Optimal. Leaf size=23 \[ \frac{1}{3} \log \left (1-2 e^x\right )-\frac{1}{3} \log \left (e^x+1\right ) \]
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Rubi [A] time = 0.0168223, antiderivative size = 23, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.188, Rules used = {2282, 616, 31} \[ \frac{1}{3} \log \left (1-2 e^x\right )-\frac{1}{3} \log \left (e^x+1\right ) \]
Antiderivative was successfully verified.
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Rule 2282
Rule 616
Rule 31
Rubi steps
\begin{align*} \int \frac{1}{1-e^{-x}+2 e^x} \, dx &=\operatorname{Subst}\left (\int \frac{1}{-1+x+2 x^2} \, dx,x,e^x\right )\\ &=\frac{2}{3} \operatorname{Subst}\left (\int \frac{1}{-1+2 x} \, dx,x,e^x\right )-\frac{2}{3} \operatorname{Subst}\left (\int \frac{1}{2+2 x} \, dx,x,e^x\right )\\ &=\frac{1}{3} \log \left (1-2 e^x\right )-\frac{1}{3} \log \left (1+e^x\right )\\ \end{align*}
Mathematica [A] time = 0.0112246, size = 16, normalized size = 0.7 \[ -\frac{2}{3} \tanh ^{-1}\left (\frac{1}{3} \left (4 e^x+1\right )\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 18, normalized size = 0.8 \begin{align*} -{\frac{\ln \left ( 1+{{\rm e}^{x}} \right ) }{3}}+{\frac{\ln \left ( 2\,{{\rm e}^{x}}-1 \right ) }{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.933128, size = 26, normalized size = 1.13 \begin{align*} -\frac{1}{3} \, \log \left (e^{\left (-x\right )} + 1\right ) + \frac{1}{3} \, \log \left (e^{\left (-x\right )} - 2\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.93344, size = 53, normalized size = 2.3 \begin{align*} \frac{1}{3} \, \log \left (2 \, e^{x} - 1\right ) - \frac{1}{3} \, \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 = 0.108245, size = 19, normalized size = 0.83 \begin{align*} \frac{\log{\left (-2 + e^{- x} \right )}}{3} - \frac{\log{\left (1 + e^{- x} \right )}}{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.04985, size = 24, normalized size = 1.04 \begin{align*} -\frac{1}{3} \, \log \left (e^{x} + 1\right ) + \frac{1}{3} \, \log \left ({\left | 2 \, e^{x} - 1 \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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