Integrand size = 10, antiderivative size = 98 \[ \int x^2 \log (\log (x) \sin (x)) \, dx=\frac {i x^4}{12}-\frac {1}{3} \operatorname {ExpIntegralEi}(3 \log (x))-\frac {1}{3} x^3 \log \left (1-e^{2 i x}\right )+\frac {1}{3} x^3 \log (\log (x) \sin (x))+\frac {1}{2} i x^2 \operatorname {PolyLog}\left (2,e^{2 i x}\right )-\frac {1}{2} x \operatorname {PolyLog}\left (3,e^{2 i x}\right )-\frac {1}{4} i \operatorname {PolyLog}\left (4,e^{2 i x}\right ) \]
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Time = 0.20 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.300, Rules used = {30, 2635, 12, 6820, 14, 3798, 2221, 2611, 6744, 2320, 6724, 2346, 2209} \[ \int x^2 \log (\log (x) \sin (x)) \, dx=-\frac {1}{3} \operatorname {ExpIntegralEi}(3 \log (x))+\frac {1}{2} i x^2 \operatorname {PolyLog}\left (2,e^{2 i x}\right )-\frac {1}{2} x \operatorname {PolyLog}\left (3,e^{2 i x}\right )-\frac {1}{4} i \operatorname {PolyLog}\left (4,e^{2 i x}\right )+\frac {i x^4}{12}-\frac {1}{3} x^3 \log \left (1-e^{2 i x}\right )+\frac {1}{3} x^3 \log (\log (x) \sin (x)) \]
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Rule 12
Rule 14
Rule 30
Rule 2209
Rule 2221
Rule 2320
Rule 2346
Rule 2611
Rule 2635
Rule 3798
Rule 6724
Rule 6744
Rule 6820
Rubi steps \begin{align*} \text {integral}& = \frac {1}{3} x^3 \log (\log (x) \sin (x))-\int \frac {x^2 (1+x \cot (x) \log (x))}{3 \log (x)} \, dx \\ & = \frac {1}{3} x^3 \log (\log (x) \sin (x))-\frac {1}{3} \int \frac {x^2 (1+x \cot (x) \log (x))}{\log (x)} \, dx \\ & = \frac {1}{3} x^3 \log (\log (x) \sin (x))-\frac {1}{3} \int x^2 \left (x \cot (x)+\frac {1}{\log (x)}\right ) \, dx \\ & = \frac {1}{3} x^3 \log (\log (x) \sin (x))-\frac {1}{3} \int \left (x^3 \cot (x)+\frac {x^2}{\log (x)}\right ) \, dx \\ & = \frac {1}{3} x^3 \log (\log (x) \sin (x))-\frac {1}{3} \int x^3 \cot (x) \, dx-\frac {1}{3} \int \frac {x^2}{\log (x)} \, dx \\ & = \frac {i x^4}{12}+\frac {1}{3} x^3 \log (\log (x) \sin (x))+\frac {2}{3} i \int \frac {e^{2 i x} x^3}{1-e^{2 i x}} \, dx-\frac {1}{3} \text {Subst}\left (\int \frac {e^{3 x}}{x} \, dx,x,\log (x)\right ) \\ & = \frac {i x^4}{12}-\frac {1}{3} \text {Ei}(3 \log (x))-\frac {1}{3} x^3 \log \left (1-e^{2 i x}\right )+\frac {1}{3} x^3 \log (\log (x) \sin (x))+\int x^2 \log \left (1-e^{2 i x}\right ) \, dx \\ & = \frac {i x^4}{12}-\frac {1}{3} \text {Ei}(3 \log (x))-\frac {1}{3} x^3 \log \left (1-e^{2 i x}\right )+\frac {1}{3} x^3 \log (\log (x) \sin (x))+\frac {1}{2} i x^2 \text {Li}_2\left (e^{2 i x}\right )-i \int x \text {Li}_2\left (e^{2 i x}\right ) \, dx \\ & = \frac {i x^4}{12}-\frac {1}{3} \text {Ei}(3 \log (x))-\frac {1}{3} x^3 \log \left (1-e^{2 i x}\right )+\frac {1}{3} x^3 \log (\log (x) \sin (x))+\frac {1}{2} i x^2 \text {Li}_2\left (e^{2 i x}\right )-\frac {1}{2} x \text {Li}_3\left (e^{2 i x}\right )+\frac {1}{2} \int \text {Li}_3\left (e^{2 i x}\right ) \, dx \\ & = \frac {i x^4}{12}-\frac {1}{3} \text {Ei}(3 \log (x))-\frac {1}{3} x^3 \log \left (1-e^{2 i x}\right )+\frac {1}{3} x^3 \log (\log (x) \sin (x))+\frac {1}{2} i x^2 \text {Li}_2\left (e^{2 i x}\right )-\frac {1}{2} x \text {Li}_3\left (e^{2 i x}\right )-\frac {1}{4} i \text {Subst}\left (\int \frac {\text {Li}_3(x)}{x} \, dx,x,e^{2 i x}\right ) \\ & = \frac {i x^4}{12}-\frac {1}{3} \text {Ei}(3 \log (x))-\frac {1}{3} x^3 \log \left (1-e^{2 i x}\right )+\frac {1}{3} x^3 \log (\log (x) \sin (x))+\frac {1}{2} i x^2 \text {Li}_2\left (e^{2 i x}\right )-\frac {1}{2} x \text {Li}_3\left (e^{2 i x}\right )-\frac {1}{4} i \text {Li}_4\left (e^{2 i x}\right ) \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.97 \[ \int x^2 \log (\log (x) \sin (x)) \, dx=\frac {1}{192} i \left (\pi ^4-16 x^4+64 i \operatorname {ExpIntegralEi}(3 \log (x))+64 i x^3 \log \left (1-e^{-2 i x}\right )-64 i x^3 \log (\log (x) \sin (x))-96 x^2 \operatorname {PolyLog}\left (2,e^{-2 i x}\right )+96 i x \operatorname {PolyLog}\left (3,e^{-2 i x}\right )+48 \operatorname {PolyLog}\left (4,e^{-2 i x}\right )\right ) \]
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 1.60 (sec) , antiderivative size = 426, normalized size of antiderivative = 4.35
method | result | size |
risch | \(-\frac {x^{3} \ln \left ({\mathrm e}^{i x}\right )}{3}+\frac {\left (-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 )+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}+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 )+i \pi \,\operatorname {csgn}\left (i {\mathrm e}^{-i x}\right ) \operatorname {csgn}\left (\ln \left (x \right ) \sin \left (x \right )\right )^{2}+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}-i \pi {\operatorname {csgn}\left (i \ln \left (x \right ) \left ({\mathrm e}^{2 i x}-1\right )\right )}^{3}+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}+i \pi \operatorname {csgn}\left (\ln \left (x \right ) \sin \left (x \right )\right )^{3}-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}+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 )-i \pi \operatorname {csgn}\left (i \ln \left (x \right ) \sin \left (x \right )\right )^{3}+i \pi \operatorname {csgn}\left (i \ln \left (x \right ) \sin \left (x \right )\right )^{2}-i \pi -2 \ln \left (2\right )\right ) x^{3}}{6}+\frac {x^{3} \ln \left ({\mathrm e}^{2 i x}-1\right )}{3}-\frac {x^{3} \ln \left ({\mathrm e}^{i x}+1\right )}{3}+i x^{2} \operatorname {Li}_{2}\left (-{\mathrm e}^{i x}\right )-2 x \,\operatorname {Li}_{3}\left (-{\mathrm e}^{i x}\right )-2 i \operatorname {Li}_{4}\left (-{\mathrm e}^{i x}\right )-\frac {x^{3} \ln \left (1-{\mathrm e}^{i x}\right )}{3}+i x^{2} \operatorname {Li}_{2}\left ({\mathrm e}^{i x}\right )-2 x \,\operatorname {Li}_{3}\left ({\mathrm e}^{i x}\right )-2 i \operatorname {Li}_{4}\left ({\mathrm e}^{i x}\right )+\frac {x^{3} \ln \left (\ln \left (x \right )\right )}{3}+\frac {\operatorname {Ei}_{1}\left (-3 \ln \left (x \right )\right )}{3}+\frac {i x^{4}}{12}\) | \(426\) |
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 234 vs. \(2 (65) = 130\).
Time = 0.33 (sec) , antiderivative size = 234, normalized size of antiderivative = 2.39 \[ \int x^2 \log (\log (x) \sin (x)) \, dx=\frac {1}{3} \, x^{3} \log \left (\log \left (x\right ) \sin \left (x\right )\right ) - \frac {1}{6} \, x^{3} \log \left (\cos \left (x\right ) + i \, \sin \left (x\right ) + 1\right ) - \frac {1}{6} \, x^{3} \log \left (\cos \left (x\right ) - i \, \sin \left (x\right ) + 1\right ) - \frac {1}{6} \, x^{3} \log \left (-\cos \left (x\right ) + i \, \sin \left (x\right ) + 1\right ) - \frac {1}{6} \, x^{3} \log \left (-\cos \left (x\right ) - i \, \sin \left (x\right ) + 1\right ) + \frac {1}{2} i \, x^{2} {\rm Li}_2\left (\cos \left (x\right ) + i \, \sin \left (x\right )\right ) - \frac {1}{2} i \, x^{2} {\rm Li}_2\left (\cos \left (x\right ) - i \, \sin \left (x\right )\right ) - \frac {1}{2} i \, x^{2} {\rm Li}_2\left (-\cos \left (x\right ) + i \, \sin \left (x\right )\right ) + \frac {1}{2} i \, x^{2} {\rm Li}_2\left (-\cos \left (x\right ) - i \, \sin \left (x\right )\right ) - x {\rm polylog}\left (3, \cos \left (x\right ) + i \, \sin \left (x\right )\right ) - x {\rm polylog}\left (3, \cos \left (x\right ) - i \, \sin \left (x\right )\right ) - x {\rm polylog}\left (3, -\cos \left (x\right ) + i \, \sin \left (x\right )\right ) - x {\rm polylog}\left (3, -\cos \left (x\right ) - i \, \sin \left (x\right )\right ) - \frac {1}{3} \, \operatorname {log\_integral}\left (x^{3}\right ) - i \, {\rm polylog}\left (4, \cos \left (x\right ) + i \, \sin \left (x\right )\right ) + i \, {\rm polylog}\left (4, \cos \left (x\right ) - i \, \sin \left (x\right )\right ) + i \, {\rm polylog}\left (4, -\cos \left (x\right ) + i \, \sin \left (x\right )\right ) - i \, {\rm polylog}\left (4, -\cos \left (x\right ) - i \, \sin \left (x\right )\right ) \]
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\[ \int x^2 \log (\log (x) \sin (x)) \, dx=\int x^{2} \log {\left (\log {\left (x \right )} \sin {\left (x \right )} \right )}\, dx \]
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Time = 0.41 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.96 \[ \int x^2 \log (\log (x) \sin (x)) \, dx=-\frac {1}{6} \, {\left (-i \, \pi + 2 \, \log \left (2\right )\right )} x^{3} - \frac {1}{4} i \, x^{4} + \frac {1}{3} \, x^{3} \log \left (\log \left (x\right )\right ) + i \, x^{2} {\rm Li}_2\left (-e^{\left (i \, x\right )}\right ) + i \, x^{2} {\rm Li}_2\left (e^{\left (i \, x\right )}\right ) - 2 \, x {\rm Li}_{3}(-e^{\left (i \, x\right )}) - 2 \, x {\rm Li}_{3}(e^{\left (i \, x\right )}) - \frac {1}{3} \, {\rm Ei}\left (3 \, \log \left (x\right )\right ) - 2 i \, {\rm Li}_{4}(-e^{\left (i \, x\right )}) - 2 i \, {\rm Li}_{4}(e^{\left (i \, x\right )}) \]
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\[ \int x^2 \log (\log (x) \sin (x)) \, dx=\int { x^{2} \log \left (\log \left (x\right ) \sin \left (x\right )\right ) \,d x } \]
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Timed out. \[ \int x^2 \log (\log (x) \sin (x)) \, dx=\int x^2\,\ln \left (\ln \left (x\right )\,\sin \left (x\right )\right ) \,d x \]
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