\[ \int \left (d+e x^2\right ) \cos ^{-1}(a x) \log \left (c x^n\right ) \, dx \]
Optimal antiderivative \[ -\frac {2 e n \left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}{27 a^{3}}-d n x \arccos \left (a x \right )-\frac {e n \,x^{3} \arccos \left (a x \right )}{9}+\frac {e n \arctanh \left (\sqrt {-a^{2} x^{2}+1}\right )}{9 a^{3}}-\frac {\left (3 a^{2} d +e \right ) n \arctanh \left (\sqrt {-a^{2} x^{2}+1}\right )}{3 a^{3}}+\frac {e \left (-a^{2} x^{2}+1\right )^{\frac {3}{2}} \ln \left (c \,x^{n}\right )}{9 a^{3}}+d x \arccos \left (a x \right ) \ln \left (c \,x^{n}\right )+\frac {e \,x^{3} \arccos \left (a x \right ) \ln \left (c \,x^{n}\right )}{3}+\frac {d n \sqrt {-a^{2} x^{2}+1}}{a}+\frac {\left (3 a^{2} d +e \right ) n \sqrt {-a^{2} x^{2}+1}}{3 a^{3}}-\frac {\left (3 a^{2} d +e \right ) \ln \left (c \,x^{n}\right ) \sqrt {-a^{2} x^{2}+1}}{3 a^{3}} \]
command
integrate((e*x^2+d)*arccos(a*x)*log(c*x^n),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 \[ \text {Timed out} \]_______________________________________________________