\[ \int \frac {1}{(d x)^{3/2} \left (a+b \log \left (c x^n\right )\right )^2} \, dx \]
Optimal antiderivative \[ -\frac {{\mathrm e}^{\frac {a}{2 b n}} \left (c \,x^{n}\right )^{\frac {1}{2 n}} \expIntegral \left (\frac {-a -b \ln \left (c \,x^{n}\right )}{2 b n}\right )}{2 b^{2} d \,n^{2} \sqrt {d x}}-\frac {1}{b d n \left (a +b \ln \left (c \,x^{n}\right )\right ) \sqrt {d x}} \]
command
int(1/(d*x)^(3/2)/(a+b*ln(c*x^n))^2,x,method=_RETURNVERBOSE)
Maple 2022.1 output
| method | result | size |
| risch | \(-\frac {2}{b n \sqrt {d x}\, \left (2 a +2 b \ln \left (c \right )+2 b \ln \left ({\mathrm e}^{n \ln \left (x \right )}\right )-i b \pi \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i {\mathrm e}^{n \ln \left (x \right )}\right ) \mathrm {csgn}\left (i c \,{\mathrm e}^{n \ln \left (x \right )}\right )+i b \pi \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,{\mathrm e}^{n \ln \left (x \right )}\right )^{2}+i b \pi \,\mathrm {csgn}\left (i {\mathrm e}^{n \ln \left (x \right )}\right ) \mathrm {csgn}\left (i c \,{\mathrm e}^{n \ln \left (x \right )}\right )^{2}-i b \pi \mathrm {csgn}\left (i c \,{\mathrm e}^{n \ln \left (x \right )}\right )^{3}\right ) d}+\frac {{\mathrm e}^{-\frac {i \left (b \pi \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i {\mathrm e}^{n \ln \left (x \right )}\right ) \mathrm {csgn}\left (i c \,{\mathrm e}^{n \ln \left (x \right )}\right )-b \pi \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,{\mathrm e}^{n \ln \left (x \right )}\right )^{2}-b \pi \,\mathrm {csgn}\left (i {\mathrm e}^{n \ln \left (x \right )}\right ) \mathrm {csgn}\left (i c \,{\mathrm e}^{n \ln \left (x \right )}\right )^{2}+b \pi \mathrm {csgn}\left (i c \,{\mathrm e}^{n \ln \left (x \right )}\right )^{3}+2 i b n \left (\ln \left (x \right )-\ln \left (d x \right )\right )+2 i b \ln \left (c \right )+2 i b \left (\ln \left ({\mathrm e}^{n \ln \left (x \right )}\right )-n \ln \left (x \right )\right )+2 i a \right )}{4 b n}} \expIntegral \left (1, \frac {\ln \left (d x \right )}{2}-\frac {i \left (b \pi \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i {\mathrm e}^{n \ln \left (x \right )}\right ) \mathrm {csgn}\left (i c \,{\mathrm e}^{n \ln \left (x \right )}\right )-b \pi \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,{\mathrm e}^{n \ln \left (x \right )}\right )^{2}-b \pi \,\mathrm {csgn}\left (i {\mathrm e}^{n \ln \left (x \right )}\right ) \mathrm {csgn}\left (i c \,{\mathrm e}^{n \ln \left (x \right )}\right )^{2}+b \pi \mathrm {csgn}\left (i c \,{\mathrm e}^{n \ln \left (x \right )}\right )^{3}+2 i b n \left (\ln \left (x \right )-\ln \left (d x \right )\right )+2 i b \ln \left (c \right )+2 i b \left (\ln \left ({\mathrm e}^{n \ln \left (x \right )}\right )-n \ln \left (x \right )\right )+2 i a \right )}{4 b n}\right )}{2 b^{2} n^{2} d}\) | \(429\) |
Maple 2021.1 output
\[ \int \frac {1}{\left (d x \right )^{\frac {3}{2}} \left (b \ln \left (c \,x^{n}\right )+a \right )^{2}}\, dx \]________________________________________________________________________________________