\[ \int \frac {x^2}{\left (a+b \log \left (c x^n\right )\right )^3} \, dx \]
Optimal antiderivative \[ \frac {9 x^{3} \expIntegral \left (\frac {3 a +3 b \ln \left (c \,x^{n}\right )}{b n}\right ) {\mathrm e}^{-\frac {3 a}{b n}} \left (c \,x^{n}\right )^{-\frac {3}{n}}}{2 b^{3} n^{3}}-\frac {x^{3}}{2 b n \left (a +b \ln \left (c \,x^{n}\right )\right )^{2}}-\frac {3 x^{3}}{2 b^{2} n^{2} \left (a +b \ln \left (c \,x^{n}\right )\right )} \]
command
int(x^2/(a+b*ln(c*x^n))^3,x,method=_RETURNVERBOSE)
Maple 2022.1 output
| method | result | size |
| risch | \(-\frac {2 b n \,x^{3}-3 i \pi b \,x^{3} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )+3 i \pi b \,x^{3} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+3 i \pi b \,x^{3} \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}-3 i \pi b \,x^{3} \mathrm {csgn}\left (i c \,x^{n}\right )^{3}+6 \ln \left (c \right ) b \,x^{3}+6 x^{3} b \ln \left (x^{n}\right )+6 x^{3} a}{\left (2 a +2 b \ln \left (c \right )+2 \ln \left (x^{n}\right ) b -i b \pi \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )+i b \pi \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+i b \pi \,\mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}-i b \pi \mathrm {csgn}\left (i c \,x^{n}\right )^{3}\right )^{2} b^{2} n^{2}}-\frac {9 x^{3} c^{-\frac {3}{n}} \left (x^{n}\right )^{-\frac {3}{n}} {\mathrm e}^{-\frac {3 \left (-i b \pi \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )+i b \pi \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+i b \pi \,\mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}-i b \pi \mathrm {csgn}\left (i c \,x^{n}\right )^{3}+2 a \right )}{2 b n}} \expIntegral \left (1, -3 \ln \left (x \right )-\frac {3 \left (-i b \pi \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )+i b \pi \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+i b \pi \,\mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}-i b \pi \mathrm {csgn}\left (i c \,x^{n}\right )^{3}+2 b \ln \left (c \right )+2 b \left (\ln \left (x^{n}\right )-n \ln \left (x \right )\right )+2 a \right )}{2 b n}\right )}{2 b^{3} n^{3}}\) | \(477\) |
Maple 2021.1 output
\[ \int \frac {x^{2}}{\left (b \ln \left (c \,x^{n}\right )+a \right )^{3}}\, dx \]________________________________________________________________________________________