Optimal. Leaf size=18 \[ e^t \log (t+1)-\frac{\text{ExpIntegralEi}(t+1)}{e} \]
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Rubi [A] time = 0.0199313, antiderivative size = 18, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 3, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375, Rules used = {2194, 2554, 2178} \[ e^t \log (t+1)-\frac{\text{ExpIntegralEi}(t+1)}{e} \]
Antiderivative was successfully verified.
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Rule 2194
Rule 2554
Rule 2178
Rubi steps
\begin{align*} \int e^t \log (1+t) \, dt &=e^t \log (1+t)-\int \frac{e^t}{1+t} \, dt\\ &=-\frac{\text{Ei}(1+t)}{e}+e^t \log (1+t)\\ \end{align*}
Mathematica [A] time = 0.0081084, size = 18, normalized size = 1. \[ e^t \log (t+1)-\frac{\text{ExpIntegralEi}(t+1)}{e} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.02, size = 19, normalized size = 1.1 \begin{align*}{{\rm e}^{t}}\ln \left ( 1+t \right ) +{{\rm e}^{-1}}{\it Ei} \left ( 1,-1-t \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.04043, size = 24, normalized size = 1.33 \begin{align*} e^{\left (-1\right )} E_{1}\left (-t - 1\right ) + e^{t} \log \left (t + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.16872, size = 58, normalized size = 3.22 \begin{align*}{\left (e^{\left (t + 1\right )} \log \left (t + 1\right ) -{\rm Ei}\left (t + 1\right )\right )} e^{\left (-1\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int e^{t} \log{\left (t + 1 \right )}\, dt \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.13445, size = 22, normalized size = 1.22 \begin{align*} -{\rm Ei}\left (t + 1\right ) e^{\left (-1\right )} + e^{t} \log \left (t + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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