\[ \int (e \csc (c+d x))^{3/2} (a+a \sec (c+d x))^2 \, dx \]
Optimal antiderivative \[ -\frac {4 a^{2} e \sqrt {e \csc \left (d x +c \right )}}{d}-\frac {2 a^{2} e \cos \left (d x +c \right ) \sqrt {e \csc \left (d x +c \right )}}{d}-\frac {2 a^{2} e \sec \left (d x +c \right ) \sqrt {e \csc \left (d x +c \right )}}{d}-\frac {2 a^{2} e \arctan \left (\sqrt {\sin }\left (d x +c \right )\right ) \sqrt {e \csc \left (d x +c \right )}\, \left (\sqrt {\sin }\left (d x +c \right )\right )}{d}+\frac {2 a^{2} e \arctanh \left (\sqrt {\sin }\left (d x +c \right )\right ) \sqrt {e \csc \left (d x +c \right )}\, \left (\sqrt {\sin }\left (d x +c \right )\right )}{d}+\frac {5 a^{2} e \sqrt {\frac {1}{2}+\frac {\sin \left (d x +c \right )}{2}}\, \EllipticE \left (\cos \left (\frac {c}{2}+\frac {\pi }{4}+\frac {d x}{2}\right ), \sqrt {2}\right ) \sqrt {e \csc \left (d x +c \right )}\, \left (\sqrt {\sin }\left (d x +c \right )\right )}{\sin \left (\frac {c}{2}+\frac {\pi }{4}+\frac {d x}{2}\right ) d}+\frac {3 a^{2} e \sin \left (d x +c \right ) \sqrt {e \csc \left (d x +c \right )}\, \tan \left (d x +c \right )}{d} \]
command
integrate((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
\[ -\frac {5 \, \sqrt {2 i} a^{2} \cos \left (d x + c\right ) e^{\frac {3}{2}} {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) + 5 \, \sqrt {-2 i} a^{2} \cos \left (d x + c\right ) e^{\frac {3}{2}} {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) + 2 \, a^{2} \arctan \left (\frac {\sin \left (d x + c\right ) - 1}{2 \, \sqrt {\sin \left (d x + c\right )}}\right ) \cos \left (d x + c\right ) e^{\frac {3}{2}} - a^{2} \cos \left (d x + c\right ) e^{\frac {3}{2}} \log \left (\frac {\cos \left (d x + c\right )^{2} + \frac {4 \, {\left (\cos \left (d x + c\right )^{2} - \sin \left (d x + c\right ) - 1\right )}}{\sqrt {\sin \left (d x + c\right )}} - 6 \, \sin \left (d x + c\right ) - 2}{\cos \left (d x + c\right )^{2} + 2 \, \sin \left (d x + c\right ) - 2}\right ) + \frac {2 \, {\left (5 \, a^{2} \cos \left (d x + c\right )^{2} e^{\frac {3}{2}} + 4 \, a^{2} \cos \left (d x + c\right ) e^{\frac {3}{2}} - a^{2} e^{\frac {3}{2}}\right )}}{\sqrt {\sin \left (d x + c\right )}}}{2 \, d \cos \left (d x + c\right )} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left ({\left (a^{2} e \csc \left (d x + c\right ) \sec \left (d x + c\right )^{2} + 2 \, a^{2} e \csc \left (d x + c\right ) \sec \left (d x + c\right ) + a^{2} e \csc \left (d x + c\right )\right )} \sqrt {e \csc \left (d x + c\right )}, x\right ) \]