\[ \int x \log (\log (x) \sin (x)) \, dx \]
Optimal antiderivative \[ \frac {i x^{3}}{6}-\frac {\expIntegral \left (2 \ln \left (x \right )\right )}{2}-\frac {x^{2} \ln \left (1-{\mathrm e}^{2 i x}\right )}{2}+\frac {x^{2} \ln \left (\ln \left (x \right ) \sin \left (x \right )\right )}{2}+\frac {i x \polylog \left (2, {\mathrm e}^{2 i x}\right )}{2}-\frac {\polylog \left (3, {\mathrm e}^{2 i x}\right )}{4} \]
command
int(x*ln(ln(x)*sin(x)),x,method=_RETURNVERBOSE)
Maple 2022.1 output
| method | result | size |
| risch | \(\frac {\expIntegral \left (1, -2 \ln \left (x \right )\right )}{2}-\frac {i \pi \mathrm {csgn}\left (i \left ({\mathrm e}^{2 i x}-1\right ) \ln \left (x \right )\right )^{3} x^{2}}{4}-\frac {i \pi \mathrm {csgn}\left (i \ln \left (x \right ) \sin \left (x \right )\right )^{3} x^{2}}{4}+\frac {i \pi \mathrm {csgn}\left (i \ln \left (x \right ) \sin \left (x \right )\right )^{2} x^{2}}{4}+\frac {i \pi \mathrm {csgn}\left (\ln \left (x \right ) \sin \left (x \right )\right )^{3} x^{2}}{4}+i x \polylog \left (2, -{\mathrm e}^{i x}\right )+i x \polylog \left (2, {\mathrm e}^{i x}\right )-\frac {i \pi \,x^{2}}{4}-\frac {\ln \left (2\right ) x^{2}}{2}-\polylog \left (3, -{\mathrm e}^{i x}\right )-\polylog \left (3, {\mathrm e}^{i x}\right )+\frac {i x^{3}}{6}-\frac {x^{2} \ln \left ({\mathrm e}^{i x}\right )}{2}-\frac {x^{2} \ln \left ({\mathrm e}^{i x}+1\right )}{2}-\frac {x^{2} \ln \left (1-{\mathrm e}^{i x}\right )}{2}+\frac {\ln \left ({\mathrm e}^{2 i x}-1\right ) x^{2}}{2}+\frac {\ln \left (\ln \left (x \right )\right ) x^{2}}{2}+\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^{2}}{4}+\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^{2}}{4}+\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^{2}}{4}+\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^{2}}{4}+\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^{2}}{4}-\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^{2}}{4}+\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^{2}}{4}-\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^{2}}{4}\) | \(434\) |
Maple 2021.1 output
\[ \int x \ln \left (\ln \left (x \right ) \sin \left (x \right )\right )\, dx \]________________________________________________________________________________________