\[ \int \frac {(a+a \cos (c+d x))^3 (A+B \cos (c+d x))}{\cos ^{\frac {11}{2}}(c+d x)} \, dx \]
Optimal antiderivative \[ -\frac {4 a^{3} \left (17 A +21 B \right ) \sqrt {\frac {\cos \left (d x +c \right )}{2}+\frac {1}{2}}\, \EllipticE \left (\sin \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )}{15 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) d}+\frac {4 a^{3} \left (11 A +13 B \right ) \sqrt {\frac {\cos \left (d x +c \right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sin \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )}{21 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) d}+\frac {4 a^{3} \left (23 A +24 B \right ) \sin \left (d x +c \right )}{105 d \cos \left (d x +c \right )^{\frac {5}{2}}}+\frac {4 a^{3} \left (11 A +13 B \right ) \sin \left (d x +c \right )}{21 d \cos \left (d x +c \right )^{\frac {3}{2}}}+\frac {2 a A \left (a +a \cos \left (d x +c \right )\right )^{2} \sin \left (d x +c \right )}{9 d \cos \left (d x +c \right )^{\frac {9}{2}}}+\frac {2 \left (13 A +9 B \right ) \left (a^{3}+a^{3} \cos \left (d x +c \right )\right ) \sin \left (d x +c \right )}{63 d \cos \left (d x +c \right )^{\frac {7}{2}}}+\frac {4 a^{3} \left (17 A +21 B \right ) \sin \left (d x +c \right )}{15 d \sqrt {\cos \left (d x +c \right )}} \]
command
integrate((a+a*cos(d*x+c))^3*(A+B*cos(d*x+c))/cos(d*x+c)^(11/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ -\frac {2 \, {\left (15 i \, \sqrt {2} {\left (11 \, A + 13 \, B\right )} a^{3} \cos \left (d x + c\right )^{5} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) - 15 i \, \sqrt {2} {\left (11 \, A + 13 \, B\right )} a^{3} \cos \left (d x + c\right )^{5} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) + 21 i \, \sqrt {2} {\left (17 \, A + 21 \, B\right )} a^{3} \cos \left (d x + c\right )^{5} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) - 21 i \, \sqrt {2} {\left (17 \, A + 21 \, B\right )} a^{3} \cos \left (d x + c\right )^{5} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) - {\left (42 \, {\left (17 \, A + 21 \, B\right )} a^{3} \cos \left (d x + c\right )^{4} + 30 \, {\left (11 \, A + 13 \, B\right )} a^{3} \cos \left (d x + c\right )^{3} + 7 \, {\left (34 \, A + 27 \, B\right )} a^{3} \cos \left (d x + c\right )^{2} + 45 \, {\left (3 \, A + B\right )} a^{3} \cos \left (d x + c\right ) + 35 \, A a^{3}\right )} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right )\right )}}{315 \, d \cos \left (d x + c\right )^{5}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {B a^{3} \cos \left (d x + c\right )^{4} + {\left (A + 3 \, B\right )} a^{3} \cos \left (d x + c\right )^{3} + 3 \, {\left (A + B\right )} a^{3} \cos \left (d x + c\right )^{2} + {\left (3 \, A + B\right )} a^{3} \cos \left (d x + c\right ) + A a^{3}}{\cos \left (d x + c\right )^{\frac {11}{2}}}, x\right ) \]