\[ \int \frac {\csc ^2(a+b x)}{(d \cos (a+b x))^{5/2}} \, dx \]
Optimal antiderivative \[ -\frac {\csc \left (b x +a \right )}{b d \left (d \cos \left (b x +a \right )\right )^{\frac {3}{2}}}+\frac {5 \sin \left (b x +a \right )}{3 b d \left (d \cos \left (b x +a \right )\right )^{\frac {3}{2}}}+\frac {5 \sqrt {\frac {\cos \left (b x +a \right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sin \left (\frac {a}{2}+\frac {b x}{2}\right ), \sqrt {2}\right ) \left (\sqrt {\cos }\left (b x +a \right )\right )}{3 \cos \left (\frac {a}{2}+\frac {b x}{2}\right ) b \,d^{2} \sqrt {d \cos \left (b x +a \right )}} \]
command
integrate(csc(b*x+a)^2/(d*cos(b*x+a))^(5/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {-5 i \, \sqrt {2} \sqrt {d} \cos \left (b x + a\right )^{2} \sin \left (b x + a\right ) {\rm weierstrassPInverse}\left (-4, 0, \cos \left (b x + a\right ) + i \, \sin \left (b x + a\right )\right ) + 5 i \, \sqrt {2} \sqrt {d} \cos \left (b x + a\right )^{2} \sin \left (b x + a\right ) {\rm weierstrassPInverse}\left (-4, 0, \cos \left (b x + a\right ) - i \, \sin \left (b x + a\right )\right ) - 2 \, \sqrt {d \cos \left (b x + a\right )} {\left (5 \, \cos \left (b x + a\right )^{2} - 2\right )}}{6 \, b d^{3} \cos \left (b x + a\right )^{2} \sin \left (b x + a\right )} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {\sqrt {d \cos \left (b x + a\right )} \csc \left (b x + a\right )^{2}}{d^{3} \cos \left (b x + a\right )^{3}}, x\right ) \]