\[ \int \frac {a+a \sin (c+d x)}{(e \cos (c+d x))^{7/2}} \, dx \]
Optimal antiderivative \[ \frac {2 a}{5 d e \left (e \cos \left (d x +c \right )\right )^{\frac {5}{2}}}+\frac {2 a \sin \left (d x +c \right )}{5 d e \left (e \cos \left (d x +c \right )\right )^{\frac {5}{2}}}+\frac {6 a \sin \left (d x +c \right )}{5 d \,e^{3} \sqrt {e \cos \left (d x +c \right )}}-\frac {6 a \sqrt {\frac {\cos \left (d x +c \right )}{2}+\frac {1}{2}}\, \EllipticE \left (\sin \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \sqrt {e \cos \left (d x +c \right )}}{5 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) d \,e^{4} \sqrt {\cos \left (d x +c \right )}} \]
command
integrate((a+a*sin(d*x+c))/(e*cos(d*x+c))^(7/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ -\frac {3 \, {\left (i \, \sqrt {2} a \cos \left (d x + c\right ) \sin \left (d x + c\right ) - i \, \sqrt {2} a \cos \left (d x + c\right )\right )} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) + 3 \, {\left (-i \, \sqrt {2} a \cos \left (d x + c\right ) \sin \left (d x + c\right ) + i \, \sqrt {2} a \cos \left (d x + c\right )\right )} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) + 2 \, {\left (3 \, a \cos \left (d x + c\right )^{2} + 3 \, a \sin \left (d x + c\right ) - 2 \, a\right )} \sqrt {\cos \left (d x + c\right )}}{5 \, {\left (d \cos \left (d x + c\right ) e^{\frac {7}{2}} \sin \left (d x + c\right ) - d \cos \left (d x + c\right ) e^{\frac {7}{2}}\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {\sqrt {e \cos \left (d x + c\right )} {\left (a \sin \left (d x + c\right ) + a\right )}}{e^{4} \cos \left (d x + c\right )^{4}}, x\right ) \]