\[ \int \frac {x}{a+b \log \left (c x^n\right )} \, dx \]
Optimal antiderivative \[ \frac {x^{2} \expIntegral \left (\frac {2 a +2 b \ln \left (c \,x^{n}\right )}{b n}\right ) {\mathrm e}^{-\frac {2 a}{b n}} \left (c \,x^{n}\right )^{-\frac {2}{n}}}{b n} \]
command
int(x/(a+b*ln(c*x^n)),x,method=_RETURNVERBOSE)
Maple 2022.1 output
| method | result | size |
| risch | \(-\frac {x^{2} c^{-\frac {2}{n}} \left (x^{n}\right )^{-\frac {2}{n}} {\mathrm e}^{-\frac {-i b \pi \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )+i b \pi \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+i b \pi \,\mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}-i b \pi \mathrm {csgn}\left (i c \,x^{n}\right )^{3}+2 a}{b n}} \expIntegral \left (1, -2 \ln \left (x \right )-\frac {-i b \pi \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )+i b \pi \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+i b \pi \,\mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}-i b \pi \mathrm {csgn}\left (i c \,x^{n}\right )^{3}+2 b \ln \left (c \right )+2 b \left (\ln \left (x^{n}\right )-n \ln \left (x \right )\right )+2 a}{b n}\right )}{b n}\) | \(242\) |
Maple 2021.1 output
\[ \int \frac {x}{b \ln \left (c \,x^{n}\right )+a}\, dx \]________________________________________________________________________________________