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