\[ \int \sec (e+f x) (a+b \sec (e+f x)) (c+d \sec (e+f x))^3 \, dx \]
Optimal antiderivative \[ \frac {\left (8 a \,c^{3}+12 a c \,d^{2}+12 b \,c^{2} d +3 b \,d^{3}\right ) \arctanh \left (\sin \left (f x +e \right )\right )}{8 f}+\frac {\left (4 a d \left (4 c^{2}+d^{2}\right )+3 b \left (c^{3}+4 c \,d^{2}\right )\right ) \tan \left (f x +e \right )}{6 f}+\frac {d \left (20 a c d +6 b \,c^{2}+9 b \,d^{2}\right ) \sec \left (f x +e \right ) \tan \left (f x +e \right )}{24 f}+\frac {\left (4 a d +3 b c \right ) \left (c +d \sec \left (f x +e \right )\right )^{2} \tan \left (f x +e \right )}{12 f}+\frac {b \left (c +d \sec \left (f x +e \right )\right )^{3} \tan \left (f x +e \right )}{4 f} \]
command
integrate(sec(f*x+e)*(a+b*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 \, b c^{2} d + 12 \, a c d^{2} + 3 \, b 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 \, b c^{2} d + 12 \, a c d^{2} + 3 \, b 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 \, b c^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{7} + 72 \, a c^{2} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{7} - 36 \, b 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} + 72 \, b c d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{7} + 24 \, a d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{7} - 15 \, b d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{7} - 72 \, b c^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} - 216 \, a c^{2} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} + 36 \, b c^{2} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} + 36 \, a c d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} - 120 \, b c d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} - 40 \, a d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} - 9 \, b d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} + 72 \, b c^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 216 \, a c^{2} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 36 \, b c^{2} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 36 \, a c d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 120 \, b c d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 40 \, a d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 9 \, b d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 24 \, b c^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 72 \, a c^{2} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 36 \, b c^{2} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 36 \, a c d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 72 \, b c d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 24 \, a d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 15 \, b 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
\[ \text {Exception raised: NotImplementedError} \]________________________________________________________________________________________