Optimal. Leaf size=85 \[ -\frac {1}{2} \text {Ei}(2 \log (x))+\frac {1}{2} i x \text {Li}_2\left (e^{2 i x}\right )-\frac {1}{4} \text {Li}_3\left (e^{2 i x}\right )+\left (-\frac {1}{6}+\frac {i}{6}\right ) x^3-\frac {1}{2} x^2 \log \left (1-e^{2 i x}\right )+\frac {1}{2} x^2 \log \left (e^x \log (x) \sin (x)\right ) \]
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Rubi [A] time = 0.14, antiderivative size = 85, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 11, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {30, 2555, 12, 14, 3717, 2190, 2531, 2282, 6589, 2309, 2178} \[ \frac {1}{2} i x \text {PolyLog}\left (2,e^{2 i x}\right )-\frac {1}{4} \text {PolyLog}\left (3,e^{2 i x}\right )-\frac {1}{2} \text {Ei}(2 \log (x))+\left (-\frac {1}{6}+\frac {i}{6}\right ) x^3-\frac {1}{2} x^2 \log \left (1-e^{2 i x}\right )+\frac {1}{2} x^2 \log \left (e^x \log (x) \sin (x)\right ) \]
Antiderivative was successfully verified.
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Rule 12
Rule 14
Rule 30
Rule 2178
Rule 2190
Rule 2282
Rule 2309
Rule 2531
Rule 2555
Rule 3717
Rule 6589
Rubi steps
\begin {align*} \int x \log \left (e^x \log (x) \sin (x)\right ) \, dx &=\frac {1}{2} x^2 \log \left (e^x \log (x) \sin (x)\right )-\int \frac {1}{2} x^2 \left (1+\cot (x)+\frac {1}{x \log (x)}\right ) \, dx\\ &=\frac {1}{2} x^2 \log \left (e^x \log (x) \sin (x)\right )-\frac {1}{2} \int x^2 \left (1+\cot (x)+\frac {1}{x \log (x)}\right ) \, dx\\ &=\frac {1}{2} x^2 \log \left (e^x \log (x) \sin (x)\right )-\frac {1}{2} \int \left (x^2 (1+\cot (x))+\frac {x}{\log (x)}\right ) \, dx\\ &=\frac {1}{2} x^2 \log \left (e^x \log (x) \sin (x)\right )-\frac {1}{2} \int x^2 (1+\cot (x)) \, dx-\frac {1}{2} \int \frac {x}{\log (x)} \, dx\\ &=\frac {1}{2} x^2 \log \left (e^x \log (x) \sin (x)\right )-\frac {1}{2} \int \left (x^2+x^2 \cot (x)\right ) \, dx-\frac {1}{2} \operatorname {Subst}\left (\int \frac {e^{2 x}}{x} \, dx,x,\log (x)\right )\\ &=-\frac {x^3}{6}-\frac {1}{2} \text {Ei}(2 \log (x))+\frac {1}{2} x^2 \log \left (e^x \log (x) \sin (x)\right )-\frac {1}{2} \int x^2 \cot (x) \, dx\\ &=\left (-\frac {1}{6}+\frac {i}{6}\right ) x^3-\frac {1}{2} \text {Ei}(2 \log (x))+\frac {1}{2} x^2 \log \left (e^x \log (x) \sin (x)\right )+i \int \frac {e^{2 i x} x^2}{1-e^{2 i x}} \, dx\\ &=\left (-\frac {1}{6}+\frac {i}{6}\right ) x^3-\frac {1}{2} \text {Ei}(2 \log (x))-\frac {1}{2} x^2 \log \left (1-e^{2 i x}\right )+\frac {1}{2} x^2 \log \left (e^x \log (x) \sin (x)\right )+\int x \log \left (1-e^{2 i x}\right ) \, dx\\ &=\left (-\frac {1}{6}+\frac {i}{6}\right ) x^3-\frac {1}{2} \text {Ei}(2 \log (x))-\frac {1}{2} x^2 \log \left (1-e^{2 i x}\right )+\frac {1}{2} x^2 \log \left (e^x \log (x) \sin (x)\right )+\frac {1}{2} i x \text {Li}_2\left (e^{2 i x}\right )-\frac {1}{2} i \int \text {Li}_2\left (e^{2 i x}\right ) \, dx\\ &=\left (-\frac {1}{6}+\frac {i}{6}\right ) x^3-\frac {1}{2} \text {Ei}(2 \log (x))-\frac {1}{2} x^2 \log \left (1-e^{2 i x}\right )+\frac {1}{2} x^2 \log \left (e^x \log (x) \sin (x)\right )+\frac {1}{2} i x \text {Li}_2\left (e^{2 i x}\right )-\frac {1}{4} \operatorname {Subst}\left (\int \frac {\text {Li}_2(x)}{x} \, dx,x,e^{2 i x}\right )\\ &=\left (-\frac {1}{6}+\frac {i}{6}\right ) x^3-\frac {1}{2} \text {Ei}(2 \log (x))-\frac {1}{2} x^2 \log \left (1-e^{2 i x}\right )+\frac {1}{2} x^2 \log \left (e^x \log (x) \sin (x)\right )+\frac {1}{2} i x \text {Li}_2\left (e^{2 i x}\right )-\frac {1}{4} \text {Li}_3\left (e^{2 i x}\right )\\ \end {align*}
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Mathematica [A] time = 0.07, size = 82, normalized size = 0.96 \[ \frac {1}{48} \left (-24 \text {Ei}(2 \log (x))-24 i x \text {Li}_2\left (e^{-2 i x}\right )-12 \text {Li}_3\left (e^{-2 i x}\right )+(-8-8 i) x^3-24 x^2 \log \left (1-e^{-2 i x}\right )+24 x^2 \log \left (e^x \log (x) \sin (x)\right )+i \pi ^3\right ) \]
Antiderivative was successfully verified.
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fricas [C] time = 0.51, size = 181, normalized size = 2.13 \[ -\frac {1}{6} \, x^{3} + \frac {1}{2} \, x^{2} \log \left (e^{x} \log \relax (x) \sin \relax (x)\right ) - \frac {1}{4} \, x^{2} \log \left (\cos \relax (x) + i \, \sin \relax (x) + 1\right ) - \frac {1}{4} \, x^{2} \log \left (\cos \relax (x) - i \, \sin \relax (x) + 1\right ) - \frac {1}{4} \, x^{2} \log \left (-\cos \relax (x) + i \, \sin \relax (x) + 1\right ) - \frac {1}{4} \, x^{2} \log \left (-\cos \relax (x) - i \, \sin \relax (x) + 1\right ) + \frac {1}{2} i \, x {\rm Li}_2\left (\cos \relax (x) + i \, \sin \relax (x)\right ) - \frac {1}{2} i \, x {\rm Li}_2\left (\cos \relax (x) - i \, \sin \relax (x)\right ) - \frac {1}{2} i \, x {\rm Li}_2\left (-\cos \relax (x) + i \, \sin \relax (x)\right ) + \frac {1}{2} i \, x {\rm Li}_2\left (-\cos \relax (x) - i \, \sin \relax (x)\right ) - \frac {1}{2} \, \operatorname {log\_integral}\left (x^{2}\right ) - \frac {1}{2} \, {\rm polylog}\left (3, \cos \relax (x) + i \, \sin \relax (x)\right ) - \frac {1}{2} \, {\rm polylog}\left (3, \cos \relax (x) - i \, \sin \relax (x)\right ) - \frac {1}{2} \, {\rm polylog}\left (3, -\cos \relax (x) + i \, \sin \relax (x)\right ) - \frac {1}{2} \, {\rm polylog}\left (3, -\cos \relax (x) - i \, \sin \relax (x)\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x \log \left (e^{x} \log \relax (x) \sin \relax (x)\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 2.35, size = 0, normalized size = 0.00 \[ \int x \ln \left ({\mathrm e}^{x} \ln \relax (x ) \sin \relax (x )\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.43, size = 70, normalized size = 0.82 \[ \frac {1}{12} \, {\left (3 i \, \pi - 6 \, \log \relax (2)\right )} x^{2} - \left (\frac {1}{3} i - \frac {1}{3}\right ) \, x^{3} + \frac {1}{2} \, x^{2} \log \left (\log \relax (x)\right ) + i \, x {\rm Li}_2\left (-e^{\left (i \, x\right )}\right ) + i \, x {\rm Li}_2\left (e^{\left (i \, x\right )}\right ) - \frac {1}{2} \, {\rm Ei}\left (2 \, \log \relax (x)\right ) - {\rm Li}_{3}(-e^{\left (i \, x\right )}) - {\rm Li}_{3}(e^{\left (i \, x\right )}) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int x\,\ln \left ({\mathrm {e}}^x\,\ln \relax (x)\,\sin \relax (x)\right ) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x \log {\left (e^{x} \log {\relax (x )} \sin {\relax (x )} \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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