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