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