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