\[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2 \log (1+e x)}{x^2} \, dx \]
Optimal antiderivative \[ 2 b^{2} e \,n^{2} \ln \left (x \right )-2 b e n \ln \left (1+\frac {1}{e x}\right ) \left (a +b \ln \left (c \,x^{n}\right )\right )-e \ln \left (1+\frac {1}{e x}\right ) \left (a +b \ln \left (c \,x^{n}\right )\right )^{2}-2 b^{2} e \,n^{2} \ln \left (e x +1\right )-\frac {2 b^{2} n^{2} \ln \left (e x +1\right )}{x}-\frac {2 b n \left (a +b \ln \left (c \,x^{n}\right )\right ) \ln \left (e x +1\right )}{x}-\frac {\left (a +b \ln \left (c \,x^{n}\right )\right )^{2} \ln \left (e x +1\right )}{x}+2 b^{2} e \,n^{2} \polylog \left (2, -\frac {1}{e x}\right )+2 b e n \left (a +b \ln \left (c \,x^{n}\right )\right ) \polylog \left (2, -\frac {1}{e x}\right )+2 b^{2} e \,n^{2} \polylog \left (3, -\frac {1}{e x}\right ) \]
command
int((a+b*ln(c*x^n))^2*ln(e*x+1)/x^2,x,method=_RETURNVERBOSE)
Maple 2022.1 output
| method | result | size |
| risch | \(\text {Expression too large to display}\) | \(3402\) |
Maple 2021.1 output
\[ \int \frac {\left (b \ln \left (c \,x^{n}\right )+a \right )^{2} \ln \left (e x +1\right )}{x^{2}}\, dx \]________________________________________________________________________________________