\[ \int \sec (e+f x) (a+a \sec (e+f x))^3 (c+d \sec (e+f x))^2 \, dx \]
Optimal antiderivative \[ \frac {a^{3} \left (20 c^{2}+30 c d +13 d^{2}\right ) \arctanh \left (\sin \left (f x +e \right )\right )}{8 f}+\frac {a^{3} \left (2 c^{4}-15 c^{3} d +72 c^{2} d^{2}+180 c \,d^{3}+76 d^{4}\right ) \tan \left (f x +e \right )}{30 d^{2} f}+\frac {a^{3} \left (4 c^{3}-30 c^{2} d +146 c \,d^{2}+195 d^{3}\right ) \sec \left (f x +e \right ) \tan \left (f x +e \right )}{120 d f}+\frac {a^{3} \left (2 c^{2}-15 c d +76 d^{2}\right ) \left (c +d \sec \left (f x +e \right )\right )^{2} \tan \left (f x +e \right )}{60 d^{2} f}-\frac {a^{3} \left (2 c -11 d \right ) \left (c +d \sec \left (f x +e \right )\right )^{3} \tan \left (f x +e \right )}{20 d^{2} f}+\frac {\left (a^{3}+a^{3} \sec \left (f x +e \right )\right ) \left (c +d \sec \left (f x +e \right )\right )^{3} \tan \left (f x +e \right )}{5 d f} \]
command
integrate(sec(f*x+e)*(a+a*sec(f*x+e))^3*(c+d*sec(f*x+e))^2,x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {15 \, {\left (20 \, a^{3} c^{2} + 30 \, a^{3} c d + 13 \, a^{3} d^{2}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1 \right |}\right ) - 15 \, {\left (20 \, a^{3} c^{2} + 30 \, a^{3} c d + 13 \, a^{3} d^{2}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1 \right |}\right ) - \frac {2 \, {\left (300 \, a^{3} c^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{9} + 450 \, a^{3} c d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{9} + 195 \, a^{3} d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{9} - 1400 \, a^{3} c^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{7} - 2100 \, a^{3} c d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{7} - 910 \, a^{3} d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{7} + 2560 \, a^{3} c^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} + 3840 \, a^{3} c d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} + 1664 \, a^{3} d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} - 2120 \, a^{3} c^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 3660 \, a^{3} c d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 1330 \, a^{3} d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 660 \, a^{3} c^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1470 \, a^{3} c d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 765 \, a^{3} d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 1\right )}^{5}}}{120 \, f} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: NotImplementedError} \]________________________________________________________________________________________