\[ \int \frac {\tan ^2(e+f x)}{(a+a \sec (e+f x))^{9/2}} \, dx \]
Optimal antiderivative \[ -\frac {2 \arctan \left (\frac {\sqrt {a}\, \tan \left (f x +e \right )}{\sqrt {a +a \sec \left (f x +e \right )}}\right )}{a^{\frac {9}{2}} f}+\frac {91 \arctan \left (\frac {\sqrt {a}\, \tan \left (f x +e \right ) \sqrt {2}}{2 \sqrt {a +a \sec \left (f x +e \right )}}\right ) \sqrt {2}}{64 a^{\frac {9}{2}} f}+\frac {\tan \left (f x +e \right )}{3 a f \left (a +a \sec \left (f x +e \right )\right )^{\frac {7}{2}}}+\frac {11 \tan \left (f x +e \right )}{24 a^{2} f \left (a +a \sec \left (f x +e \right )\right )^{\frac {5}{2}}}+\frac {27 \tan \left (f x +e \right )}{32 a^{3} f \left (a +a \sec \left (f x +e \right )\right )^{\frac {3}{2}}} \]
command
integrate(tan(f*x+e)^2/(a+a*sec(f*x+e))^(9/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {\sqrt {-a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + a} {\left (2 \, {\left (\frac {4 \, \sqrt {2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2}}{a^{5} \mathrm {sgn}\left (\cos \left (f x + e\right )\right )} - \frac {19 \, \sqrt {2}}{a^{5} \mathrm {sgn}\left (\cos \left (f x + e\right )\right )}\right )} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + \frac {111 \, \sqrt {2}}{a^{5} \mathrm {sgn}\left (\cos \left (f x + e\right )\right )}\right )} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )}{192 \, f} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: NotImplementedError} \]________________________________________________________________________________________