Optimal. Leaf size=22 \[ \text {Ei}(x)-e^x-e^x \log (x)+e^x x \log (x) \]
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Rubi [A] time = 0.05, antiderivative size = 22, normalized size of antiderivative = 1.00, 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 2178
Rule 2194
Rule 2199
Rule 2554
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*}
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Mathematica [A] time = 0.01, size = 17, normalized size = 0.77 \[ \text {Ei}(x)-e^x+e^x (x-1) \log (x) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 15, normalized size = 0.68 \[ {\left (x - 1\right )} e^{x} \log \relax (x) + {\rm Ei}\relax (x) - e^{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.15, size = 15, normalized size = 0.68 \[ {\left (x - 1\right )} e^{x} \log \relax (x) + {\rm Ei}\relax (x) - e^{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 21, normalized size = 0.95 \[ \left (x -1\right ) {\mathrm e}^{x} \ln \relax (x )-\Ei \left (1, -x \right )-{\mathrm e}^{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.52, size = 15, normalized size = 0.68 \[ {\left (x - 1\right )} e^{x} \log \relax (x) + {\rm Ei}\relax (x) - e^{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.22, size = 28, normalized size = 1.27 \[ \mathrm {ei}\relax (x)-\frac {x\,{\mathrm {e}}^x+x\,{\mathrm {e}}^x\,\ln \relax (x)-x^2\,{\mathrm {e}}^x\,\ln \relax (x)}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 3.34, size = 17, normalized size = 0.77 \[ \left (x e^{x} - e^{x}\right ) \log {\relax (x )} - e^{x} + \operatorname {Ei}{\relax (x )} \]
Verification of antiderivative is not currently implemented for this CAS.
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