\[ \int \frac {\sin (e+f x)}{\sqrt {a+b \tan ^2(e+f x)}} \, dx \]
Optimal antiderivative \[ -\frac {\cos \left (f x +e \right ) \sqrt {a -b +b \left (\sec ^{2}\left (f x +e \right )\right )}}{\left (a -b \right ) f} \]
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
\[ \frac {\sqrt {b} \mathrm {sgn}\left (f\right ) \mathrm {sgn}\left (\cos \left (f x + e\right )\right )}{a {\left | f \right |} - b {\left | f \right |}} - \frac {\sqrt {a \cos \left (f x + e\right )^{2} - b \cos \left (f x + e\right )^{2} + b}}{a {\left | f \right |} \mathrm {sgn}\left (f\right ) \mathrm {sgn}\left (\cos \left (f x + e\right )\right ) - b {\left | f \right |} \mathrm {sgn}\left (f\right ) \mathrm {sgn}\left (\cos \left (f x + e\right )\right )} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: NotImplementedError} \]________________________________________________________________________________________