\[ \int \frac {(a+b \cos (c+d x))^3 (A+B \cos (c+d x))}{\cos ^{\frac {7}{2}}(c+d x)} \, dx \]
Optimal antiderivative \[ -\frac {2 \left (3 a^{3} A +15 A a \,b^{2}+15 a^{2} b B -5 b^{3} B \right ) \sqrt {\frac {\cos \left (d x +c \right )}{2}+\frac {1}{2}}\, \EllipticE \left (\sin \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )}{5 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) d}+\frac {2 \left (3 A \,a^{2} b +3 A \,b^{3}+a^{3} B +9 B a \,b^{2}\right ) \sqrt {\frac {\cos \left (d x +c \right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sin \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )}{3 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) d}+\frac {2 a^{2} \left (9 A b +5 B a \right ) \sin \left (d x +c \right )}{15 d \cos \left (d x +c \right )^{\frac {3}{2}}}+\frac {2 a A \left (a +b \cos \left (d x +c \right )\right )^{2} \sin \left (d x +c \right )}{5 d \cos \left (d x +c \right )^{\frac {5}{2}}}+\frac {2 a \left (3 a^{2} A +14 A \,b^{2}+15 a b B \right ) \sin \left (d x +c \right )}{5 d \sqrt {\cos \left (d x +c \right )}} \]
command
integrate((a+b*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
\[ -\frac {5 \, \sqrt {2} {\left (i \, B a^{3} + 3 i \, A a^{2} b + 9 i \, B a b^{2} + 3 i \, A b^{3}\right )} \cos \left (d x + c\right )^{3} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) + 5 \, \sqrt {2} {\left (-i \, B a^{3} - 3 i \, A a^{2} b - 9 i \, B a b^{2} - 3 i \, A b^{3}\right )} \cos \left (d x + c\right )^{3} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) + 3 \, \sqrt {2} {\left (3 i \, A a^{3} + 15 i \, B a^{2} b + 15 i \, A a b^{2} - 5 i \, B b^{3}\right )} \cos \left (d x + c\right )^{3} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) + 3 \, \sqrt {2} {\left (-3 i \, A a^{3} - 15 i \, B a^{2} b - 15 i \, A a b^{2} + 5 i \, B b^{3}\right )} \cos \left (d x + c\right )^{3} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) - 2 \, {\left (3 \, A a^{3} + 9 \, {\left (A a^{3} + 5 \, B a^{2} b + 5 \, A a b^{2}\right )} \cos \left (d x + c\right )^{2} + 5 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} \cos \left (d x + c\right )\right )} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right )}{15 \, d \cos \left (d x + c\right )^{3}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {B b^{3} \cos \left (d x + c\right )^{4} + A a^{3} + {\left (3 \, B a b^{2} + A b^{3}\right )} \cos \left (d x + c\right )^{3} + 3 \, {\left (B a^{2} b + A a b^{2}\right )} \cos \left (d x + c\right )^{2} + {\left (B a^{3} + 3 \, A a^{2} b\right )} \cos \left (d x + c\right )}{\cos \left (d x + c\right )^{\frac {7}{2}}}, x\right ) \]