\[ \int \frac {1}{x^4 \left (a+b \log \left (c x^n\right )\right )} \, dx \]
Optimal antiderivative \[ \frac {{\mathrm e}^{\frac {3 a}{b n}} \left (c \,x^{n}\right )^{\frac {3}{n}} \expIntegral \left (-\frac {3 \left (a +b \ln \left (c \,x^{n}\right )\right )}{b n}\right )}{b n \,x^{3}} \]
command
int(1/x^4/(a+b*ln(c*x^n)),x,method=_RETURNVERBOSE)
Maple 2022.1 output
| method | result | size |
| risch | \(-\frac {c^{\frac {3}{n}} \left (x^{n}\right )^{\frac {3}{n}} {\mathrm e}^{\frac {-\frac {3 i b \pi \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )}{2}+\frac {3 i b \pi \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}}{2}+\frac {3 i b \pi \,\mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}}{2}-\frac {3 i b \pi \mathrm {csgn}\left (i c \,x^{n}\right )^{3}}{2}+3 a}{b n}} \expIntegral \left (1, 3 \ln \left (x \right )-\frac {3 \left (i b \pi \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )-i b \pi \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}-i b \pi \,\mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+i b \pi \mathrm {csgn}\left (i c \,x^{n}\right )^{3}-2 b \ln \left (c \right )-2 b \left (\ln \left (x^{n}\right )-n \ln \left (x \right )\right )-2 a \right )}{2 b n}\right )}{b n \,x^{3}}\) | \(242\) |
Maple 2021.1 output
\[ \int \frac {1}{\left (b \ln \left (c \,x^{n}\right )+a \right ) x^{4}}\, dx \]________________________________________________________________________________________