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