\[ \int \sec (e+f x) (a+a \sec (e+f x)) (c+d \sec (e+f x))^3 \, dx \]
Optimal antiderivative \[ \frac {a \left (8 c^{3}+12 c^{2} d +12 c \,d^{2}+3 d^{3}\right ) \arctanh \left (\sin \left (f x +e \right )\right )}{8 f}+\frac {a \left (3 c^{3}+16 c^{2} d +12 c \,d^{2}+4 d^{3}\right ) \tan \left (f x +e \right )}{6 f}+\frac {a d \left (6 c^{2}+20 c d +9 d^{2}\right ) \sec \left (f x +e \right ) \tan \left (f x +e \right )}{24 f}+\frac {a \left (3 c +4 d \right ) \left (c +d \sec \left (f x +e \right )\right )^{2} \tan \left (f x +e \right )}{12 f}+\frac {a \left (c +d \sec \left (f x +e \right )\right )^{3} \tan \left (f x +e \right )}{4 f} \]
command
integrate(sec(f*x+e)*(a+a*sec(f*x+e))*(c+d*sec(f*x+e))^3,x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {3 \, {\left (8 \, a c^{3} + 12 \, a c^{2} d + 12 \, a c d^{2} + 3 \, a d^{3}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1 \right |}\right ) - 3 \, {\left (8 \, a c^{3} + 12 \, a c^{2} d + 12 \, a c d^{2} + 3 \, a d^{3}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1 \right |}\right ) - \frac {2 \, {\left (24 \, a c^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{7} + 36 \, a c^{2} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{7} + 36 \, a c d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{7} + 9 \, a d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{7} - 72 \, a c^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} - 180 \, a c^{2} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} - 84 \, a c d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} - 49 \, a d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} + 72 \, a c^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 252 \, a c^{2} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 156 \, a c d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 31 \, a d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 24 \, a c^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 108 \, a c^{2} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 108 \, a c d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 39 \, a d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 1\right )}^{4}}}{24 \, f} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: NotImplementedError} \]________________________________________________________________________________________