Optimal. Leaf size=12 \[ e^x-\log \left (e^x+1\right ) \]
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Rubi [A] time = 0.0196965, antiderivative size = 12, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {2248, 43} \[ e^x-\log \left (e^x+1\right ) \]
Antiderivative was successfully verified.
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Rule 2248
Rule 43
Rubi steps
\begin{align*} \int \frac{e^{2 x}}{1+e^x} \, dx &=\operatorname{Subst}\left (\int \frac{x}{1+x} \, dx,x,e^x\right )\\ &=\operatorname{Subst}\left (\int \left (1+\frac{1}{-1-x}\right ) \, dx,x,e^x\right )\\ &=e^x-\log \left (1+e^x\right )\\ \end{align*}
Mathematica [A] time = 0.0074975, size = 12, normalized size = 1. \[ e^x-\log \left (e^x+1\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0., size = 11, normalized size = 0.9 \begin{align*}{{\rm e}^{x}}-\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 = 0.954929, size = 14, normalized size = 1.17 \begin{align*} e^{x} - \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 = 2.00244, size = 27, normalized size = 2.25 \begin{align*} e^{x} - \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 = 0.082269, size = 8, normalized size = 0.67 \begin{align*} e^{x} - \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 = 1.0726, size = 14, normalized size = 1.17 \begin{align*} e^{x} - \log \left (e^{x} + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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