln \left ( \tanh \left ( x/2 \right ) \right ) +2\,\ln \left ( -1+\tanh \left ( x/2 \right ) \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.948572, size = 22, normalized size = 1. \begin{align*} \frac{2}{e^{x} - 1} - 2 \, \log \left (e^{x} - 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.98946, size = 115, normalized size = 5.23 \begin{align*} -\frac{2 \,{\left ({\left (\cosh \left (x\right ) + \sinh \left (x\right ) - 1\right )} \log \left (\cosh \left (x\right ) + \sinh \left (x\right ) - 1\right ) - 1\right )}}{\cosh \left (x\right ) + \sinh \left (x\right ) - 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{e^{x}}{\cosh{\left (x \right )} - 1}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.11136, size = 23, normalized size = 1.05 \begin{align*} \frac{2}{e^{x} - 1} - 2 \, \log \left ({\left | e^{x} - 1 \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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