\[ \int \sec (e+f x) (a+a \sec (e+f x)) (c+d \sec (e+f x))^2 \, dx \]
Optimal antiderivative \[ \frac {a \left (2 c^{2}+2 c d +d^{2}\right ) \arctanh \left (\sin \left (f x +e \right )\right )}{2 f}+\frac {2 a \left (c^{2}+3 c d +d^{2}\right ) \tan \left (f x +e \right )}{3 f}+\frac {a d \left (2 c +3 d \right ) \sec \left (f x +e \right ) \tan \left (f x +e \right )}{6 f}+\frac {a \left (c +d \sec \left (f x +e \right )\right )^{2} \tan \left (f x +e \right )}{3 f} \]
command
integrate(sec(f*x+e)*(a+a*sec(f*x+e))*(c+d*sec(f*x+e))^2,x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {3 \, {\left (2 \, a c^{2} + 2 \, a c d + a d^{2}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1 \right |}\right ) - 3 \, {\left (2 \, a c^{2} + 2 \, a c d + a d^{2}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1 \right |}\right ) - \frac {2 \, {\left (6 \, a c^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} + 6 \, a c d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} + 3 \, a d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} - 12 \, a c^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 24 \, a c d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 4 \, a d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 6 \, a c^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 18 \, a c d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 9 \, a d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 1\right )}^{3}}}{6 \, f} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: NotImplementedError} \]________________________________________________________________________________________