Optimal. Leaf size=22 \[ \text{ExpIntegralEi}(x)-e^x-e^x \log (x)+e^x x \log (x) \]
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Rubi [A] time = 0.0521265, antiderivative size = 22, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 7, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.714, Rules used = {2176, 2194, 2554, 2199, 2178} \[ \text{ExpIntegralEi}(x)-e^x-e^x \log (x)+e^x x \log (x) \]
Antiderivative was successfully verified.
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Rule 2176
Rule 2194
Rule 2554
Rule 2199
Rule 2178
Rubi steps
\begin{align*} \int e^x x \log (x) \, dx &=-e^x \log (x)+e^x x \log (x)-\int \frac{e^x (-1+x)}{x} \, dx\\ &=-e^x \log (x)+e^x x \log (x)-\int \left (e^x-\frac{e^x}{x}\right ) \, dx\\ &=-e^x \log (x)+e^x x \log (x)-\int e^x \, dx+\int \frac{e^x}{x} \, dx\\ &=-e^x+\text{Ei}(x)-e^x \log (x)+e^x x \log (x)\\ \end{align*}
Mathematica [A] time = 0.0110205, size = 17, normalized size = 0.77 \[ \text{ExpIntegralEi}(x)-e^x+e^x (x-1) \log (x) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.013, size = 21, normalized size = 1. \begin{align*} \left ( -1+x \right ){{\rm e}^{x}}\ln \left ( x \right ) -{\it Ei} \left ( 1,-x \right ) -{{\rm e}^{x}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.02932, size = 20, normalized size = 0.91 \begin{align*}{\left (x - 1\right )} e^{x} \log \left (x\right ) +{\rm Ei}\left (x\right ) - e^{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.85356, size = 46, normalized size = 2.09 \begin{align*}{\left (x - 1\right )} e^{x} \log \left (x\right ) +{\rm Ei}\left (x\right ) - e^{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 3.19639, size = 17, normalized size = 0.77 \begin{align*} \left (x e^{x} - e^{x}\right ) \log{\left (x \right )} - e^{x} + \operatorname{Ei}{\left (x \right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.07831, size = 20, normalized size = 0.91 \begin{align*}{\left (x - 1\right )} e^{x} \log \left (x\right ) +{\rm Ei}\left (x\right ) - e^{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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