\[ \int (a+b \tan (e+f x))^3 (c+d \tan (e+f x))^2 \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right ) \, dx \]
Optimal antiderivative \[ -\left (a^{3} \left (c^{2} C +2 B c d -C \,d^{2}-A \left (c^{2}-d^{2}\right )\right )-3 a \,b^{2} \left (c^{2} C +2 B c d -C \,d^{2}-A \left (c^{2}-d^{2}\right )\right )+3 a^{2} b \left (2 c \left (A -C \right ) d +B \left (c^{2}-d^{2}\right )\right )-b^{3} \left (2 c \left (A -C \right ) d +B \left (c^{2}-d^{2}\right )\right )\right ) x +\frac {\left (3 a^{2} b \left (c^{2} C +2 B c d -C \,d^{2}-A \left (c^{2}-d^{2}\right )\right )-b^{3} \left (c^{2} C +2 B c d -C \,d^{2}-A \left (c^{2}-d^{2}\right )\right )-a^{3} \left (2 c \left (A -C \right ) d +B \left (c^{2}-d^{2}\right )\right )+3 a \,b^{2} \left (2 c \left (A -C \right ) d +B \left (c^{2}-d^{2}\right )\right )\right ) \ln \left (\cos \left (f x +e \right )\right )}{f}+\frac {d \left (3 a^{2} b \left (A c -B d -c C \right )-b^{3} \left (A c -B d -c C \right )+a^{3} \left (B c +\left (A -C \right ) d \right )-3 a \,b^{2} \left (B c +\left (A -C \right ) d \right )\right ) \tan \left (f x +e \right )}{f}+\frac {\left (a^{3} B -3 a \,b^{2} B +3 a^{2} b \left (A -C \right )-b^{3} \left (A -C \right )\right ) \left (c +d \tan \left (f x +e \right )\right )^{2}}{2 f}+\frac {\left (4 a^{3} C \,d^{3}-3 a^{2} b \,d^{2} \left (-16 B d +3 c C \right )+3 a \,b^{2} d \left (2 c^{2} C -5 B c d +20 \left (A -C \right ) d^{2}\right )-b^{3} \left (c^{3} C -2 B \,c^{2} d +5 c \left (A -C \right ) d^{2}+20 B \,d^{3}\right )\right ) \left (c +d \tan \left (f x +e \right )\right )^{3}}{60 d^{4} f}+\frac {b \left (5 b \left (A b +a B -b C \right ) d^{2}+\left (-a d +b c \right ) \left (-2 b B d -a C d +b c C \right )\right ) \tan \left (f x +e \right ) \left (c +d \tan \left (f x +e \right )\right )^{3}}{20 d^{3} f}-\frac {\left (-2 b B d -a C d +b c C \right ) \left (a +b \tan \left (f x +e \right )\right )^{2} \left (c +d \tan \left (f x +e \right )\right )^{3}}{10 d^{2} f}+\frac {C \left (a +b \tan \left (f x +e \right )\right )^{3} \left (c +d \tan \left (f x +e \right )\right )^{3}}{6 d f} \]
command
integrate((a+b*tan(f*x+e))^3*(c+d*tan(f*x+e))^2*(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 \[ \text {Timed out} \]_______________________________________________________