\[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^3 \log \left (d \left (\frac {1}{d}+f x^2\right )\right )}{x} \, dx \]
Optimal antiderivative \[ -\frac {\left (a +b \ln \left (c \,x^{n}\right )\right )^{3} \polylog \left (2, -d f \,x^{2}\right )}{2}+\frac {3 b n \left (a +b \ln \left (c \,x^{n}\right )\right )^{2} \polylog \left (3, -d f \,x^{2}\right )}{4}-\frac {3 b^{2} n^{2} \left (a +b \ln \left (c \,x^{n}\right )\right ) \polylog \left (4, -d f \,x^{2}\right )}{4}+\frac {3 b^{3} n^{3} \polylog \left (5, -d f \,x^{2}\right )}{8} \]
command
int((a+b*ln(c*x^n))^3*ln(d*(1/d+f*x^2))/x,x,method=_RETURNVERBOSE)
Maple 2022.1 output
| method | result | size |
| risch | \(\text {Expression too large to display}\) | \(23414\) |
Maple 2021.1 output
\[ \int \frac {\left (b \ln \left (c \,x^{n}\right )+a \right )^{3} \ln \left (\left (f \,x^{2}+\frac {1}{d}\right ) d \right )}{x}\, dx \]________________________________________________________________________________________