\[ \int (a+b \tan (e+f x)) (c+d \tan (e+f x))^3 \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right ) \, dx \]
Optimal antiderivative \[ \left (a \left (A \,c^{3}-3 A c \,d^{2}-3 B \,c^{2} d +B \,d^{3}-c^{3} C +3 c C \,d^{2}\right )-b \left (\left (A -C \right ) d \left (3 c^{2}-d^{2}\right )+B \left (c^{3}-3 c \,d^{2}\right )\right )\right ) x -\frac {\left (A \left (3 a \,c^{2} d -a \,d^{3}+b \,c^{3}-3 b c \,d^{2}\right )-b \left (3 B \,c^{2} d -B \,d^{3}+c^{3} C -3 c C \,d^{2}\right )+a \left (B \,c^{3}-3 B c \,d^{2}-3 c^{2} C d +C \,d^{3}\right )\right ) \ln \left (\cos \left (f x +e \right )\right )}{f}+\frac {d \left (a \left (B \,c^{2}-B \,d^{2}-2 c C d \right )-b \left (2 B c d +c^{2} C -C \,d^{2}\right )+A \left (2 a c d +b \left (c^{2}-d^{2}\right )\right )\right ) \tan \left (f x +e \right )}{f}+\frac {\left (a A d +A b c +a B c -b B d -a C d -b c C \right ) \left (c +d \tan \left (f x +e \right )\right )^{2}}{2 f}+\frac {\left (A b +a B -b C \right ) \left (c +d \tan \left (f x +e \right )\right )^{3}}{3 f}-\frac {\left (-5 b B d -5 a C d +b c C \right ) \left (c +d \tan \left (f x +e \right )\right )^{4}}{20 d^{2} f}+\frac {b C \tan \left (f x +e \right ) \left (c +d \tan \left (f x +e \right )\right )^{4}}{5 d f} \]
command
integrate((a+b*tan(f*x+e))*(c+d*tan(f*x+e))^3*(A+B*tan(f*x+e)+C*tan(f*x+e)^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 \[ \text {Timed out} \]_______________________________________________________