\[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^3 \log (1+e x)}{x^3} \, dx \]
Optimal antiderivative \[ -\frac {45 b^{3} e \,n^{3}}{8 x}-\frac {3 b^{3} e^{2} n^{3} \ln \left (x \right )}{8}-\frac {21 b^{2} e \,n^{2} \left (a +b \ln \left (c \,x^{n}\right )\right )}{4 x}+\frac {3 b^{2} e^{2} n^{2} \ln \left (1+\frac {1}{e x}\right ) \left (a +b \ln \left (c \,x^{n}\right )\right )}{4}-\frac {9 b e n \left (a +b \ln \left (c \,x^{n}\right )\right )^{2}}{4 x}+\frac {3 b \,e^{2} n \ln \left (1+\frac {1}{e x}\right ) \left (a +b \ln \left (c \,x^{n}\right )\right )^{2}}{4}-\frac {e \left (a +b \ln \left (c \,x^{n}\right )\right )^{3}}{2 x}+\frac {e^{2} \ln \left (1+\frac {1}{e x}\right ) \left (a +b \ln \left (c \,x^{n}\right )\right )^{3}}{2}+\frac {3 b^{3} e^{2} n^{3} \ln \left (e x +1\right )}{8}-\frac {3 b^{3} n^{3} \ln \left (e x +1\right )}{8 x^{2}}-\frac {3 b^{2} n^{2} \left (a +b \ln \left (c \,x^{n}\right )\right ) \ln \left (e x +1\right )}{4 x^{2}}-\frac {3 b n \left (a +b \ln \left (c \,x^{n}\right )\right )^{2} \ln \left (e x +1\right )}{4 x^{2}}-\frac {\left (a +b \ln \left (c \,x^{n}\right )\right )^{3} \ln \left (e x +1\right )}{2 x^{2}}-\frac {3 b^{3} e^{2} n^{3} \polylog \left (2, -\frac {1}{e x}\right )}{4}-\frac {3 b^{2} e^{2} n^{2} \left (a +b \ln \left (c \,x^{n}\right )\right ) \polylog \left (2, -\frac {1}{e x}\right )}{2}-\frac {3 b \,e^{2} n \left (a +b \ln \left (c \,x^{n}\right )\right )^{2} \polylog \left (2, -\frac {1}{e x}\right )}{2}-\frac {3 b^{3} e^{2} n^{3} \polylog \left (3, -\frac {1}{e x}\right )}{2}-3 b^{2} e^{2} n^{2} \left (a +b \ln \left (c \,x^{n}\right )\right ) \polylog \left (3, -\frac {1}{e x}\right )-3 b^{3} e^{2} n^{3} \polylog \left (4, -\frac {1}{e x}\right ) \]
command
int((a+b*ln(c*x^n))^3*ln(e*x+1)/x^3,x,method=_RETURNVERBOSE)
Maple 2022.1 output
| method | result | size |
| risch | \(\text {Expression too large to display}\) | \(17975\) |
Maple 2021.1 output
\[ \int \frac {\left (b \ln \left (c \,x^{n}\right )+a \right )^{3} \ln \left (e x +1\right )}{x^{3}}\, dx \]________________________________________________________________________________________