\[ \int \frac {\sin (e+f x)}{\left (a+b \tan ^2(e+f x)\right )^{5/2}} \, dx \]
Optimal antiderivative \[ -\frac {\cos \left (f x +e \right )}{\left (a -b \right ) f \left (a -b +b \left (\sec ^{2}\left (f x +e \right )\right )\right )^{\frac {3}{2}}}-\frac {4 b \sec \left (f x +e \right )}{3 \left (a -b \right )^{2} f \left (a -b +b \left (\sec ^{2}\left (f x +e \right )\right )\right )^{\frac {3}{2}}}-\frac {8 b \sec \left (f x +e \right )}{3 \left (a -b \right )^{3} f \sqrt {a -b +b \left (\sec ^{2}\left (f x +e \right )\right )}} \]
command
integrate(sin(f*x+e)/(a+b*tan(f*x+e)^2)^(5/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ -\frac {f^{4} {\left (\frac {3 \, \sqrt {a \cos \left (f x + e\right )^{2} - b \cos \left (f x + e\right )^{2} + b}}{a {\left | f \right |} \mathrm {sgn}\left (f\right ) \mathrm {sgn}\left (\cos \left (f x + e\right )\right ) - b {\left | f \right |} \mathrm {sgn}\left (f\right ) \mathrm {sgn}\left (\cos \left (f x + e\right )\right )} + \frac {6 \, {\left (a \cos \left (f x + e\right )^{2} - b \cos \left (f x + e\right )^{2} + b\right )} b - b^{2}}{{\left (a {\left | f \right |} \mathrm {sgn}\left (f\right ) \mathrm {sgn}\left (\cos \left (f x + e\right )\right ) - b {\left | f \right |} \mathrm {sgn}\left (f\right ) \mathrm {sgn}\left (\cos \left (f x + e\right )\right )\right )} {\left (a \cos \left (f x + e\right )^{2} - b \cos \left (f x + e\right )^{2} + b\right )}^{\frac {3}{2}}}\right )}}{3 \, {\left (a f^{2} - b f^{2}\right )}^{2}} + \frac {8 \, \sqrt {b} \mathrm {sgn}\left (f\right ) \mathrm {sgn}\left (\cos \left (f x + e\right )\right )}{3 \, {\left (a^{3} {\left | f \right |} - 3 \, a^{2} b {\left | f \right |} + 3 \, a b^{2} {\left | f \right |} - b^{3} {\left | f \right |}\right )}} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: NotImplementedError} \]________________________________________________________________________________________