\[ \int \frac {(a+b \tan (e+f x))^2}{(d \sec (e+f x))^{9/2}} \, dx \]
Optimal antiderivative \[ -\frac {10 a b}{63 f \left (d \sec \left (f x +e \right )\right )^{\frac {9}{2}}}+\frac {2 \left (7 a^{2}+2 b^{2}\right ) \sin \left (f x +e \right )}{63 d f \left (d \sec \left (f x +e \right )\right )^{\frac {7}{2}}}+\frac {2 \left (7 a^{2}+2 b^{2}\right ) \sin \left (f x +e \right )}{45 d^{3} f \left (d \sec \left (f x +e \right )\right )^{\frac {3}{2}}}+\frac {2 \left (7 a^{2}+2 b^{2}\right ) \sqrt {\frac {\cos \left (f x +e \right )}{2}+\frac {1}{2}}\, \EllipticE \left (\sin \left (\frac {f x}{2}+\frac {e}{2}\right ), \sqrt {2}\right )}{15 \cos \left (\frac {f x}{2}+\frac {e}{2}\right ) d^{4} f \sqrt {\cos \left (f x +e \right )}\, \sqrt {d \sec \left (f x +e \right )}}-\frac {2 b \left (a +b \tan \left (f x +e \right )\right )}{7 f \left (d \sec \left (f x +e \right )\right )^{\frac {9}{2}}} \]
command
integrate((a+b*tan(f*x+e))^2/(d*sec(f*x+e))^(9/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ -\frac {3 \, \sqrt {2} {\left (-7 i \, a^{2} - 2 i \, b^{2}\right )} \sqrt {d} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (f x + e\right ) + i \, \sin \left (f x + e\right )\right )\right ) + 3 \, \sqrt {2} {\left (7 i \, a^{2} + 2 i \, b^{2}\right )} \sqrt {d} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (f x + e\right ) - i \, \sin \left (f x + e\right )\right )\right ) + 2 \, {\left (10 \, a b \cos \left (f x + e\right )^{5} - {\left (5 \, {\left (a^{2} - b^{2}\right )} \cos \left (f x + e\right )^{4} + {\left (7 \, a^{2} + 2 \, b^{2}\right )} \cos \left (f x + e\right )^{2}\right )} \sin \left (f x + e\right )\right )} \sqrt {\frac {d}{\cos \left (f x + e\right )}}}{45 \, d^{5} f} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {{\left (b^{2} \tan \left (f x + e\right )^{2} + 2 \, a b \tan \left (f x + e\right ) + a^{2}\right )} \sqrt {d \sec \left (f x + e\right )}}{d^{5} \sec \left (f x + e\right )^{5}}, x\right ) \]