\[ \int \sec (e+f x) (a+a \sec (e+f x))^2 (c+d \sec (e+f x))^4 \, dx \]
Optimal antiderivative \[ \frac {a^{2} \left (24 c^{4}+64 c^{3} d +84 c^{2} d^{2}+48 c \,d^{3}+11 d^{4}\right ) \arctanh \left (\sin \left (f x +e \right )\right )}{16 f}-\frac {a^{2} \left (4 c^{5}-48 c^{4} d -311 c^{3} d^{2}-448 c^{2} d^{3}-288 c \,d^{4}-64 d^{5}\right ) \tan \left (f x +e \right )}{60 d f}-\frac {a^{2} \left (8 c^{4}-96 c^{3} d -438 c^{2} d^{2}-464 c \,d^{3}-165 d^{4}\right ) \sec \left (f x +e \right ) \tan \left (f x +e \right )}{240 f}-\frac {a^{2} \left (4 c^{3}-48 c^{2} d -123 c \,d^{2}-64 d^{3}\right ) \left (c +d \sec \left (f x +e \right )\right )^{2} \tan \left (f x +e \right )}{120 d f}-\frac {a^{2} \left (4 c^{2}-48 c d -55 d^{2}\right ) \left (c +d \sec \left (f x +e \right )\right )^{3} \tan \left (f x +e \right )}{120 d f}-\frac {a^{2} \left (c -12 d \right ) \left (c +d \sec \left (f x +e \right )\right )^{4} \tan \left (f x +e \right )}{30 d f}+\frac {a^{2} \left (c +d \sec \left (f x +e \right )\right )^{5} \tan \left (f x +e \right )}{6 d f} \]
command
integrate(sec(f*x+e)*(a+a*sec(f*x+e))^2*(c+d*sec(f*x+e))^4,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 \[ \text {Exception raised: NotImplementedError} \]_______________________________________________________