Integrand size = 6, antiderivative size = 52 \[ \int \log (\log (x) \sin (x)) \, dx=\frac {i x^2}{2}-x \log \left (1-e^{2 i x}\right )+x \log (\log (x) \sin (x))-\operatorname {LogIntegral}(x)+\frac {1}{2} i \operatorname {PolyLog}\left (2,e^{2 i x}\right ) \]
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Time = 0.04 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {2629, 3798, 2221, 2317, 2438, 2335} \[ \int \log (\log (x) \sin (x)) \, dx=-\operatorname {LogIntegral}(x)+\frac {1}{2} i \operatorname {PolyLog}\left (2,e^{2 i x}\right )+\frac {i x^2}{2}-x \log \left (1-e^{2 i x}\right )+x \log (\log (x) \sin (x)) \]
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Rule 2221
Rule 2317
Rule 2335
Rule 2438
Rule 2629
Rule 3798
Rubi steps \begin{align*} \text {integral}& = 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 \text {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*}
Time = 0.03 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.90 \[ \int \log (\log (x) \sin (x)) \, dx=-x \log \left (1-e^{2 i x}\right )+x \log (\log (x) \sin (x))-\operatorname {LogIntegral}(x)+\frac {1}{2} i \left (x^2+\operatorname {PolyLog}\left (2,e^{2 i x}\right )\right ) \]
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.92 (sec) , antiderivative size = 368, normalized size of antiderivative = 7.08
method | result | size |
risch | \(-x \ln \left ({\mathrm e}^{i x}\right )+\frac {i x^{2}}{2}+\frac {i \pi \,\operatorname {csgn}\left (\ln \left (x \right ) \sin \left (x \right )\right ) \operatorname {csgn}\left (i \ln \left (x \right ) \sin \left (x \right )\right ) x}{2}-\frac {i \pi x}{2}+\frac {i \pi \,\operatorname {csgn}\left (i {\mathrm e}^{-i x}\right ) \operatorname {csgn}\left (\ln \left (x \right ) \sin \left (x \right )\right )^{2} x}{2}+\frac {i \pi \operatorname {csgn}\left (\ln \left (x \right ) \sin \left (x \right )\right )^{3} x}{2}-i \ln \left ({\mathrm e}^{i x}\right ) \ln \left ({\mathrm e}^{2 i x}-1\right )-x \ln \left (2\right )+\frac {i \pi \,\operatorname {csgn}\left (i {\mathrm e}^{-i x}\right ) \operatorname {csgn}\left (i \ln \left (x \right ) \left ({\mathrm e}^{2 i x}-1\right )\right ) \operatorname {csgn}\left (\ln \left (x \right ) \sin \left (x \right )\right ) x}{2}+i \ln \left ({\mathrm e}^{i x}\right ) \ln \left ({\mathrm e}^{i x}+1\right )-\frac {i \pi {\operatorname {csgn}\left (i \ln \left (x \right ) \left ({\mathrm e}^{2 i x}-1\right )\right )}^{3} x}{2}-\frac {i \pi \,\operatorname {csgn}\left (i \left ({\mathrm e}^{2 i x}-1\right )\right ) \operatorname {csgn}\left (i \ln \left (x \right )\right ) \operatorname {csgn}\left (i \ln \left (x \right ) \left ({\mathrm e}^{2 i x}-1\right )\right ) x}{2}-\frac {i \pi \,\operatorname {csgn}\left (\ln \left (x \right ) \sin \left (x \right )\right ) \operatorname {csgn}\left (i \ln \left (x \right ) \sin \left (x \right )\right )^{2} x}{2}+\frac {i \pi \,\operatorname {csgn}\left (i \ln \left (x \right ) \left ({\mathrm e}^{2 i x}-1\right )\right ) \operatorname {csgn}\left (\ln \left (x \right ) \sin \left (x \right )\right )^{2} x}{2}+i \operatorname {dilog}\left ({\mathrm e}^{i x}+1\right )-i \operatorname {dilog}\left ({\mathrm e}^{i x}\right )+\frac {i \pi \,\operatorname {csgn}\left (i \ln \left (x \right )\right ) {\operatorname {csgn}\left (i \ln \left (x \right ) \left ({\mathrm e}^{2 i x}-1\right )\right )}^{2} x}{2}+\frac {i \pi \,\operatorname {csgn}\left (i \left ({\mathrm e}^{2 i x}-1\right )\right ) {\operatorname {csgn}\left (i \ln \left (x \right ) \left ({\mathrm e}^{2 i x}-1\right )\right )}^{2} x}{2}+\frac {i \pi \operatorname {csgn}\left (i \ln \left (x \right ) \sin \left (x \right )\right )^{2} x}{2}-\frac {i \pi \operatorname {csgn}\left (i \ln \left (x \right ) \sin \left (x \right )\right )^{3} x}{2}+\ln \left (\ln \left (x \right )\right ) x +\operatorname {Ei}_{1}\left (-\ln \left (x \right )\right )\) | \(368\) |
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 109 vs. \(2 (37) = 74\).
Time = 0.32 (sec) , antiderivative size = 109, normalized size of antiderivative = 2.10 \[ \int \log (\log (x) \sin (x)) \, dx=x \log \left (\log \left (x\right ) \sin \left (x\right )\right ) - \frac {1}{2} \, x \log \left (\cos \left (x\right ) + i \, \sin \left (x\right ) + 1\right ) - \frac {1}{2} \, x \log \left (\cos \left (x\right ) - i \, \sin \left (x\right ) + 1\right ) - \frac {1}{2} \, x \log \left (-\cos \left (x\right ) + i \, \sin \left (x\right ) + 1\right ) - \frac {1}{2} \, x \log \left (-\cos \left (x\right ) - i \, \sin \left (x\right ) + 1\right ) + \frac {1}{2} i \, {\rm Li}_2\left (\cos \left (x\right ) + i \, \sin \left (x\right )\right ) - \frac {1}{2} i \, {\rm Li}_2\left (\cos \left (x\right ) - i \, \sin \left (x\right )\right ) - \frac {1}{2} i \, {\rm Li}_2\left (-\cos \left (x\right ) + i \, \sin \left (x\right )\right ) + \frac {1}{2} i \, {\rm Li}_2\left (-\cos \left (x\right ) - i \, \sin \left (x\right )\right ) - \operatorname {log\_integral}\left (x\right ) \]
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\[ \int \log (\log (x) \sin (x)) \, dx=\int \log {\left (\log {\left (x \right )} \sin {\left (x \right )} \right )}\, dx \]
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none
Time = 0.35 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.83 \[ \int \log (\log (x) \sin (x)) \, dx=\frac {1}{2} \, {\left (i \, \pi - 2 \, \log \left (2\right )\right )} x - \frac {1}{2} i \, x^{2} + x \log \left (\log \left (x\right )\right ) - {\rm Ei}\left (\log \left (x\right )\right ) + i \, {\rm Li}_2\left (-e^{\left (i \, x\right )}\right ) + i \, {\rm Li}_2\left (e^{\left (i \, x\right )}\right ) \]
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\[ \int \log (\log (x) \sin (x)) \, dx=\int { \log \left (\log \left (x\right ) \sin \left (x\right )\right ) \,d x } \]
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Timed out. \[ \int \log (\log (x) \sin (x)) \, dx=\int \ln \left (\ln \left (x\right )\,\sin \left (x\right )\right ) \,d x \]
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