\[ \int \frac {\sin (e+f x)}{\left (a+b \tan ^2(e+f x)\right )^{3/2}} \, dx \]
Optimal antiderivative \[ -\frac {\cos \left (f x +e \right )}{\left (a -b \right ) f \sqrt {a -b +b \left (\sec ^{2}\left (f x +e \right )\right )}}-\frac {2 b \sec \left (f x +e \right )}{\left (a -b \right )^{2} f \sqrt {a -b +b \left (\sec ^{2}\left (f x +e \right )\right )}} \]
command
integrate(sin(f*x+e)/(a+b*tan(f*x+e)^2)^(3/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ -\frac {f^{2} {\left (\frac {\sqrt {a \cos \left (f x + e\right )^{2} - b \cos \left (f x + e\right )^{2} + b}}{a {\left | f \right |} \mathrm {sgn}\left (f\right ) \mathrm {sgn}\left (\cos \left (f x + e\right )\right ) - b {\left | f \right |} \mathrm {sgn}\left (f\right ) \mathrm {sgn}\left (\cos \left (f x + e\right )\right )} + \frac {b}{{\left (a {\left | f \right |} \mathrm {sgn}\left (f\right ) \mathrm {sgn}\left (\cos \left (f x + e\right )\right ) - b {\left | f \right |} \mathrm {sgn}\left (f\right ) \mathrm {sgn}\left (\cos \left (f x + e\right )\right )\right )} \sqrt {a \cos \left (f x + e\right )^{2} - b \cos \left (f x + e\right )^{2} + b}}\right )}}{a f^{2} - b f^{2}} + \frac {2 \, \sqrt {b} \mathrm {sgn}\left (f\right ) \mathrm {sgn}\left (\cos \left (f x + e\right )\right )}{a^{2} {\left | f \right |} - 2 \, a b {\left | f \right |} + b^{2} {\left | f \right |}} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: NotImplementedError} \]________________________________________________________________________________________