Optimal. Leaf size=6 \[ \log \left (1+e^x\right ) \]
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Rubi [A]
time = 0.01, antiderivative size = 6, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {2278, 31}
\begin {gather*} \log \left (e^x+1\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 2278
Rubi steps
\begin {align*} \int \frac {e^x}{1+e^x} \, dx &=\text {Subst}\left (\int \frac {1}{1+x} \, dx,x,e^x\right )\\ &=\log \left (1+e^x\right )\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 6, normalized size = 1.00 \begin {gather*} \log \left (1+e^x\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.00, size = 6, normalized size = 1.00
method | result | size |
derivativedivides | \(\ln \left (1+{\mathrm e}^{x}\right )\) | \(6\) |
default | \(\ln \left (1+{\mathrm e}^{x}\right )\) | \(6\) |
norman | \(\ln \left (1+{\mathrm e}^{x}\right )\) | \(6\) |
risch | \(\ln \left (1+{\mathrm e}^{x}\right )\) | \(6\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 1.30, size = 5, normalized size = 0.83 \begin {gather*} \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 = 0.43, size = 5, normalized size = 0.83 \begin {gather*} \log \left (e^{x} + 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.02, size = 5, normalized size = 0.83 \begin {gather*} \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 = 0.58, size = 5, normalized size = 0.83 \begin {gather*} \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 = 0.00, size = 5, normalized size = 0.83 \begin {gather*} \ln \left ({\mathrm {e}}^x+1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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