Optimal. Leaf size=57 \[ -\text {li}(x)+\frac {1}{2} i \text {Li}_2\left (e^{2 i x}\right )+\left (-\frac {1}{2}+\frac {i}{2}\right ) x^2-x \log \left (1-e^{2 i x}\right )+x \log \left (e^x \log (x) \sin (x)\right ) \]
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Rubi [A] time = 0.06, antiderivative size = 57, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {2549, 3717, 2190, 2279, 2391, 2298} \[ \frac {1}{2} i \text {PolyLog}\left (2,e^{2 i x}\right )-\text {li}(x)+\left (-\frac {1}{2}+\frac {i}{2}\right ) x^2-x \log \left (1-e^{2 i x}\right )+x \log \left (e^x \log (x) \sin (x)\right ) \]
Antiderivative was successfully verified.
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Rule 2190
Rule 2279
Rule 2298
Rule 2391
Rule 2549
Rule 3717
Rubi steps
\begin {align*} \int \log \left (e^x \log (x) \sin (x)\right ) \, dx &=x \log \left (e^x \log (x) \sin (x)\right )-\int \left (x+x \cot (x)+\frac {1}{\log (x)}\right ) \, dx\\ &=-\frac {x^2}{2}+x \log \left (e^x \log (x) \sin (x)\right )-\int x \cot (x) \, dx-\int \frac {1}{\log (x)} \, dx\\ &=\left (-\frac {1}{2}+\frac {i}{2}\right ) x^2+x \log \left (e^x \log (x) \sin (x)\right )-\text {li}(x)+2 i \int \frac {e^{2 i x} x}{1-e^{2 i x}} \, dx\\ &=\left (-\frac {1}{2}+\frac {i}{2}\right ) x^2-x \log \left (1-e^{2 i x}\right )+x \log \left (e^x \log (x) \sin (x)\right )-\text {li}(x)+\int \log \left (1-e^{2 i x}\right ) \, dx\\ &=\left (-\frac {1}{2}+\frac {i}{2}\right ) x^2-x \log \left (1-e^{2 i x}\right )+x \log \left (e^x \log (x) \sin (x)\right )-\text {li}(x)-\frac {1}{2} i \operatorname {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{2 i x}\right )\\ &=\left (-\frac {1}{2}+\frac {i}{2}\right ) x^2-x \log \left (1-e^{2 i x}\right )+x \log \left (e^x \log (x) \sin (x)\right )-\text {li}(x)+\frac {1}{2} i \text {Li}_2\left (e^{2 i x}\right )\\ \end {align*}
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Mathematica [A] time = 0.03, size = 56, normalized size = 0.98 \[ \frac {1}{2} \left (-2 \text {li}(x)+i \text {Li}_2\left (e^{2 i x}\right )+(-1+i) x^2-2 x \log \left (1-e^{2 i x}\right )+2 x \log \left (e^x \log (x) \sin (x)\right )\right ) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.49, size = 116, normalized size = 2.04 \[ -\frac {1}{2} \, x^{2} + x \log \left (e^{x} \log \relax (x) \sin \relax (x)\right ) - \frac {1}{2} \, x \log \left (\cos \relax (x) + i \, \sin \relax (x) + 1\right ) - \frac {1}{2} \, x \log \left (\cos \relax (x) - i \, \sin \relax (x) + 1\right ) - \frac {1}{2} \, x \log \left (-\cos \relax (x) + i \, \sin \relax (x) + 1\right ) - \frac {1}{2} \, x \log \left (-\cos \relax (x) - i \, \sin \relax (x) + 1\right ) + \frac {1}{2} i \, {\rm Li}_2\left (\cos \relax (x) + i \, \sin \relax (x)\right ) - \frac {1}{2} i \, {\rm Li}_2\left (\cos \relax (x) - i \, \sin \relax (x)\right ) - \frac {1}{2} i \, {\rm Li}_2\left (-\cos \relax (x) + i \, \sin \relax (x)\right ) + \frac {1}{2} i \, {\rm Li}_2\left (-\cos \relax (x) - i \, \sin \relax (x)\right ) - \operatorname {log\_integral}\relax (x) \]
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 \[ \int \log \left (e^{x} \log \relax (x) \sin \relax (x)\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 1.04, size = 583, normalized size = 10.23 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.29, size = 43, normalized size = 0.75 \[ \frac {1}{2} \, {\left (i \, \pi - 2 \, \log \relax (2)\right )} x - \left (\frac {1}{2} i - \frac {1}{2}\right ) \, x^{2} + x \log \left (\log \relax (x)\right ) - {\rm Ei}\left (\log \relax (x)\right ) + i \, {\rm Li}_2\left (-e^{\left (i \, x\right )}\right ) + i \, {\rm Li}_2\left (e^{\left (i \, x\right )}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \ln \left ({\mathrm {e}}^x\,\ln \relax (x)\,\sin \relax (x)\right ) \,d x \]
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 \[ \int \log {\left (e^{x} \log {\relax (x )} \sin {\relax (x )} \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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