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