Optimal. Leaf size=57 \[ \frac{1}{2} i \text{PolyLog}\left (2,e^{2 i x}\right )-\text{li}(x)+\left (-\frac{1}{2}+\frac{i}{2}\right ) x^2-x \log \left (1-e^{2 i x}\right )+x \log \left (e^x \log (x) \sin (x)\right ) \]
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Rubi [A] time = 0.0642922, antiderivative size = 57, normalized size of antiderivative = 1., 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 2549
Rule 3717
Rule 2190
Rule 2279
Rule 2391
Rule 2298
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*}
Mathematica [A] time = 0.0297676, size = 56, normalized size = 0.98 \[ \frac{1}{2} \left (i \text{PolyLog}\left (2,e^{2 i x}\right )-2 \text{li}(x)+(-1+i) x^2-2 x \log \left (1-e^{2 i x}\right )+2 x \log \left (e^x \log (x) \sin (x)\right )\right ) \]
Antiderivative was successfully verified.
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Maple [C] time = 0.244, size = 583, normalized size = 10.2 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.80966, size = 58, normalized size = 1.02 \begin{align*} \frac{1}{2} \,{\left (i \, \pi - 2 \, \log \left (2\right )\right )} x - \left (\frac{1}{2} i - \frac{1}{2}\right ) \, x^{2} + x \log \left (\log \left (x\right )\right ) -{\rm Ei}\left (\log \left (x\right )\right ) + i \,{\rm Li}_2\left (-e^{\left (i \, x\right )}\right ) + i \,{\rm Li}_2\left (e^{\left (i \, x\right )}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.42055, size = 447, normalized size = 7.84 \begin{align*} -\frac{1}{2} \, x^{2} + x \log \left (e^{x} \log \left (x\right ) \sin \left (x\right )\right ) - \frac{1}{2} \, x \log \left (\cos \left (x\right ) + i \, \sin \left (x\right ) + 1\right ) - \frac{1}{2} \, x \log \left (\cos \left (x\right ) - i \, \sin \left (x\right ) + 1\right ) - \frac{1}{2} \, x \log \left (-\cos \left (x\right ) + i \, \sin \left (x\right ) + 1\right ) - \frac{1}{2} \, x \log \left (-\cos \left (x\right ) - i \, \sin \left (x\right ) + 1\right ) + \frac{1}{2} i \,{\rm Li}_2\left (\cos \left (x\right ) + i \, \sin \left (x\right )\right ) - \frac{1}{2} i \,{\rm Li}_2\left (\cos \left (x\right ) - i \, \sin \left (x\right )\right ) - \frac{1}{2} i \,{\rm Li}_2\left (-\cos \left (x\right ) + i \, \sin \left (x\right )\right ) + \frac{1}{2} i \,{\rm Li}_2\left (-\cos \left (x\right ) - i \, \sin \left (x\right )\right ) - \logintegral \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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