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