Optimal. Leaf size=18 \[ -e^x+\left (1+e^x\right ) \log \left (1+e^x\right ) \]
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Rubi [A]
time = 0.02, antiderivative size = 22, normalized size of antiderivative = 1.22, number of steps
used = 4, number of rules used = 4, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {2225, 2634,
2280, 45} \begin {gather*} -e^x+e^x \log \left (e^x+1\right )+\log \left (e^x+1\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 2225
Rule 2280
Rule 2634
Rubi steps
\begin {align*} \int e^x \log \left (1+e^x\right ) \, dx &=e^x \log \left (1+e^x\right )-\int \frac {e^{2 x}}{1+e^x} \, dx\\ &=e^x \log \left (1+e^x\right )-\text {Subst}\left (\int \frac {x}{1+x} \, dx,x,e^x\right )\\ &=e^x \log \left (1+e^x\right )-\text {Subst}\left (\int \left (1+\frac {1}{-1-x}\right ) \, dx,x,e^x\right )\\ &=-e^x+\log \left (1+e^x\right )+e^x \log \left (1+e^x\right )\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 18, normalized size = 1.00 \begin {gather*} -e^x+\left (1+e^x\right ) \log \left (1+e^x\right ) \end {gather*}
Antiderivative was successfully verified.
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Mathics [F(-1)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [A]
time = 0.01, size = 17, normalized size = 0.94
method | result | size |
derivativedivides | \(\left (1+{\mathrm e}^{x}\right ) \ln \left (1+{\mathrm e}^{x}\right )-1-{\mathrm e}^{x}\) | \(17\) |
default | \(\left (1+{\mathrm e}^{x}\right ) \ln \left (1+{\mathrm e}^{x}\right )-1-{\mathrm e}^{x}\) | \(17\) |
norman | \(-{\mathrm e}^{x}+\ln \left (1+{\mathrm e}^{x}\right )+{\mathrm e}^{x} \ln \left (1+{\mathrm e}^{x}\right )\) | \(19\) |
risch | \(-{\mathrm e}^{x}+\ln \left (1+{\mathrm e}^{x}\right )+{\mathrm e}^{x} \ln \left (1+{\mathrm e}^{x}\right )\) | \(19\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 16, normalized size = 0.89 \begin {gather*} {\left (e^{x} + 1\right )} \log \left (e^{x} + 1\right ) - e^{x} - 1 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.33, size = 15, normalized size = 0.83 \begin {gather*} {\left (e^{x} + 1\right )} \log \left (e^{x} + 1\right ) - e^{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.00, size = 15, normalized size = 0.83 \begin {gather*} -\mathrm {e}^{x}-1+\left (\mathrm {e}^{x}+1\right ) \ln \left (\mathrm {e}^{x}+1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.24, size = 18, normalized size = 1.00 \begin {gather*} \ln \left ({\mathrm {e}}^x+1\right )-{\mathrm {e}}^x+{\mathrm {e}}^x\,\ln \left ({\mathrm {e}}^x+1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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