Optimal. Leaf size=12 \[ e^x-\log \left (e^x+1\right ) \]
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Rubi [A] time = 0.02, antiderivative size = 12, normalized size of antiderivative = 1.00, 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 43
Rule 2248
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*}
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Mathematica [A] time = 0.01, size = 12, normalized size = 1.00 \[ e^x-\log \left (e^x+1\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.41, size = 10, normalized size = 0.83 \[ e^{x} - \log \left (e^{x} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.21, size = 10, normalized size = 0.83 \[ e^{x} - \log \left (e^{x} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 11, normalized size = 0.92 \[ {\mathrm e}^{x}-\ln \left ({\mathrm e}^{x}+1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.88, size = 10, normalized size = 0.83 \[ e^{x} - \log \left (e^{x} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.54, size = 10, normalized size = 0.83 \[ {\mathrm {e}}^x-\ln \left ({\mathrm {e}}^x+1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.09, size = 8, normalized size = 0.67 \[ e^{x} - \log {\left (e^{x} + 1 \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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