\[ \int \sec (e+f x) (a+a \sec (e+f x))^2 (c+d \sec (e+f x))^2 \, dx \]
Optimal antiderivative \[ \frac {a^{2} \left (12 c^{2}+16 c d +7 d^{2}\right ) \arctanh \left (\sin \left (f x +e \right )\right )}{8 f}-\frac {a^{2} \left (c^{3}-8 c^{2} d -20 c \,d^{2}-8 d^{3}\right ) \tan \left (f x +e \right )}{6 d f}-\frac {a^{2} \left (2 c \left (c -8 d \right )-21 d^{2}\right ) \sec \left (f x +e \right ) \tan \left (f x +e \right )}{24 f}-\frac {a^{2} \left (c -8 d \right ) \left (c +d \sec \left (f x +e \right )\right )^{2} \tan \left (f x +e \right )}{12 d f}+\frac {a^{2} \left (c +d \sec \left (f x +e \right )\right )^{3} \tan \left (f x +e \right )}{4 d f} \]
command
integrate(sec(f*x+e)*(a+a*sec(f*x+e))^2*(c+d*sec(f*x+e))^2,x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {3 \, {\left (12 \, a^{2} c^{2} + 16 \, a^{2} c d + 7 \, a^{2} d^{2}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1 \right |}\right ) - 3 \, {\left (12 \, a^{2} c^{2} + 16 \, a^{2} c d + 7 \, a^{2} d^{2}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1 \right |}\right ) - \frac {2 \, {\left (36 \, a^{2} c^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{7} + 48 \, a^{2} c d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{7} + 21 \, a^{2} d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{7} - 132 \, a^{2} c^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} - 176 \, a^{2} c d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} - 77 \, a^{2} d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} + 156 \, a^{2} c^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 272 \, a^{2} c d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 83 \, a^{2} d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 60 \, a^{2} c^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 144 \, a^{2} c d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 75 \, a^{2} d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 1\right )}^{4}}}{24 \, f} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: NotImplementedError} \]________________________________________________________________________________________