Optimal. Leaf size=52 \[ -\text {li}(x)+\frac {1}{2} i \text {Li}_2\left (e^{2 i x}\right )+\frac {i x^2}{2}-x \log \left (1-e^{2 i x}\right )+x \log (\log (x) \sin (x)) \]
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Rubi [A] time = 0.06, antiderivative size = 52, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {2549, 3717, 2190, 2279, 2391, 2298} \[ \frac {1}{2} i \text {PolyLog}\left (2,e^{2 i x}\right )-\text {li}(x)+\frac {i x^2}{2}-x \log \left (1-e^{2 i x}\right )+x \log (\log (x) \sin (x)) \]
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 (\log (x) \sin (x)) \, dx &=x \log (\log (x) \sin (x))-\int \left (x \cot (x)+\frac {1}{\log (x)}\right ) \, dx\\ &=x \log (\log (x) \sin (x))-\int x \cot (x) \, dx-\int \frac {1}{\log (x)} \, dx\\ &=\frac {i x^2}{2}+x \log (\log (x) \sin (x))-\text {li}(x)+2 i \int \frac {e^{2 i x} x}{1-e^{2 i x}} \, dx\\ &=\frac {i x^2}{2}-x \log \left (1-e^{2 i x}\right )+x \log (\log (x) \sin (x))-\text {li}(x)+\int \log \left (1-e^{2 i x}\right ) \, dx\\ &=\frac {i x^2}{2}-x \log \left (1-e^{2 i x}\right )+x \log (\log (x) \sin (x))-\text {li}(x)-\frac {1}{2} i \operatorname {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{2 i x}\right )\\ &=\frac {i x^2}{2}-x \log \left (1-e^{2 i x}\right )+x \log (\log (x) \sin (x))-\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 = 47, normalized size = 0.90 \[ -\text {li}(x)+\frac {1}{2} i \left (x^2+\text {Li}_2\left (e^{2 i x}\right )\right )-x \log \left (1-e^{2 i x}\right )+x \log (\log (x) \sin (x)) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.46, size = 109, normalized size = 2.10 \[ x \log \left (\log \relax (x) \sin \relax (x)\right ) - \frac {1}{2} \, x \log \left (\cos \relax (x) + i \, \sin \relax (x) + 1\right ) - \frac {1}{2} \, x \log \left (\cos \relax (x) - i \, \sin \relax (x) + 1\right ) - \frac {1}{2} \, x \log \left (-\cos \relax (x) + i \, \sin \relax (x) + 1\right ) - \frac {1}{2} \, x \log \left (-\cos \relax (x) - i \, \sin \relax (x) + 1\right ) + \frac {1}{2} i \, {\rm Li}_2\left (\cos \relax (x) + i \, \sin \relax (x)\right ) - \frac {1}{2} i \, {\rm Li}_2\left (\cos \relax (x) - i \, \sin \relax (x)\right ) - \frac {1}{2} i \, {\rm Li}_2\left (-\cos \relax (x) + i \, \sin \relax (x)\right ) + \frac {1}{2} i \, {\rm Li}_2\left (-\cos \relax (x) - i \, \sin \relax (x)\right ) - \operatorname {log\_integral}\relax (x) \]
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 \[ \int \log \left (\log \relax (x) \sin \relax (x)\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.84, size = 368, normalized size = 7.08 \[ -\frac {i \pi x \,\mathrm {csgn}\left (i \left ({\mathrm e}^{2 i x}-1\right )\right ) \mathrm {csgn}\left (i \ln \relax (x )\right ) \mathrm {csgn}\left (i \left ({\mathrm e}^{2 i x}-1\right ) \ln \relax (x )\right )}{2}+\frac {i \pi x \,\mathrm {csgn}\left (i \left ({\mathrm e}^{2 i x}-1\right )\right ) \mathrm {csgn}\left (i \left ({\mathrm e}^{2 i x}-1\right ) \ln \relax (x )\right )^{2}}{2}+\frac {i \pi x \,\mathrm {csgn}\left (i {\mathrm e}^{-i x}\right ) \mathrm {csgn}\left (\ln \relax (x ) \sin \relax (x )\right )^{2}}{2}+\frac {i \pi x \,\mathrm {csgn}\left (i {\mathrm e}^{-i x}\right ) \mathrm {csgn}\left (\ln \relax (x ) \sin \relax (x )\right ) \mathrm {csgn}\left (i \left ({\mathrm e}^{2 i x}-1\right ) \ln \relax (x )\right )}{2}+\frac {i \pi x \,\mathrm {csgn}\left (i \ln \relax (x )\right ) \mathrm {csgn}\left (i \left ({\mathrm e}^{2 i x}-1\right ) \ln \relax (x )\right )^{2}}{2}+\frac {i \pi x \mathrm {csgn}\left (\ln \relax (x ) \sin \relax (x )\right )^{3}}{2}+\frac {i \pi x \mathrm {csgn}\left (\ln \relax (x ) \sin \relax (x )\right )^{2} \mathrm {csgn}\left (i \left ({\mathrm e}^{2 i x}-1\right ) \ln \relax (x )\right )}{2}-\frac {i \pi x \,\mathrm {csgn}\left (\ln \relax (x ) \sin \relax (x )\right ) \mathrm {csgn}\left (i \ln \relax (x ) \sin \relax (x )\right )^{2}}{2}-\frac {i \pi x \mathrm {csgn}\left (i \left ({\mathrm e}^{2 i x}-1\right ) \ln \relax (x )\right )^{3}}{2}-\frac {i \pi x \mathrm {csgn}\left (i \ln \relax (x ) \sin \relax (x )\right )^{3}}{2}+\frac {i \pi x \,\mathrm {csgn}\left (\ln \relax (x ) \sin \relax (x )\right ) \mathrm {csgn}\left (i \ln \relax (x ) \sin \relax (x )\right )}{2}+\frac {i \pi x \mathrm {csgn}\left (i \ln \relax (x ) \sin \relax (x )\right )^{2}}{2}+\frac {i x^{2}}{2}-x \ln \left ({\mathrm e}^{i x}\right )+x \ln \left (\ln \relax (x )\right )+i \ln \left ({\mathrm e}^{i x}+1\right ) \ln \left ({\mathrm e}^{i x}\right )-i \ln \left ({\mathrm e}^{2 i x}-1\right ) \ln \left ({\mathrm e}^{i x}\right )-\frac {i \pi x}{2}-\ln \relax (2) x +\Ei \left (1, -\ln \relax (x )\right )+i \dilog \left ({\mathrm e}^{i x}+1\right )-i \dilog \left ({\mathrm e}^{i x}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.76, size = 43, normalized size = 0.83 \[ \frac {1}{2} \, {\left (i \, \pi - 2 \, \log \relax (2)\right )} x - \frac {1}{2} i \, x^{2} + x \log \left (\log \relax (x)\right ) - {\rm Ei}\left (\log \relax (x)\right ) + i \, {\rm Li}_2\left (-e^{\left (i \, x\right )}\right ) + i \, {\rm Li}_2\left (e^{\left (i \, x\right )}\right ) \]
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 \[ \int \ln \left (\ln \relax (x)\,\sin \relax (x)\right ) \,d x \]
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 \[ \int \log {\left (\log {\relax (x )} \sin {\relax (x )} \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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