\[ \int x^2 \log (\log (x) \sin (x)) \, dx \]
Optimal antiderivative \[ \frac {i x^{4}}{12}-\frac {\expIntegral \left (3 \ln \left (x \right )\right )}{3}-\frac {x^{3} \ln \left (1-{\mathrm e}^{2 i x}\right )}{3}+\frac {x^{3} \ln \left (\ln \left (x \right ) \sin \left (x \right )\right )}{3}+\frac {i x^{2} \polylog \left (2, {\mathrm e}^{2 i x}\right )}{2}-\frac {x \polylog \left (3, {\mathrm e}^{2 i x}\right )}{2}-\frac {i \polylog \left (4, {\mathrm e}^{2 i x}\right )}{4} \]
command
int(x^2*ln(ln(x)*sin(x)),x,method=_RETURNVERBOSE)
Maple 2022.1 output
| method | result | size |
| risch | \(i x^{2} \polylog \left (2, -{\mathrm e}^{i x}\right )+i x^{2} \polylog \left (2, {\mathrm e}^{i x}\right )-\frac {i \pi \,x^{3}}{6}+\frac {i x^{4}}{12}-2 x \polylog \left (3, -{\mathrm e}^{i x}\right )-2 x \polylog \left (3, {\mathrm e}^{i x}\right )-2 i \polylog \left (4, -{\mathrm e}^{i x}\right )-2 i \polylog \left (4, {\mathrm e}^{i x}\right )-\frac {x^{3} \ln \left ({\mathrm e}^{i x}\right )}{3}-\frac {x^{3} \ln \left ({\mathrm e}^{i x}+1\right )}{3}-\frac {x^{3} \ln \left (1-{\mathrm e}^{i x}\right )}{3}+\frac {x^{3} \ln \left ({\mathrm e}^{2 i x}-1\right )}{3}+\frac {i \pi \,\mathrm {csgn}\left (i {\mathrm e}^{-i x}\right ) \mathrm {csgn}\left (i \left ({\mathrm e}^{2 i x}-1\right ) \ln \left (x \right )\right ) \mathrm {csgn}\left (\ln \left (x \right ) \sin \left (x \right )\right ) x^{3}}{6}-\frac {i \pi \,\mathrm {csgn}\left (i \left ({\mathrm e}^{2 i x}-1\right )\right ) \mathrm {csgn}\left (i \ln \left (x \right )\right ) \mathrm {csgn}\left (i \left ({\mathrm e}^{2 i x}-1\right ) \ln \left (x \right )\right ) x^{3}}{6}-\frac {\ln \left (2\right ) x^{3}}{3}+\frac {i \pi \mathrm {csgn}\left (\ln \left (x \right ) \sin \left (x \right )\right )^{3} x^{3}}{6}-\frac {i \pi \mathrm {csgn}\left (i \ln \left (x \right ) \sin \left (x \right )\right )^{3} x^{3}}{6}-\frac {i \pi \mathrm {csgn}\left (i \left ({\mathrm e}^{2 i x}-1\right ) \ln \left (x \right )\right )^{3} x^{3}}{6}+\frac {i \pi \mathrm {csgn}\left (i \ln \left (x \right ) \sin \left (x \right )\right )^{2} x^{3}}{6}+\frac {i \pi \,\mathrm {csgn}\left (i \ln \left (x \right )\right ) \mathrm {csgn}\left (i \left ({\mathrm e}^{2 i x}-1\right ) \ln \left (x \right )\right )^{2} x^{3}}{6}+\frac {i \pi \,\mathrm {csgn}\left (i \left ({\mathrm e}^{2 i x}-1\right ) \ln \left (x \right )\right ) \mathrm {csgn}\left (\ln \left (x \right ) \sin \left (x \right )\right )^{2} x^{3}}{6}+\frac {i \pi \,\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 \left (x \right )\right )^{2} x^{3}}{6}+\frac {i \pi \,\mathrm {csgn}\left (i {\mathrm e}^{-i x}\right ) \mathrm {csgn}\left (\ln \left (x \right ) \sin \left (x \right )\right )^{2} x^{3}}{6}+\frac {i \pi \,\mathrm {csgn}\left (\ln \left (x \right ) \sin \left (x \right )\right ) \mathrm {csgn}\left (i \ln \left (x \right ) \sin \left (x \right )\right ) x^{3}}{6}-\frac {i \pi \,\mathrm {csgn}\left (\ln \left (x \right ) \sin \left (x \right )\right ) \mathrm {csgn}\left (i \ln \left (x \right ) \sin \left (x \right )\right )^{2} x^{3}}{6}+\frac {x^{3} \ln \left (\ln \left (x \right )\right )}{3}+\frac {\expIntegral \left (1, -3 \ln \left (x \right )\right )}{3}\) | \(462\) |
Maple 2021.1 output
\[ \int x^{2} \ln \left (\ln \left (x \right ) \sin \left (x \right )\right )\, dx \]________________________________________________________________________________________