\[ \int \frac {1}{x^2 \left (a+b \log \left (c x^n\right )\right )^3} \, dx \]
Optimal antiderivative \[ \frac {{\mathrm e}^{\frac {a}{b n}} \left (c \,x^{n}\right )^{\frac {1}{n}} \expIntegral \left (\frac {-a -b \ln \left (c \,x^{n}\right )}{b n}\right )}{2 b^{3} n^{3} x}-\frac {1}{2 b n x \left (a +b \ln \left (c \,x^{n}\right )\right )^{2}}+\frac {1}{2 b^{2} n^{2} x \left (a +b \ln \left (c \,x^{n}\right )\right )} \]
command
int(1/x^2/(a+b*ln(c*x^n))^3,x,method=_RETURNVERBOSE)
Maple 2022.1 output
| method | result | size |
| risch | \(\frac {-2 b n +2 a +2 b \ln \left (c \right )+2 \ln \left (x^{n}\right ) b -i b \pi \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )+i b \pi \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+i b \pi \,\mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}-i b \pi \mathrm {csgn}\left (i c \,x^{n}\right )^{3}}{\left (2 a +2 b \ln \left (c \right )+2 \ln \left (x^{n}\right ) b -i b \pi \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )+i b \pi \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+i b \pi \,\mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}-i b \pi \mathrm {csgn}\left (i c \,x^{n}\right )^{3}\right )^{2} b^{2} n^{2} x}-\frac {c^{\frac {1}{n}} \left (x^{n}\right )^{\frac {1}{n}} {\mathrm e}^{\frac {-i b \pi \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )+i b \pi \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+i b \pi \,\mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}-i b \pi \mathrm {csgn}\left (i c \,x^{n}\right )^{3}+2 a}{2 b n}} \expIntegral \left (1, \ln \left (x \right )+\frac {-i b \pi \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )+i b \pi \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+i b \pi \,\mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}-i b \pi \mathrm {csgn}\left (i c \,x^{n}\right )^{3}+2 b \ln \left (c \right )+2 b \left (\ln \left (x^{n}\right )-n \ln \left (x \right )\right )+2 a}{2 b n}\right )}{2 b^{3} n^{3} x}\) | \(449\) |
Maple 2021.1 output
\[ \int \frac {1}{\left (b \ln \left (c \,x^{n}\right )+a \right )^{3} x^{2}}\, dx \]________________________________________________________________________________________