\[ \int \frac {\left (a+b \log \left (c x^n\right )\right ) \log (1+e x)}{x} \, dx \]
Optimal antiderivative \[ -\left (a +b \ln \left (c \,x^{n}\right )\right ) \polylog \left (2, -e x \right )+b n \polylog \left (3, -e x \right ) \]
command
int((a+b*ln(c*x^n))*ln(e*x+1)/x,x,method=_RETURNVERBOSE)
Maple 2022.1 output
| method | result | size |
| risch | \(-\ln \left (x \right ) \polylog \left (2, -e x \right ) b n +\ln \left (x \right ) \dilog \left (e x +1\right ) b n -\ln \left (x^{n}\right ) \dilog \left (e x +1\right ) b +b n \polylog \left (3, -e x \right )-\left (-\frac {i b \pi \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )}{2}+\frac {i b \pi \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}}{2}+\frac {i b \pi \,\mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}}{2}-\frac {i b \pi \mathrm {csgn}\left (i c \,x^{n}\right )^{3}}{2}+b \ln \left (c \right )+a \right ) \dilog \left (e x +1\right )\) | \(143\) |
Maple 2021.1 output
\[ \int \frac {\left (b \ln \left (c \,x^{n}\right )+a \right ) \ln \left (e x +1\right )}{x}\, dx \]________________________________________________________________________________________