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