\[ \int \sec (e+f x) (a+b \sec (e+f x)) (c+d \sec (e+f x))^4 \, dx \]
Optimal antiderivative \[ \frac {\left (8 a \,c^{4}+24 a \,c^{2} d^{2}+3 a \,d^{4}+16 b \,c^{3} d +12 b c \,d^{3}\right ) \arctanh \left (\sin \left (f x +e \right )\right )}{8 f}+\frac {\left (95 a \,c^{3} d +80 a c \,d^{3}+12 b \,c^{4}+112 b \,c^{2} d^{2}+16 b \,d^{4}\right ) \tan \left (f x +e \right )}{30 f}+\frac {d \left (130 a \,c^{2} d +45 a \,d^{3}+24 b \,c^{3}+116 b c \,d^{2}\right ) \sec \left (f x +e \right ) \tan \left (f x +e \right )}{120 f}+\frac {\left (35 a c d +12 b \,c^{2}+16 b \,d^{2}\right ) \left (c +d \sec \left (f x +e \right )\right )^{2} \tan \left (f x +e \right )}{60 f}+\frac {\left (5 a d +4 b c \right ) \left (c +d \sec \left (f x +e \right )\right )^{3} \tan \left (f x +e \right )}{20 f}+\frac {b \left (c +d \sec \left (f x +e \right )\right )^{4} \tan \left (f x +e \right )}{5 f} \]
command
integrate(sec(f*x+e)*(a+b*sec(f*x+e))*(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 \[ \text {Exception raised: NotImplementedError} \]_______________________________________________________