\[ \int \frac {(a+a \sin (c+d x))^3}{(e \cos (c+d x))^{11/2}} \, dx \]
Optimal antiderivative \[ \frac {2 a^{6} \left (e \cos \left (d x +c \right )\right )^{\frac {3}{2}}}{9 d \,e^{7} \left (a -a \sin \left (d x +c \right )\right )^{3}}+\frac {2 a^{5} \left (e \cos \left (d x +c \right )\right )^{\frac {3}{2}}}{15 d \,e^{7} \left (a -a \sin \left (d x +c \right )\right )^{2}}+\frac {2 a^{6} \left (e \cos \left (d x +c \right )\right )^{\frac {3}{2}}}{15 d \,e^{7} \left (a^{3}-a^{3} \sin \left (d x +c \right )\right )}-\frac {2 a^{3} \sqrt {\frac {\cos \left (d x +c \right )}{2}+\frac {1}{2}}\, \EllipticE \left (\sin \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \sqrt {e \cos \left (d x +c \right )}}{15 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) d \,e^{6} \sqrt {\cos \left (d x +c \right )}} \]
command
integrate((a+a*sin(d*x+c))^3/(e*cos(d*x+c))^(11/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {3 \, {\left (-i \, \sqrt {2} a^{3} \cos \left (d x + c\right )^{3} - 3 i \, \sqrt {2} a^{3} \cos \left (d x + c\right )^{2} + 2 i \, \sqrt {2} a^{3} \cos \left (d x + c\right ) + 4 i \, \sqrt {2} a^{3} + {\left (i \, \sqrt {2} a^{3} \cos \left (d x + c\right )^{2} - 2 i \, \sqrt {2} a^{3} \cos \left (d x + c\right ) - 4 i \, \sqrt {2} a^{3}\right )} \sin \left (d x + c\right )\right )} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) + 3 \, {\left (i \, \sqrt {2} a^{3} \cos \left (d x + c\right )^{3} + 3 i \, \sqrt {2} a^{3} \cos \left (d x + c\right )^{2} - 2 i \, \sqrt {2} a^{3} \cos \left (d x + c\right ) - 4 i \, \sqrt {2} a^{3} + {\left (-i \, \sqrt {2} a^{3} \cos \left (d x + c\right )^{2} + 2 i \, \sqrt {2} a^{3} \cos \left (d x + c\right ) + 4 i \, \sqrt {2} a^{3}\right )} \sin \left (d x + c\right )\right )} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) + 2 \, {\left (3 \, a^{3} \cos \left (d x + c\right )^{3} - 6 \, a^{3} \cos \left (d x + c\right )^{2} - 14 \, a^{3} \cos \left (d x + c\right ) - 5 \, a^{3} + {\left (3 \, a^{3} \cos \left (d x + c\right )^{2} + 9 \, a^{3} \cos \left (d x + c\right ) - 5 \, a^{3}\right )} \sin \left (d x + c\right )\right )} \sqrt {\cos \left (d x + c\right )}}{45 \, {\left (d \cos \left (d x + c\right )^{3} e^{\frac {11}{2}} + 3 \, d \cos \left (d x + c\right )^{2} e^{\frac {11}{2}} - 2 \, d \cos \left (d x + c\right ) e^{\frac {11}{2}} - 4 \, d e^{\frac {11}{2}} - {\left (d \cos \left (d x + c\right )^{2} e^{\frac {11}{2}} - 2 \, d \cos \left (d x + c\right ) e^{\frac {11}{2}} - 4 \, d e^{\frac {11}{2}}\right )} \sin \left (d x + c\right )\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (-\frac {{\left (3 \, a^{3} \cos \left (d x + c\right )^{2} - 4 \, a^{3} + {\left (a^{3} \cos \left (d x + c\right )^{2} - 4 \, a^{3}\right )} \sin \left (d x + c\right )\right )} \sqrt {e \cos \left (d x + c\right )}}{e^{6} \cos \left (d x + c\right )^{6}}, x\right ) \]