\[ \int \frac {(a+a \sec (c+d x))^2}{(e \csc (c+d x))^{5/2}} \, dx \]
Optimal antiderivative \[ -\frac {4 a^{2} \sin \left (d x +c \right )}{3 d \,e^{2} \sqrt {e \csc \left (d x +c \right )}}-\frac {2 a^{2} \cos \left (d x +c \right ) \sin \left (d x +c \right )}{5 d \,e^{2} \sqrt {e \csc \left (d x +c \right )}}-\frac {2 a^{2} \arctan \left (\sqrt {\sin }\left (d x +c \right )\right )}{d \,e^{2} \sqrt {e \csc \left (d x +c \right )}\, \sqrt {\sin \left (d x +c \right )}}+\frac {2 a^{2} \arctanh \left (\sqrt {\sin }\left (d x +c \right )\right )}{d \,e^{2} \sqrt {e \csc \left (d x +c \right )}\, \sqrt {\sin \left (d x +c \right )}}+\frac {9 a^{2} \sqrt {\frac {1}{2}+\frac {\sin \left (d x +c \right )}{2}}\, \EllipticE \left (\cos \left (\frac {c}{2}+\frac {\pi }{4}+\frac {d x}{2}\right ), \sqrt {2}\right )}{5 \sin \left (\frac {c}{2}+\frac {\pi }{4}+\frac {d x}{2}\right ) d \,e^{2} \sqrt {e \csc \left (d x +c \right )}\, \sqrt {\sin \left (d x +c \right )}}+\frac {a^{2} \tan \left (d x +c \right )}{d \,e^{2} \sqrt {e \csc \left (d x +c \right )}} \]
command
integrate((a+a*sec(d*x+c))^2/(e*csc(d*x+c))^(5/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ -\frac {{\left (27 \, \sqrt {2 i} a^{2} \cos \left (d x + c\right ) {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) + 27 \, \sqrt {-2 i} a^{2} \cos \left (d x + c\right ) {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) - 30 \, a^{2} \arctan \left (\frac {76 \, \cos \left (d x + c\right )^{2} + \frac {425 \, {\left (\cos \left (d x + c\right )^{2} + \sin \left (d x + c\right ) - 1\right )}}{\sqrt {\sin \left (d x + c\right )}} - 152 \, \sin \left (d x + c\right ) - 152}{2 \, {\left (16 \, \cos \left (d x + c\right )^{2} + 393 \, \sin \left (d x + c\right ) - 32\right )}}\right ) \cos \left (d x + c\right ) - 15 \, a^{2} \cos \left (d x + c\right ) \log \left (\frac {\cos \left (d x + c\right )^{2} + \frac {4 \, {\left (\cos \left (d x + c\right )^{2} - \sin \left (d x + c\right ) - 1\right )}}{\sqrt {\sin \left (d x + c\right )}} - 6 \, \sin \left (d x + c\right ) - 2}{\cos \left (d x + c\right )^{2} + 2 \, \sin \left (d x + c\right ) - 2}\right ) - \frac {2 \, {\left (6 \, a^{2} \cos \left (d x + c\right )^{4} + 20 \, a^{2} \cos \left (d x + c\right )^{3} - 21 \, a^{2} \cos \left (d x + c\right )^{2} - 20 \, a^{2} \cos \left (d x + c\right ) + 15 \, a^{2}\right )}}{\sqrt {\sin \left (d x + c\right )}}\right )} e^{\left (-\frac {5}{2}\right )}}{30 \, d \cos \left (d x + c\right )} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {{\left (a^{2} \sec \left (d x + c\right )^{2} + 2 \, a^{2} \sec \left (d x + c\right ) + a^{2}\right )} \sqrt {e \csc \left (d x + c\right )}}{e^{3} \csc \left (d x + c\right )^{3}}, x\right ) \]