Optimal. Leaf size=4 \[ \tanh ^{-1}\left (e^x\right ) \]
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Rubi [A] time = 0.0196363, antiderivative size = 4, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {2249, 206} \[ \tanh ^{-1}\left (e^x\right ) \]
Antiderivative was successfully verified.
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Rule 2249
Rule 206
Rubi steps
\begin{align*} \int \frac{e^x}{1-e^{2 x}} \, dx &=\operatorname{Subst}\left (\int \frac{1}{1-x^2} \, dx,x,e^x\right )\\ &=\tanh ^{-1}\left (e^x\right )\\ \end{align*}
Mathematica [A] time = 0.0030205, size = 4, normalized size = 1. \[ \tanh ^{-1}\left (e^x\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.018, size = 4, normalized size = 1. \begin{align*}{\it Artanh} \left ({{\rm e}^{x}} \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 0.962192, size = 20, normalized size = 5. \begin{align*} \frac{1}{2} \, \log \left (e^{x} + 1\right ) - \frac{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 [B] time = 0.735926, size = 50, normalized size = 12.5 \begin{align*} \frac{1}{2} \, \log \left (e^{x} + 1\right ) - \frac{1}{2} \, \log \left (e^{x} - 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.099919, size = 15, normalized size = 3.75 \begin{align*} - \frac{\log{\left (e^{x} - 1 \right )}}{2} + \frac{\log{\left (e^{x} + 1 \right )}}{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.27548, size = 22, normalized size = 5.5 \begin{align*} \frac{1}{2} \, \log \left (e^{x} + 1\right ) - \frac{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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