\[ \int \frac {(a+b \cos (c+d x))^3 \left (A+C \cos ^2(c+d x)\right )}{\sqrt {\sec (c+d x)}} \, dx \]
Optimal antiderivative \[ \frac {2 b \left (8 a^{2} C +3 b^{2} \left (11 A +9 C \right )\right ) \sin \left (d x +c \right )}{231 d \sec \left (d x +c \right )^{\frac {5}{2}}}+\frac {2 a \left (99 A \,b^{2}+8 a^{2} C +77 b^{2} C \right ) \sin \left (d x +c \right )}{165 d \sec \left (d x +c \right )^{\frac {3}{2}}}+\frac {4 a C \left (a +b \cos \left (d x +c \right )\right )^{2} \sin \left (d x +c \right )}{33 d \sec \left (d x +c \right )^{\frac {3}{2}}}+\frac {2 C \left (a +b \cos \left (d x +c \right )\right )^{3} \sin \left (d x +c \right )}{11 d \sec \left (d x +c \right )^{\frac {3}{2}}}+\frac {2 b \left (33 a^{2} \left (7 A +5 C \right )+5 b^{2} \left (11 A +9 C \right )\right ) \sin \left (d x +c \right )}{231 d \sqrt {\sec \left (d x +c \right )}}+\frac {2 a \left (a^{2} \left (5 A +3 C \right )+b^{2} \left (9 A +7 C \right )\right ) \sqrt {\frac {\cos \left (d x +c \right )}{2}+\frac {1}{2}}\, \EllipticE \left (\sin \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \left (\sqrt {\cos }\left (d x +c \right )\right ) \left (\sqrt {\sec }\left (d x +c \right )\right )}{5 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) d}+\frac {2 b \left (33 a^{2} \left (7 A +5 C \right )+5 b^{2} \left (11 A +9 C \right )\right ) \sqrt {\frac {\cos \left (d x +c \right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sin \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \left (\sqrt {\cos }\left (d x +c \right )\right ) \left (\sqrt {\sec }\left (d x +c \right )\right )}{231 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) d} \]
command
integrate((a+b*cos(d*x+c))^3*(A+C*cos(d*x+c)^2)/sec(d*x+c)^(1/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ -\frac {5 \, \sqrt {2} {\left (33 i \, {\left (7 \, A + 5 \, C\right )} a^{2} b + 5 i \, {\left (11 \, A + 9 \, C\right )} b^{3}\right )} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) + 5 \, \sqrt {2} {\left (-33 i \, {\left (7 \, A + 5 \, C\right )} a^{2} b - 5 i \, {\left (11 \, A + 9 \, C\right )} b^{3}\right )} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) + 231 \, \sqrt {2} {\left (-i \, {\left (5 \, A + 3 \, C\right )} a^{3} - i \, {\left (9 \, A + 7 \, C\right )} a b^{2}\right )} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) + 231 \, \sqrt {2} {\left (i \, {\left (5 \, A + 3 \, C\right )} a^{3} + i \, {\left (9 \, A + 7 \, C\right )} a b^{2}\right )} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) - \frac {2 \, {\left (105 \, C b^{3} \cos \left (d x + c\right )^{5} + 385 \, C a b^{2} \cos \left (d x + c\right )^{4} + 15 \, {\left (33 \, C a^{2} b + {\left (11 \, A + 9 \, C\right )} b^{3}\right )} \cos \left (d x + c\right )^{3} + 77 \, {\left (3 \, C a^{3} + {\left (9 \, A + 7 \, C\right )} a b^{2}\right )} \cos \left (d x + c\right )^{2} + 5 \, {\left (33 \, {\left (7 \, A + 5 \, C\right )} a^{2} b + 5 \, {\left (11 \, A + 9 \, C\right )} b^{3}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{\sqrt {\cos \left (d x + c\right )}}}{1155 \, d} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {C b^{3} \cos \left (d x + c\right )^{5} + 3 \, C a b^{2} \cos \left (d x + c\right )^{4} + 3 \, A a^{2} b \cos \left (d x + c\right ) + A a^{3} + {\left (3 \, C a^{2} b + A b^{3}\right )} \cos \left (d x + c\right )^{3} + {\left (C a^{3} + 3 \, A a b^{2}\right )} \cos \left (d x + c\right )^{2}}{\sqrt {\sec \left (d x + c\right )}}, x\right ) \]