\[ \int \frac {1}{(e \csc (c+d x))^{5/2} (a+a \sec (c+d x))^2} \, dx \]
Optimal antiderivative \[ -\frac {2 \cot \left (d x +c \right )}{a^{2} d \,e^{2} \sqrt {e \csc \left (d x +c \right )}}-\frac {2 \left (\cos ^{2}\left (d x +c \right )\right ) \cot \left (d x +c \right )}{a^{2} d \,e^{2} \sqrt {e \csc \left (d x +c \right )}}+\frac {4 \csc \left (d x +c \right )}{a^{2} d \,e^{2} \sqrt {e \csc \left (d x +c \right )}}+\frac {4 \sin \left (d x +c \right )}{3 a^{2} d \,e^{2} \sqrt {e \csc \left (d x +c \right )}}-\frac {12 \cos \left (d x +c \right ) \sin \left (d x +c \right )}{5 a^{2} d \,e^{2} \sqrt {e \csc \left (d x +c \right )}}+\frac {44 \sqrt {\frac {1}{2}+\frac {\sin \left (d x +c \right )}{2}}\, \EllipticE \left (\cos \left (\frac {c}{2}+\frac {\pi }{4}+\frac {d x}{2}\right ), \sqrt {2}\right )}{5 \sin \left (\frac {c}{2}+\frac {\pi }{4}+\frac {d x}{2}\right ) a^{2} d \,e^{2} \sqrt {e \csc \left (d x +c \right )}\, \sqrt {\sin \left (d x +c \right )}} \]
command
integrate(1/(e*csc(d*x+c))^(5/2)/(a+a*sec(d*x+c))^2,x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ -\frac {2 \, {\left (33 \, \sqrt {2 i} {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) + 33 \, \sqrt {-2 i} {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) - \frac {3 \, \cos \left (d x + c\right )^{3} - 10 \, \cos \left (d x + c\right )^{2} - 33 \, \cos \left (d x + c\right ) + 40}{\sqrt {\sin \left (d x + c\right )}}\right )} e^{\left (-\frac {5}{2}\right )}}{15 \, a^{2} d} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {\sqrt {e \csc \left (d x + c\right )}}{a^{2} e^{3} \csc \left (d x + c\right )^{3} \sec \left (d x + c\right )^{2} + 2 \, a^{2} e^{3} \csc \left (d x + c\right )^{3} \sec \left (d x + c\right ) + a^{2} e^{3} \csc \left (d x + c\right )^{3}}, x\right ) \]