Optimal. Leaf size=6 \[ -\tanh ^{-1}\left (e^x\right ) \]
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Rubi [A] time = 0.0095504, antiderivative size = 6, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {2282, 207} \[ -\tanh ^{-1}\left (e^x\right ) \]
Antiderivative was successfully verified.
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Rule 2282
Rule 207
Rubi steps
\begin{align*} \int \frac{1}{-e^{-x}+e^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 = 0.0027007, size = 6, normalized size = 1. \[ -\tanh ^{-1}\left (e^x\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.002, size = 6, 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 = 0.935972, size = 26, normalized size = 4.33 \begin{align*} -\frac{1}{2} \, \log \left (e^{\left (-x\right )} + 1\right ) + \frac{1}{2} \, \log \left (e^{\left (-x\right )} - 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.90714, size = 51, normalized size = 8.5 \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 = 0.101435, size = 19, normalized size = 3.17 \begin{align*} \frac{\log{\left (-1 + e^{- x} \right )}}{2} - \frac{\log{\left (1 + e^{- x} \right )}}{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.05135, size = 22, normalized size = 3.67 \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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