\[ \int \frac {\sec (e+f x) (c+d \sec (e+f x))^2}{(a+b \sec (e+f x))^2} \, dx \]
Optimal antiderivative \[ \frac {d^{2} \arctanh \left (\sin \left (f x +e \right )\right )}{b^{2} f}+\frac {2 \left (-a d +b c \right )^{2} \arctanh \left (\frac {\sqrt {a -b}\, \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{\sqrt {a +b}}\right )}{a \left (a -b \right )^{\frac {3}{2}} \left (a +b \right )^{\frac {3}{2}} f}-\frac {\left (-a d +b c \right )^{2} \sin \left (f x +e \right )}{b \left (a^{2}-b^{2}\right ) f \left (b +a \cos \left (f x +e \right )\right )}+\frac {2 \left (-a^{2} d^{2}+b^{2} c^{2}\right ) \arctanh \left (\frac {\sqrt {a -b}\, \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{\sqrt {a +b}}\right )}{a \,b^{2} f \sqrt {a -b}\, \sqrt {a +b}} \]
command
integrate(sec(f*x+e)*(c+d*sec(f*x+e))^2/(a+b*sec(f*x+e))^2,x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {\frac {d^{2} \log \left ({\left | \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1 \right |}\right )}{b^{2}} - \frac {d^{2} \log \left ({\left | \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1 \right |}\right )}{b^{2}} - \frac {2 \, {\left (a b^{2} c^{2} - 2 \, b^{3} c d - a^{3} d^{2} + 2 \, a b^{2} d^{2}\right )} {\left (\pi \left \lfloor \frac {f x + e}{2 \, \pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (2 \, a - 2 \, b\right ) + \arctan \left (\frac {a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - b \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )}{\sqrt {-a^{2} + b^{2}}}\right )\right )}}{{\left (a^{2} b^{2} - b^{4}\right )} \sqrt {-a^{2} + b^{2}}} + \frac {2 \, {\left (b^{2} c^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 2 \, a b c d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + a^{2} d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}}{{\left (a^{2} b - b^{3}\right )} {\left (a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - b \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - a - b\right )}}}{f} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: NotImplementedError} \]________________________________________________________________________________________