\[ \int \frac {(a+a \cos (c+d x))^2 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\sec ^{\frac {3}{2}}(c+d x)} \, dx \]
Optimal antiderivative \[ \frac {2 a^{2} \left (99 A +121 B +89 C \right ) \sin \left (d x +c \right )}{693 d \sec \left (d x +c \right )^{\frac {5}{2}}}+\frac {2 C \left (a +a \cos \left (d x +c \right )\right )^{2} \sin \left (d x +c \right )}{11 d \sec \left (d x +c \right )^{\frac {5}{2}}}+\frac {2 \left (11 B +4 C \right ) \left (a^{2}+a^{2} \cos \left (d x +c \right )\right ) \sin \left (d x +c \right )}{99 d \sec \left (d x +c \right )^{\frac {5}{2}}}+\frac {4 a^{2} \left (9 A +8 B +7 C \right ) \sin \left (d x +c \right )}{45 d \sec \left (d x +c \right )^{\frac {3}{2}}}+\frac {4 a^{2} \left (66 A +55 B +50 C \right ) \sin \left (d x +c \right )}{231 d \sqrt {\sec \left (d x +c \right )}}+\frac {4 a^{2} \left (9 A +8 B +7 C \right ) \sqrt {\frac {\cos \left (d x +c \right )}{2}+\frac {1}{2}}\, \EllipticE \left (\sin \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \left (\sqrt {\cos }\left (d x +c \right )\right ) \left (\sqrt {\sec }\left (d x +c \right )\right )}{15 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) d}+\frac {4 a^{2} \left (66 A +55 B +50 C \right ) \sqrt {\frac {\cos \left (d x +c \right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sin \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \left (\sqrt {\cos }\left (d x +c \right )\right ) \left (\sqrt {\sec }\left (d x +c \right )\right )}{231 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) d} \]
command
integrate((a+a*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
\[ -\frac {2 \, {\left (15 i \, \sqrt {2} {\left (66 \, A + 55 \, B + 50 \, C\right )} a^{2} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) - 15 i \, \sqrt {2} {\left (66 \, A + 55 \, B + 50 \, C\right )} a^{2} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) - 231 i \, \sqrt {2} {\left (9 \, A + 8 \, B + 7 \, C\right )} a^{2} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) + 231 i \, \sqrt {2} {\left (9 \, A + 8 \, B + 7 \, C\right )} a^{2} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) - \frac {{\left (315 \, C a^{2} \cos \left (d x + c\right )^{5} + 385 \, {\left (B + 2 \, C\right )} a^{2} \cos \left (d x + c\right )^{4} + 45 \, {\left (11 \, A + 22 \, B + 20 \, C\right )} a^{2} \cos \left (d x + c\right )^{3} + 154 \, {\left (9 \, A + 8 \, B + 7 \, C\right )} a^{2} \cos \left (d x + c\right )^{2} + 30 \, {\left (66 \, A + 55 \, B + 50 \, C\right )} a^{2} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{\sqrt {\cos \left (d x + c\right )}}\right )}}{3465 \, d} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {C a^{2} \cos \left (d x + c\right )^{4} + {\left (B + 2 \, C\right )} a^{2} \cos \left (d x + c\right )^{3} + {\left (A + 2 \, B + C\right )} a^{2} \cos \left (d x + c\right )^{2} + {\left (2 \, A + B\right )} a^{2} \cos \left (d x + c\right ) + A a^{2}}{\sec \left (d x + c\right )^{\frac {3}{2}}}, x\right ) \]