\[ \int \frac {(a+b \cos (c+d x))^3 (A+B \cos (c+d x))}{\sqrt {\cos (c+d x)}} \, dx \]
Optimal antiderivative \[ \frac {2 \left (15 A \,a^{2} b +3 A \,b^{3}+5 a^{3} B +9 B a \,b^{2}\right ) \sqrt {\frac {\cos \left (d x +c \right )}{2}+\frac {1}{2}}\, \EllipticE \left (\sin \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )}{5 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) d}+\frac {2 \left (21 a^{3} A +21 A a \,b^{2}+21 a^{2} b B +5 b^{3} B \right ) \sqrt {\frac {\cos \left (d x +c \right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sin \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )}{21 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) d}+\frac {2 b^{2} \left (7 A b +11 B a \right ) \left (\cos ^{\frac {3}{2}}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{35 d}+\frac {2 b \left (21 A a b +18 B \,a^{2}+5 b^{2} B \right ) \sin \left (d x +c \right ) \left (\sqrt {\cos }\left (d x +c \right )\right )}{21 d}+\frac {2 b B \left (a +b \cos \left (d x +c \right )\right )^{2} \sin \left (d x +c \right ) \left (\sqrt {\cos }\left (d x +c \right )\right )}{7 d} \]
command
integrate((a+b*cos(d*x+c))^3*(A+B*cos(d*x+c))/cos(d*x+c)^(1/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {2 \, {\left (15 \, B b^{3} \cos \left (d x + c\right )^{2} + 105 \, B a^{2} b + 105 \, A a b^{2} + 25 \, B b^{3} + 21 \, {\left (3 \, B a b^{2} + A b^{3}\right )} \cos \left (d x + c\right )\right )} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right ) - 5 \, \sqrt {2} {\left (21 i \, A a^{3} + 21 i \, B a^{2} b + 21 i \, A a b^{2} + 5 i \, B b^{3}\right )} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) - 5 \, \sqrt {2} {\left (-21 i \, A a^{3} - 21 i \, B a^{2} b - 21 i \, A a b^{2} - 5 i \, B b^{3}\right )} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) - 21 \, \sqrt {2} {\left (-5 i \, B a^{3} - 15 i \, A a^{2} b - 9 i \, B a b^{2} - 3 i \, A b^{3}\right )} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) - 21 \, \sqrt {2} {\left (5 i \, B a^{3} + 15 i \, A a^{2} b + 9 i \, B a b^{2} + 3 i \, A b^{3}\right )} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right )}{105 \, d} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {B b^{3} \cos \left (d x + c\right )^{4} + A a^{3} + {\left (3 \, B a b^{2} + A b^{3}\right )} \cos \left (d x + c\right )^{3} + 3 \, {\left (B a^{2} b + A a b^{2}\right )} \cos \left (d x + c\right )^{2} + {\left (B a^{3} + 3 \, A a^{2} b\right )} \cos \left (d x + c\right )}{\sqrt {\cos \left (d x + c\right )}}, x\right ) \]