\[ \int \frac {1}{(b \sec (e+f x))^{3/2} (a \sin (e+f x))^{13/2}} \, dx \]
Optimal antiderivative \[ -\frac {2}{11 a b f \left (a \sin \left (f x +e \right )\right )^{\frac {11}{2}} \sqrt {b \sec \left (f x +e \right )}}+\frac {2}{77 a^{3} b f \left (a \sin \left (f x +e \right )\right )^{\frac {7}{2}} \sqrt {b \sec \left (f x +e \right )}}+\frac {4}{77 a^{5} b f \left (a \sin \left (f x +e \right )\right )^{\frac {3}{2}} \sqrt {b \sec \left (f x +e \right )}}+\frac {4 \sqrt {\frac {1}{2}+\frac {\sin \left (2 f x +2 e \right )}{2}}\, \EllipticF \left (\cos \left (e +\frac {\pi }{4}+f x \right ), \sqrt {2}\right ) \sqrt {b \sec \left (f x +e \right )}\, \left (\sqrt {\sin }\left (2 f x +2 e \right )\right )}{77 \sin \left (e +\frac {\pi }{4}+f x \right ) a^{6} b^{2} f \sqrt {a \sin \left (f x +e \right )}} \]
command
integrate(1/(b*sec(f*x+e))^(3/2)/(a*sin(f*x+e))^(13/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {2 \, {\left (2 \, {\left (\cos \left (f x + e\right )^{6} - 3 \, \cos \left (f x + e\right )^{4} + 3 \, \cos \left (f x + e\right )^{2} - 1\right )} \sqrt {i \, a b} {\rm ellipticF}\left (\cos \left (f x + e\right ) + i \, \sin \left (f x + e\right ), -1\right ) + 2 \, {\left (\cos \left (f x + e\right )^{6} - 3 \, \cos \left (f x + e\right )^{4} + 3 \, \cos \left (f x + e\right )^{2} - 1\right )} \sqrt {-i \, a b} {\rm ellipticF}\left (\cos \left (f x + e\right ) - i \, \sin \left (f x + e\right ), -1\right ) - {\left (2 \, \cos \left (f x + e\right )^{5} - 5 \, \cos \left (f x + e\right )^{3} - 4 \, \cos \left (f x + e\right )\right )} \sqrt {a \sin \left (f x + e\right )} \sqrt {\frac {b}{\cos \left (f x + e\right )}}\right )}}{77 \, {\left (a^{7} b^{2} f \cos \left (f x + e\right )^{6} - 3 \, a^{7} b^{2} f \cos \left (f x + e\right )^{4} + 3 \, a^{7} b^{2} f \cos \left (f x + e\right )^{2} - a^{7} b^{2} f\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (-\frac {\sqrt {b \sec \left (f x + e\right )} \sqrt {a \sin \left (f x + e\right )}}{{\left (a^{7} b^{2} \cos \left (f x + e\right )^{6} - 3 \, a^{7} b^{2} \cos \left (f x + e\right )^{4} + 3 \, a^{7} b^{2} \cos \left (f x + e\right )^{2} - a^{7} b^{2}\right )} \sec \left (f x + e\right )^{2} \sin \left (f x + e\right )}, x\right ) \]