\[ \int \frac {1}{(e \csc (c+d x))^{3/2} (a+a \sec (c+d x))^2} \, dx \]
Optimal antiderivative \[ \frac {4}{a^{2} d e \sqrt {e \csc \left (d x +c \right )}}-\frac {4 \cos \left (d x +c \right )}{3 a^{2} d e \sqrt {e \csc \left (d x +c \right )}}-\frac {2 \cos \left (d x +c \right ) \left (\cot ^{2}\left (d x +c \right )\right )}{3 a^{2} d e \sqrt {e \csc \left (d x +c \right )}}-\frac {2 \cot \left (d x +c \right ) \csc \left (d x +c \right )}{3 a^{2} d e \sqrt {e \csc \left (d x +c \right )}}+\frac {4 \left (\csc ^{2}\left (d x +c \right )\right )}{3 a^{2} d e \sqrt {e \csc \left (d x +c \right )}}+\frac {4 \sqrt {\frac {1}{2}+\frac {\sin \left (d x +c \right )}{2}}\, \EllipticF \left (\cos \left (\frac {c}{2}+\frac {\pi }{4}+\frac {d x}{2}\right ), \sqrt {2}\right )}{\sin \left (\frac {c}{2}+\frac {\pi }{4}+\frac {d x}{2}\right ) a^{2} d e \sqrt {e \csc \left (d x +c \right )}\, \sqrt {\sin \left (d x +c \right )}} \]
command
integrate(1/(e*csc(d*x+c))^(3/2)/(a+a*sec(d*x+c))^2,x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ -\frac {2 \, {\left (3 \, \sqrt {2 i} {\left (-i \, \cos \left (d x + c\right ) - i\right )} {\rm weierstrassPInverse}\left (4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) + 3 \, \sqrt {-2 i} {\left (i \, \cos \left (d x + c\right ) + i\right )} {\rm weierstrassPInverse}\left (4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) + {\left (\cos \left (d x + c\right )^{2} - 5 \, \cos \left (d x + c\right ) - 8\right )} \sqrt {\sin \left (d x + c\right )}\right )}}{3 \, {\left (a^{2} d \cos \left (d x + c\right ) e^{\frac {3}{2}} + a^{2} d e^{\frac {3}{2}}\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {\sqrt {e \csc \left (d x + c\right )}}{a^{2} e^{2} \csc \left (d x + c\right )^{2} \sec \left (d x + c\right )^{2} + 2 \, a^{2} e^{2} \csc \left (d x + c\right )^{2} \sec \left (d x + c\right ) + a^{2} e^{2} \csc \left (d x + c\right )^{2}}, x\right ) \]