Optimal. Leaf size=12 \[ x-2 \log \left (1-e^x\right ) \]
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Rubi [A] time = 0.0210943, antiderivative size = 12, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {2282, 72} \[ x-2 \log \left (1-e^x\right ) \]
Antiderivative was successfully verified.
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Rule 2282
Rule 72
Rubi steps
\begin{align*} \int \frac{1+e^x}{1-e^x} \, dx &=\operatorname{Subst}\left (\int \frac{1+x}{(1-x) x} \, dx,x,e^x\right )\\ &=\operatorname{Subst}\left (\int \left (-\frac{2}{-1+x}+\frac{1}{x}\right ) \, dx,x,e^x\right )\\ &=x-2 \log \left (1-e^x\right )\\ \end{align*}
Mathematica [A] time = 0.0084745, size = 12, normalized size = 1. \[ x-2 \log \left (1-e^x\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 12, normalized size = 1. \begin{align*} \ln \left ({{\rm e}^{x}} \right ) -2\,\ln \left ( -1+{{\rm e}^{x}} \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.942234, size = 12, normalized size = 1. \begin{align*} x - 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 = 1.91797, size = 27, normalized size = 2.25 \begin{align*} x - 2 \, \log \left (e^{x} - 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.079297, size = 8, normalized size = 0.67 \begin{align*} x - 2 \log{\left (e^{x} - 1 \right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.05154, size = 14, normalized size = 1.17 \begin{align*} x - 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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