Optimal. Leaf size=8 \[ e \log \left (1+e^x\right ) \]
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Rubi [A]
time = 0.02, antiderivative size = 8, normalized size of antiderivative = 1.00, 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 = {2279, 2278, 31}
\begin {gather*} e \log \left (e^x+1\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 2278
Rule 2279
Rubi steps
\begin {align*} \int \frac {e^{1+x}}{1+e^x} \, dx &=e \int \frac {e^x}{1+e^x} \, dx\\ &=e \text {Subst}\left (\int \frac {1}{1+x} \, dx,x,e^x\right )\\ &=e \log \left (1+e^x\right )\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 8, normalized size = 1.00 \begin {gather*} e \log \left (1+e^x\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.01, size = 9, normalized size = 1.12
method | result | size |
default | \({\mathrm e} \ln \left (1+{\mathrm e}^{x}\right )\) | \(9\) |
norman | \({\mathrm e} \ln \left (1+{\mathrm e}^{x}\right )\) | \(9\) |
risch | \({\mathrm e} \ln \left (1+{\mathrm e}^{x}\right )\) | \(9\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 2.05, size = 8, normalized size = 1.00 \begin {gather*} e \log \left (e^{x} + 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.64, size = 11, normalized size = 1.38 \begin {gather*} e \log \left (e + e^{\left (x + 1\right )}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.03, size = 8, normalized size = 1.00 \begin {gather*} e \log {\left (e^{x} + 1 \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.46, size = 8, normalized size = 1.00 \begin {gather*} e \log \left (e^{x} + 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.18, size = 8, normalized size = 1.00 \begin {gather*} \mathrm {e}\,\ln \left ({\mathrm {e}}^x+1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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