Optimal. Leaf size=80 \[ \frac{1}{2} i x \text{PolyLog}\left (2,e^{2 i x}\right )-\frac{1}{4} \text{PolyLog}\left (3,e^{2 i x}\right )-\frac{1}{2} \text{Ei}(2 \log (x))+\frac{i x^3}{6}-\frac{1}{2} x^2 \log \left (1-e^{2 i x}\right )+\frac{1}{2} x^2 \log (\log (x) \sin (x)) \]
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Rubi [A] time = 0.181542, antiderivative size = 80, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 12, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 1.5, Rules used = {30, 2555, 12, 6688, 14, 3717, 2190, 2531, 2282, 6589, 2309, 2178} \[ \frac{1}{2} i x \text{PolyLog}\left (2,e^{2 i x}\right )-\frac{1}{4} \text{PolyLog}\left (3,e^{2 i x}\right )-\frac{1}{2} \text{Ei}(2 \log (x))+\frac{i x^3}{6}-\frac{1}{2} x^2 \log \left (1-e^{2 i x}\right )+\frac{1}{2} x^2 \log (\log (x) \sin (x)) \]
Antiderivative was successfully verified.
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Rule 30
Rule 2555
Rule 12
Rule 6688
Rule 14
Rule 3717
Rule 2190
Rule 2531
Rule 2282
Rule 6589
Rule 2309
Rule 2178
Rubi steps
\begin{align*} \int x \log (\log (x) \sin (x)) \, dx &=\frac{1}{2} x^2 \log (\log (x) \sin (x))-\int \frac{x (1+x \cot (x) \log (x))}{2 \log (x)} \, dx\\ &=\frac{1}{2} x^2 \log (\log (x) \sin (x))-\frac{1}{2} \int \frac{x (1+x \cot (x) \log (x))}{\log (x)} \, dx\\ &=\frac{1}{2} x^2 \log (\log (x) \sin (x))-\frac{1}{2} \int x \left (x \cot (x)+\frac{1}{\log (x)}\right ) \, dx\\ &=\frac{1}{2} x^2 \log (\log (x) \sin (x))-\frac{1}{2} \int \left (x^2 \cot (x)+\frac{x}{\log (x)}\right ) \, dx\\ &=\frac{1}{2} x^2 \log (\log (x) \sin (x))-\frac{1}{2} \int x^2 \cot (x) \, dx-\frac{1}{2} \int \frac{x}{\log (x)} \, dx\\ &=\frac{i x^3}{6}+\frac{1}{2} x^2 \log (\log (x) \sin (x))+i \int \frac{e^{2 i x} x^2}{1-e^{2 i x}} \, dx-\frac{1}{2} \operatorname{Subst}\left (\int \frac{e^{2 x}}{x} \, dx,x,\log (x)\right )\\ &=\frac{i x^3}{6}-\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 (\log (x) \sin (x))+\int x \log \left (1-e^{2 i x}\right ) \, dx\\ &=\frac{i x^3}{6}-\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 (\log (x) \sin (x))+\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\\ &=\frac{i x^3}{6}-\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 (\log (x) \sin (x))+\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 )\\ &=\frac{i x^3}{6}-\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 (\log (x) \sin (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 )\\ \end{align*}
Mathematica [A] time = 0.0389492, size = 79, normalized size = 0.99 \[ \frac{1}{48} \left (-24 i x \text{PolyLog}\left (2,e^{-2 i x}\right )-12 \text{PolyLog}\left (3,e^{-2 i x}\right )-24 \text{Ei}(2 \log (x))-8 i x^3-24 x^2 \log \left (1-e^{-2 i x}\right )+24 x^2 \log (\log (x) \sin (x))+i \pi ^3\right ) \]
Antiderivative was successfully verified.
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Maple [F] time = 0.207, size = 0, normalized size = 0. \begin{align*} \int x\ln \left ( \ln \left ( x \right ) \sin \left ( x \right ) \right ) \, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 2.03011, size = 95, normalized size = 1.19 \begin{align*} \frac{1}{12} \,{\left (3 i \, \pi - 6 \, \log \left (2\right )\right )} x^{2} - \frac{1}{3} i \, x^{3} + \frac{1}{2} \, x^{2} \log \left (\log \left (x\right )\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 \left (x\right )\right ) -{\rm Li}_{3}(-e^{\left (i \, x\right )}) -{\rm Li}_{3}(e^{\left (i \, x\right )}) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [C] time = 2.51811, size = 662, normalized size = 8.28 \begin{align*} \frac{1}{2} \, x^{2} \log \left (\log \left (x\right ) \sin \left (x\right )\right ) - \frac{1}{4} \, x^{2} \log \left (\cos \left (x\right ) + i \, \sin \left (x\right ) + 1\right ) - \frac{1}{4} \, x^{2} \log \left (\cos \left (x\right ) - i \, \sin \left (x\right ) + 1\right ) - \frac{1}{4} \, x^{2} \log \left (-\cos \left (x\right ) + i \, \sin \left (x\right ) + 1\right ) - \frac{1}{4} \, x^{2} \log \left (-\cos \left (x\right ) - i \, \sin \left (x\right ) + 1\right ) + \frac{1}{2} i \, x{\rm Li}_2\left (\cos \left (x\right ) + i \, \sin \left (x\right )\right ) - \frac{1}{2} i \, x{\rm Li}_2\left (\cos \left (x\right ) - i \, \sin \left (x\right )\right ) - \frac{1}{2} i \, x{\rm Li}_2\left (-\cos \left (x\right ) + i \, \sin \left (x\right )\right ) + \frac{1}{2} i \, x{\rm Li}_2\left (-\cos \left (x\right ) - i \, \sin \left (x\right )\right ) - \frac{1}{2} \, \logintegral \left (x^{2}\right ) - \frac{1}{2} \,{\rm polylog}\left (3, \cos \left (x\right ) + i \, \sin \left (x\right )\right ) - \frac{1}{2} \,{\rm polylog}\left (3, \cos \left (x\right ) - i \, \sin \left (x\right )\right ) - \frac{1}{2} \,{\rm polylog}\left (3, -\cos \left (x\right ) + i \, \sin \left (x\right )\right ) - \frac{1}{2} \,{\rm polylog}\left (3, -\cos \left (x\right ) - i \, \sin \left (x\right )\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x \log{\left (\log{\left (x \right )} \sin{\left (x \right )} \right )}\, dx \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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