\[ \int \frac {(a+b \cos (c+d x))^3 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\sec ^{\frac {3}{2}}(c+d x)} \, dx \]
Optimal antiderivative \[ \frac {2 b \left (143 A \,b^{2}+195 a b B +24 a^{2} C +121 b^{2} C \right ) \sin \left (d x +c \right )}{1287 d \sec \left (d x +c \right )^{\frac {7}{2}}}+\frac {2 \left (338 a^{2} b B +117 b^{3} B +24 a^{3} C +39 a \,b^{2} \left (11 A +9 C \right )\right ) \sin \left (d x +c \right )}{1001 d \sec \left (d x +c \right )^{\frac {5}{2}}}+\frac {2 \left (13 b B +6 a C \right ) \left (a +b \cos \left (d x +c \right )\right )^{2} \sin \left (d x +c \right )}{143 d \sec \left (d x +c \right )^{\frac {5}{2}}}+\frac {2 C \left (a +b \cos \left (d x +c \right )\right )^{3} \sin \left (d x +c \right )}{13 d \sec \left (d x +c \right )^{\frac {5}{2}}}+\frac {2 \left (117 a^{3} B +273 B a \,b^{2}+39 a^{2} b \left (9 A +7 C \right )+7 b^{3} \left (13 A +11 C \right )\right ) \sin \left (d x +c \right )}{585 d \sec \left (d x +c \right )^{\frac {3}{2}}}+\frac {2 \left (165 a^{2} b B +45 b^{3} B +11 a^{3} \left (7 A +5 C \right )+15 a \,b^{2} \left (11 A +9 C \right )\right ) \sin \left (d x +c \right )}{231 d \sqrt {\sec \left (d x +c \right )}}+\frac {2 \left (117 a^{3} B +273 B a \,b^{2}+39 a^{2} b \left (9 A +7 C \right )+7 b^{3} \left (13 A +11 C \right )\right ) \sqrt {\frac {\cos \left (d x +c \right )}{2}+\frac {1}{2}}\, \EllipticE \left (\sin \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \left (\sqrt {\cos }\left (d x +c \right )\right ) \left (\sqrt {\sec }\left (d x +c \right )\right )}{195 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) d}+\frac {2 \left (165 a^{2} b B +45 b^{3} B +11 a^{3} \left (7 A +5 C \right )+15 a \,b^{2} \left (11 A +9 C \right )\right ) \sqrt {\frac {\cos \left (d x +c \right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sin \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \left (\sqrt {\cos }\left (d x +c \right )\right ) \left (\sqrt {\sec }\left (d x +c \right )\right )}{231 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) d} \]
command
integrate((a+b*cos(d*x+c))^3*(A+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 {195 \, \sqrt {2} {\left (11 i \, {\left (7 \, A + 5 \, C\right )} a^{3} + 165 i \, B a^{2} b + 15 i \, {\left (11 \, A + 9 \, C\right )} a b^{2} + 45 i \, B b^{3}\right )} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) + 195 \, \sqrt {2} {\left (-11 i \, {\left (7 \, A + 5 \, C\right )} a^{3} - 165 i \, B a^{2} b - 15 i \, {\left (11 \, A + 9 \, C\right )} a b^{2} - 45 i \, B 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 (-117 i \, B a^{3} - 39 i \, {\left (9 \, A + 7 \, C\right )} a^{2} b - 273 i \, B a b^{2} - 7 i \, {\left (13 \, A + 11 \, C\right )} 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 ) + 231 \, \sqrt {2} {\left (117 i \, B a^{3} + 39 i \, {\left (9 \, A + 7 \, C\right )} a^{2} b + 273 i \, B a b^{2} + 7 i \, {\left (13 \, A + 11 \, C\right )} 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 ) - \frac {2 \, {\left (3465 \, C b^{3} \cos \left (d x + c\right )^{6} + 4095 \, {\left (3 \, C a b^{2} + B b^{3}\right )} \cos \left (d x + c\right )^{5} + 385 \, {\left (39 \, C a^{2} b + 39 \, B a b^{2} + {\left (13 \, A + 11 \, C\right )} b^{3}\right )} \cos \left (d x + c\right )^{4} + 585 \, {\left (11 \, C a^{3} + 33 \, B a^{2} b + 3 \, {\left (11 \, A + 9 \, C\right )} a b^{2} + 9 \, B b^{3}\right )} \cos \left (d x + c\right )^{3} + 77 \, {\left (117 \, B a^{3} + 39 \, {\left (9 \, A + 7 \, C\right )} a^{2} b + 273 \, B a b^{2} + 7 \, {\left (13 \, A + 11 \, C\right )} b^{3}\right )} \cos \left (d x + c\right )^{2} + 195 \, {\left (11 \, {\left (7 \, A + 5 \, C\right )} a^{3} + 165 \, B a^{2} b + 15 \, {\left (11 \, A + 9 \, C\right )} a b^{2} + 45 \, B b^{3}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{\sqrt {\cos \left (d x + c\right )}}}{45045 \, d} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {C b^{3} \cos \left (d x + c\right )^{5} + {\left (3 \, C a b^{2} + B b^{3}\right )} \cos \left (d x + c\right )^{4} + A a^{3} + {\left (3 \, C a^{2} b + 3 \, B a b^{2} + A b^{3}\right )} \cos \left (d x + c\right )^{3} + {\left (C a^{3} + 3 \, B a^{2} b + 3 \, A a b^{2}\right )} \cos \left (d x + c\right )^{2} + {\left (B a^{3} + 3 \, A a^{2} b\right )} \cos \left (d x + c\right )}{\sec \left (d x + c\right )^{\frac {3}{2}}}, x\right ) \]