\[ \int \frac {(a+b \cos (c+d x))^2 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\sec ^{\frac {3}{2}}(c+d x)} \, dx \]
Optimal antiderivative \[ \frac {2 b \left (11 b B +4 a C \right ) \sin \left (d x +c \right )}{99 d \sec \left (d x +c \right )^{\frac {7}{2}}}+\frac {2 \left (11 A \,b^{2}+22 a b B +4 a^{2} C +9 b^{2} C \right ) \sin \left (d x +c \right )}{77 d \sec \left (d x +c \right )^{\frac {5}{2}}}+\frac {2 C \left (a +b \cos \left (d x +c \right )\right )^{2} \sin \left (d x +c \right )}{11 d \sec \left (d x +c \right )^{\frac {5}{2}}}+\frac {2 \left (18 A a b +9 B \,a^{2}+7 b^{2} B +14 a b C \right ) \sin \left (d x +c \right )}{45 d \sec \left (d x +c \right )^{\frac {3}{2}}}+\frac {2 \left (110 a b B +11 a^{2} \left (7 A +5 C \right )+5 b^{2} \left (11 A +9 C \right )\right ) \sin \left (d x +c \right )}{231 d \sqrt {\sec \left (d x +c \right )}}+\frac {2 \left (18 A a b +9 B \,a^{2}+7 b^{2} B +14 a b C \right ) \sqrt {\frac {\cos \left (d x +c \right )}{2}+\frac {1}{2}}\, \EllipticE \left (\sin \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \left (\sqrt {\cos }\left (d x +c \right )\right ) \left (\sqrt {\sec }\left (d x +c \right )\right )}{15 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) d}+\frac {2 \left (110 a b B +11 a^{2} \left (7 A +5 C \right )+5 b^{2} \left (11 A +9 C \right )\right ) \sqrt {\frac {\cos \left (d x +c \right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sin \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \left (\sqrt {\cos }\left (d x +c \right )\right ) \left (\sqrt {\sec }\left (d x +c \right )\right )}{231 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) d} \]
command
integrate((a+b*cos(d*x+c))^2*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)/sec(d*x+c)^(3/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ -\frac {15 \, \sqrt {2} {\left (11 i \, {\left (7 \, A + 5 \, C\right )} a^{2} + 110 i \, B a b + 5 i \, {\left (11 \, A + 9 \, C\right )} b^{2}\right )} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) + 15 \, \sqrt {2} {\left (-11 i \, {\left (7 \, A + 5 \, C\right )} a^{2} - 110 i \, B a b - 5 i \, {\left (11 \, A + 9 \, C\right )} b^{2}\right )} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) + 231 \, \sqrt {2} {\left (-9 i \, B a^{2} - 2 i \, {\left (9 \, A + 7 \, C\right )} a b - 7 i \, B b^{2}\right )} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) + 231 \, \sqrt {2} {\left (9 i \, B a^{2} + 2 i \, {\left (9 \, A + 7 \, C\right )} a b + 7 i \, B b^{2}\right )} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) - \frac {2 \, {\left (315 \, C b^{2} \cos \left (d x + c\right )^{5} + 385 \, {\left (2 \, C a b + B b^{2}\right )} \cos \left (d x + c\right )^{4} + 45 \, {\left (11 \, C a^{2} + 22 \, B a b + {\left (11 \, A + 9 \, C\right )} b^{2}\right )} \cos \left (d x + c\right )^{3} + 77 \, {\left (9 \, B a^{2} + 2 \, {\left (9 \, A + 7 \, C\right )} a b + 7 \, B b^{2}\right )} \cos \left (d x + c\right )^{2} + 15 \, {\left (11 \, {\left (7 \, A + 5 \, C\right )} a^{2} + 110 \, B a b + 5 \, {\left (11 \, A + 9 \, C\right )} b^{2}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{\sqrt {\cos \left (d x + c\right )}}}{3465 \, d} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {C b^{2} \cos \left (d x + c\right )^{4} + {\left (2 \, C a b + B b^{2}\right )} \cos \left (d x + c\right )^{3} + A a^{2} + {\left (C a^{2} + 2 \, B a b + A b^{2}\right )} \cos \left (d x + c\right )^{2} + {\left (B a^{2} + 2 \, A a b\right )} \cos \left (d x + c\right )}{\sec \left (d x + c\right )^{\frac {3}{2}}}, x\right ) \]