\[ \int \sqrt {a+a \sec (e+f x)} (c-c \sec (e+f x)) \, dx \]
Optimal antiderivative \[ \frac {2 c \arctan \left (\frac {\sqrt {a}\, \tan \left (f x +e \right )}{\sqrt {a +a \sec \left (f x +e \right )}}\right ) \sqrt {a}}{f}-\frac {2 a c \tan \left (f x +e \right )}{f \sqrt {a +a \sec \left (f x +e \right )}} \]
command
integrate((c-c*sec(f*x+e))*(a+a*sec(f*x+e))^(1/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {\frac {2 \, \sqrt {2} \sqrt {-a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + a} a c \mathrm {sgn}\left (\cos \left (f x + e\right )\right ) \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )}{a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - a} - \frac {\sqrt {-a} a c \log \left (\frac {{\left | 2 \, {\left (\sqrt {-a} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - \sqrt {-a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + a}\right )}^{2} - 4 \, \sqrt {2} {\left | a \right |} - 6 \, a \right |}}{{\left | 2 \, {\left (\sqrt {-a} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - \sqrt {-a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + a}\right )}^{2} + 4 \, \sqrt {2} {\left | a \right |} - 6 \, a \right |}}\right ) \mathrm {sgn}\left (\cos \left (f x + e\right )\right )}{{\left | a \right |}}}{f} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: NotImplementedError} \]________________________________________________________________________________________