\[ \int \frac {\sqrt {a+a \sec (e+f x)}}{c+d \sec (e+f x)} \, dx \]
Optimal antiderivative \[ \frac {2 \arctan \left (\frac {\sqrt {a}\, \tan \left (f x +e \right )}{\sqrt {a +a \sec \left (f x +e \right )}}\right ) \sqrt {a}}{c f}-\frac {2 \arctan \left (\frac {\sqrt {a}\, \sqrt {d}\, \tan \left (f x +e \right )}{\sqrt {c +d}\, \sqrt {a +a \sec \left (f x +e \right )}}\right ) \sqrt {a}\, \sqrt {d}}{c f \sqrt {c +d}} \]
command
integrate((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
\[ -\frac {\sqrt {2} {\left (\frac {2 \, \sqrt {2} \sqrt {-a} d \arctan \left (\frac {\sqrt {2} {\left ({\left (\sqrt {-a} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - \sqrt {-a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + a}\right )}^{2} c - {\left (\sqrt {-a} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - \sqrt {-a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + a}\right )}^{2} d + a c + 3 \, a d\right )}}{4 \, \sqrt {-c d - d^{2}} a}\right )}{\sqrt {-c d - d^{2}} c} + \frac {\sqrt {2} \sqrt {-a} a \log \left (\frac {{\left | 2 \, {\left (\sqrt {-a} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - \sqrt {-a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + a}\right )}^{2} - 4 \, \sqrt {2} {\left | a \right |} - 6 \, a \right |}}{{\left | 2 \, {\left (\sqrt {-a} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - \sqrt {-a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + a}\right )}^{2} + 4 \, \sqrt {2} {\left | a \right |} - 6 \, a \right |}}\right )}{c {\left | a \right |}}\right )} \mathrm {sgn}\left (\cos \left (f x + e\right )\right )}{2 \, f} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: NotImplementedError} \]________________________________________________________________________________________