\[ \int \frac {\tan ^2(e+f x)}{(a+a \sin (e+f x))^{5/2}} \, dx \]
Optimal antiderivative \[ -\frac {\sec \left (f x +e \right )}{6 f \left (a +a \sin \left (f x +e \right )\right )^{\frac {5}{2}}}-\frac {11 \cos \left (f x +e \right )}{128 a f \left (a +a \sin \left (f x +e \right )\right )^{\frac {3}{2}}}+\frac {17 \sec \left (f x +e \right )}{48 a f \left (a +a \sin \left (f x +e \right )\right )^{\frac {3}{2}}}-\frac {11 \arctanh \left (\frac {\cos \left (f x +e \right ) \sqrt {a}\, \sqrt {2}}{2 \sqrt {a +a \sin \left (f x +e \right )}}\right ) \sqrt {2}}{256 a^{\frac {5}{2}} f}+\frac {11 \sec \left (f x +e \right )}{96 a^{2} f \sqrt {a +a \sin \left (f x +e \right )}} \]
command
integrate(tan(f*x+e)^2/(a+a*sin(f*x+e))^(5/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {\frac {48 \, \sqrt {2}}{a^{\frac {5}{2}} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \sin \left (\frac {3}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )} - \frac {\sqrt {2} {\left (15 \, \sqrt {a} \sin \left (\frac {3}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} - 56 \, \sqrt {a} \sin \left (\frac {3}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 33 \, \sqrt {a} \sin \left (\frac {3}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}}{{\left (\sin \left (\frac {3}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 1\right )}^{3} a^{3} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}}{768 \, f} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: NotImplementedError} \]________________________________________________________________________________________