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