Optimal. Leaf size=12 \[ e^{-x}-\tanh ^{-1}\left (e^x\right ) \]
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Rubi [A] time = 0.0133264, antiderivative size = 12, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {2282, 325, 207} \[ e^{-x}-\tanh ^{-1}\left (e^x\right ) \]
Antiderivative was successfully verified.
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Rule 2282
Rule 325
Rule 207
Rubi steps
\begin{align*} \int \frac{1}{-e^x+e^{3 x}} \, dx &=\operatorname{Subst}\left (\int \frac{1}{x^2 \left (-1+x^2\right )} \, dx,x,e^x\right )\\ &=e^{-x}+\operatorname{Subst}\left (\int \frac{1}{-1+x^2} \, dx,x,e^x\right )\\ &=e^{-x}-\tanh ^{-1}\left (e^x\right )\\ \end{align*}
Mathematica [C] time = 0.0059167, size = 19, normalized size = 1.58 \[ e^{-x} \, _2F_1\left (-\frac{1}{2},1;\frac{1}{2};e^{2 x}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.026, size = 20, normalized size = 1.7 \begin{align*}{\frac{\ln \left ( -1+{{\rm e}^{x}} \right ) }{2}}+ \left ({{\rm e}^{x}} \right ) ^{-1}-{\frac{\ln \left ( 1+{{\rm e}^{x}} \right ) }{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.967865, size = 26, normalized size = 2.17 \begin{align*} e^{\left (-x\right )} - \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 = 0.821906, size = 74, normalized size = 6.17 \begin{align*} -\frac{1}{2} \,{\left (e^{x} \log \left (e^{x} + 1\right ) - e^{x} \log \left (e^{x} - 1\right ) - 2\right )} e^{\left (-x\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.111915, size = 20, normalized size = 1.67 \begin{align*} \frac{\log{\left (e^{x} - 1 \right )}}{2} - \frac{\log{\left (e^{x} + 1 \right )}}{2} + e^{- x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.22295, size = 27, normalized size = 2.25 \begin{align*} e^{\left (-x\right )} - \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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