Optimal. Leaf size=18 \[ \left (e^x+1\right ) \log \left (e^x+1\right )-e^x \]
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Rubi [A] time = 0.0295601, antiderivative size = 22, normalized size of antiderivative = 1.22, number of steps used = 4, number of rules used = 4, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, 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 2194
Rule 2554
Rule 2248
Rule 43
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*}
Mathematica [A] time = 0.005832, size = 18, normalized size = 1. \[ \left (e^x+1\right ) \log \left (e^x+1\right )-e^x \]
Antiderivative was successfully verified.
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Maple [A] time = 0.002, size = 17, normalized size = 0.9 \begin{align*} \left ( 1+{{\rm e}^{x}} \right ) \ln \left ( 1+{{\rm e}^{x}} \right ) -1-{{\rm e}^{x}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.931435, size = 22, normalized size = 1.22 \begin{align*}{\left (e^{x} + 1\right )} \log \left (e^{x} + 1\right ) - e^{x} - 1 \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.92823, size = 41, normalized size = 2.28 \begin{align*}{\left (e^{x} + 1\right )} \log \left (e^{x} + 1\right ) - e^{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.04817, size = 22, normalized size = 1.22 \begin{align*}{\left (e^{x} + 1\right )} \log \left (e^{x} + 1\right ) - e^{x} - 1 \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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