\[ \int \frac {(a+a \cos (c+d x))^3 (A+B \cos (c+d x))}{\cos ^{\frac {7}{2}}(c+d x)} \, dx \]
Optimal antiderivative \[ -\frac {4 a^{3} \left (9 A +5 B \right ) \sqrt {\frac {\cos \left (d x +c \right )}{2}+\frac {1}{2}}\, \EllipticE \left (\sin \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )}{5 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) d}+\frac {4 a^{3} \left (3 A +5 B \right ) \sqrt {\frac {\cos \left (d x +c \right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sin \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )}{3 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) d}+\frac {2 a A \left (a +a \cos \left (d x +c \right )\right )^{2} \sin \left (d x +c \right )}{5 d \cos \left (d x +c \right )^{\frac {5}{2}}}+\frac {2 \left (9 A +5 B \right ) \left (a^{3}+a^{3} \cos \left (d x +c \right )\right ) \sin \left (d x +c \right )}{15 d \cos \left (d x +c \right )^{\frac {3}{2}}}+\frac {4 a^{3} \left (21 A +20 B \right ) \sin \left (d x +c \right )}{15 d \sqrt {\cos \left (d x +c \right )}} \]
command
integrate((a+a*cos(d*x+c))^3*(A+B*cos(d*x+c))/cos(d*x+c)^(7/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ -\frac {2 \, {\left (5 i \, \sqrt {2} {\left (3 \, A + 5 \, B\right )} a^{3} \cos \left (d x + c\right )^{3} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) - 5 i \, \sqrt {2} {\left (3 \, A + 5 \, B\right )} a^{3} \cos \left (d x + c\right )^{3} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) + 3 i \, \sqrt {2} {\left (9 \, A + 5 \, B\right )} a^{3} \cos \left (d x + c\right )^{3} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) - 3 i \, \sqrt {2} {\left (9 \, A + 5 \, B\right )} a^{3} \cos \left (d x + c\right )^{3} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) - {\left (9 \, {\left (6 \, A + 5 \, B\right )} a^{3} \cos \left (d x + c\right )^{2} + 5 \, {\left (3 \, A + B\right )} a^{3} \cos \left (d x + c\right ) + 3 \, A a^{3}\right )} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right )\right )}}{15 \, d \cos \left (d x + c\right )^{3}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {B a^{3} \cos \left (d x + c\right )^{4} + {\left (A + 3 \, B\right )} a^{3} \cos \left (d x + c\right )^{3} + 3 \, {\left (A + B\right )} a^{3} \cos \left (d x + c\right )^{2} + {\left (3 \, A + B\right )} a^{3} \cos \left (d x + c\right ) + A a^{3}}{\cos \left (d x + c\right )^{\frac {7}{2}}}, x\right ) \]