\[ \int (e x)^m \sin ^2\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx \]
Optimal antiderivative \[ \frac {2 b^{2} d^{2} n^{2} \left (e x \right )^{1+m}}{e \left (1+m \right ) \left (\left (1+m \right )^{2}+4 b^{2} d^{2} n^{2}\right )}-\frac {2 b d n \left (e x \right )^{1+m} \cos \left (d \left (a +b \ln \left (c \,x^{n}\right )\right )\right ) \sin \left (d \left (a +b \ln \left (c \,x^{n}\right )\right )\right )}{e \left (\left (1+m \right )^{2}+4 b^{2} d^{2} n^{2}\right )}+\frac {\left (1+m \right ) \left (e x \right )^{1+m} \left (\sin ^{2}\left (d \left (a +b \ln \left (c \,x^{n}\right )\right )\right )\right )}{e \left (\left (1+m \right )^{2}+4 b^{2} d^{2} n^{2}\right )} \]
command
integrate((e*x)^m*sin(d*(a+b*log(c*x^n)))^2,x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \text {output too large to display} \]
Giac 1.7.0 via sagemath 9.3 output \[ \text {Timed out} \]_______________________________________________________