\[ \int \frac {(a+a \sin (c+d x))^4}{(e \cos (c+d x))^{13/2}} \, dx \]
Optimal antiderivative \[ -\frac {2 a^{4} \sqrt {\frac {\cos \left (d x +c \right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sin \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \left (\sqrt {\cos }\left (d x +c \right )\right )}{77 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) d \,e^{6} \sqrt {e \cos \left (d x +c \right )}}+\frac {4 a^{7} \sqrt {e \cos \left (d x +c \right )}}{11 d \,e^{7} \left (a -a \sin \left (d x +c \right )\right )^{3}}-\frac {2 a^{8} \sqrt {e \cos \left (d x +c \right )}}{77 d \,e^{7} \left (a^{2}-a^{2} \sin \left (d x +c \right )\right )^{2}}-\frac {2 a^{8} \sqrt {e \cos \left (d x +c \right )}}{77 d \,e^{7} \left (a^{4}-a^{4} \sin \left (d x +c \right )\right )} \]
command
integrate((a+a*sin(d*x+c))^4/(e*cos(d*x+c))^(13/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ -\frac {{\left (-3 i \, \sqrt {2} a^{4} \cos \left (d x + c\right )^{2} + 4 i \, \sqrt {2} a^{4} + {\left (i \, \sqrt {2} a^{4} \cos \left (d x + c\right )^{2} - 4 i \, \sqrt {2} a^{4}\right )} \sin \left (d x + c\right )\right )} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) + {\left (3 i \, \sqrt {2} a^{4} \cos \left (d x + c\right )^{2} - 4 i \, \sqrt {2} a^{4} + {\left (-i \, \sqrt {2} a^{4} \cos \left (d x + c\right )^{2} + 4 i \, \sqrt {2} a^{4}\right )} \sin \left (d x + c\right )\right )} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) + 2 \, {\left (a^{4} \cos \left (d x + c\right )^{2} + 3 \, a^{4} \sin \left (d x + c\right ) + 11 \, a^{4}\right )} \sqrt {\cos \left (d x + c\right )}}{77 \, {\left (3 \, d \cos \left (d x + c\right )^{2} e^{\frac {13}{2}} - 4 \, d e^{\frac {13}{2}} - {\left (d \cos \left (d x + c\right )^{2} e^{\frac {13}{2}} - 4 \, d e^{\frac {13}{2}}\right )} \sin \left (d x + c\right )\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {{\left (a^{4} \cos \left (d x + c\right )^{4} - 8 \, a^{4} \cos \left (d x + c\right )^{2} + 8 \, a^{4} - 4 \, {\left (a^{4} \cos \left (d x + c\right )^{2} - 2 \, a^{4}\right )} \sin \left (d x + c\right )\right )} \sqrt {e \cos \left (d x + c\right )}}{e^{7} \cos \left (d x + c\right )^{7}}, x\right ) \]