\[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2 \log (1+e x)}{x^3} \, dx \]
Optimal antiderivative \[ -\frac {7 b^{2} e \,n^{2}}{4 x}-\frac {b^{2} e^{2} n^{2} \ln \left (x \right )}{4}-\frac {3 b e n \left (a +b \ln \left (c \,x^{n}\right )\right )}{2 x}+\frac {b \,e^{2} n \ln \left (1+\frac {1}{e x}\right ) \left (a +b \ln \left (c \,x^{n}\right )\right )}{2}-\frac {e \left (a +b \ln \left (c \,x^{n}\right )\right )^{2}}{2 x}+\frac {e^{2} \ln \left (1+\frac {1}{e x}\right ) \left (a +b \ln \left (c \,x^{n}\right )\right )^{2}}{2}+\frac {b^{2} e^{2} n^{2} \ln \left (e x +1\right )}{4}-\frac {b^{2} n^{2} \ln \left (e x +1\right )}{4 x^{2}}-\frac {b n \left (a +b \ln \left (c \,x^{n}\right )\right ) \ln \left (e x +1\right )}{2 x^{2}}-\frac {\left (a +b \ln \left (c \,x^{n}\right )\right )^{2} \ln \left (e x +1\right )}{2 x^{2}}-\frac {b^{2} e^{2} n^{2} \polylog \left (2, -\frac {1}{e x}\right )}{2}-b \,e^{2} n \left (a +b \ln \left (c \,x^{n}\right )\right ) \polylog \left (2, -\frac {1}{e x}\right )-b^{2} e^{2} n^{2} \polylog \left (3, -\frac {1}{e x}\right ) \]
command
int((a+b*ln(c*x^n))^2*ln(e*x+1)/x^3,x,method=_RETURNVERBOSE)
Maple 2022.1 output
| method | result | size |
| risch | \(\text {Expression too large to display}\) | \(4445\) |
Maple 2021.1 output
\[ \int \frac {\left (b \ln \left (c \,x^{n}\right )+a \right )^{2} \ln \left (e x +1\right )}{x^{3}}\, dx \]________________________________________________________________________________________