\[ \int \frac {1}{x^4 \left (a+b \log \left (c x^n\right )\right )^2} \, dx \]
Optimal antiderivative \[ -\frac {3 \,{\mathrm e}^{\frac {3 a}{b n}} \left (c \,x^{n}\right )^{\frac {3}{n}} \expIntegral \left (-\frac {3 \left (a +b \ln \left (c \,x^{n}\right )\right )}{b n}\right )}{b^{2} n^{2} x^{3}}-\frac {1}{b n \,x^{3} \left (a +b \ln \left (c \,x^{n}\right )\right )} \]
command
int(1/x^4/(a+b*ln(c*x^n))^2,x,method=_RETURNVERBOSE)
Maple 2022.1 output
| method | result | size |
| risch | \(-\frac {2}{x^{3} \left (2 a +2 b \ln \left (c \right )+2 \ln \left (x^{n}\right ) b -i b \pi \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )+i b \pi \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+i b \pi \,\mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}-i b \pi \mathrm {csgn}\left (i c \,x^{n}\right )^{3}\right ) b n}+\frac {3 c^{\frac {3}{n}} \left (x^{n}\right )^{\frac {3}{n}} {\mathrm e}^{\frac {-\frac {3 i b \pi \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )}{2}+\frac {3 i b \pi \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}}{2}+\frac {3 i b \pi \,\mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}}{2}-\frac {3 i b \pi \mathrm {csgn}\left (i c \,x^{n}\right )^{3}}{2}+3 a}{b n}} \expIntegral \left (1, 3 \ln \left (x \right )+\frac {-\frac {3 i b \pi \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )}{2}+\frac {3 i b \pi \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}}{2}+\frac {3 i b \pi \,\mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}}{2}-\frac {3 i b \pi \mathrm {csgn}\left (i c \,x^{n}\right )^{3}}{2}+3 b \ln \left (c \right )+3 b \left (\ln \left (x^{n}\right )-n \ln \left (x \right )\right )+3 a}{b n}\right )}{b^{2} n^{2} x^{3}}\) | \(354\) |
Maple 2021.1 output
\[ \int \frac {1}{\left (b \ln \left (c \,x^{n}\right )+a \right )^{2} x^{4}}\, dx \]________________________________________________________________________________________