Optimal. Leaf size=18 \[ \left (e^x+1\right ) \log \left (e^x+1\right )-e^x \]
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Rubi [A] time = 0.03, 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 = {2194, 2554, 2248, 43} \[ -e^x+e^x \log \left (e^x+1\right )+\log \left (e^x+1\right ) \]
Antiderivative was successfully verified.
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Rule 43
Rule 2194
Rule 2248
Rule 2554
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 )-\operatorname {Subst}\left (\int \frac {x}{1+x} \, dx,x,e^x\right )\\ &=e^x \log \left (1+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 )+e^x \log \left (1+e^x\right )\\ \end {align*}
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Mathematica [A] time = 0.01, size = 18, normalized size = 1.00 \[ \left (e^x+1\right ) \log \left (e^x+1\right )-e^x \]
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 15, normalized size = 0.83 \[ {\left (e^{x} + 1\right )} \log \left (e^{x} + 1\right ) - e^{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.04, size = 16, normalized size = 0.89 \[ {\left (e^{x} + 1\right )} \log \left (e^{x} + 1\right ) - e^{x} - 1 \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 17, normalized size = 0.94 \[ -{\mathrm e}^{x}+\left ({\mathrm e}^{x}+1\right ) \ln \left ({\mathrm e}^{x}+1\right )-1 \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.42, size = 16, normalized size = 0.89 \[ {\left (e^{x} + 1\right )} \log \left (e^{x} + 1\right ) - e^{x} - 1 \]
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 \[ \ln \left ({\mathrm {e}}^x+1\right )-{\mathrm {e}}^x+{\mathrm {e}}^x\,\ln \left ({\mathrm {e}}^x+1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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