\[ \int \frac {(a+b \cos (c+d x))^4 \left (A+C \cos ^2(c+d x)\right )}{\sqrt {\sec (c+d x)}} \, dx \]
Optimal antiderivative \[ \frac {4 a b \left (1573 A \,b^{2}+96 a^{2} C +1259 b^{2} C \right ) \sin \left (d x +c \right )}{9009 d \sec \left (d x +c \right )^{\frac {5}{2}}}+\frac {2 \left (192 a^{4} C +77 b^{4} \left (13 A +11 C \right )+11 a^{2} b^{2} \left (637 A +491 C \right )\right ) \sin \left (d x +c \right )}{6435 d \sec \left (d x +c \right )^{\frac {3}{2}}}+\frac {2 \left (48 a^{2} C +11 b^{2} \left (13 A +11 C \right )\right ) \left (a +b \cos \left (d x +c \right )\right )^{2} \sin \left (d x +c \right )}{1287 d \sec \left (d x +c \right )^{\frac {3}{2}}}+\frac {16 a C \left (a +b \cos \left (d x +c \right )\right )^{3} \sin \left (d x +c \right )}{143 d \sec \left (d x +c \right )^{\frac {3}{2}}}+\frac {2 C \left (a +b \cos \left (d x +c \right )\right )^{4} \sin \left (d x +c \right )}{13 d \sec \left (d x +c \right )^{\frac {3}{2}}}+\frac {8 a b \left (11 a^{2} \left (7 A +5 C \right )+5 b^{2} \left (11 A +9 C \right )\right ) \sin \left (d x +c \right )}{231 d \sqrt {\sec \left (d x +c \right )}}+\frac {2 \left (39 a^{4} \left (5 A +3 C \right )+78 a^{2} b^{2} \left (9 A +7 C \right )+7 b^{4} \left (13 A +11 C \right )\right ) \sqrt {\frac {\cos \left (d x +c \right )}{2}+\frac {1}{2}}\, \EllipticE \left (\sin \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \left (\sqrt {\cos }\left (d x +c \right )\right ) \left (\sqrt {\sec }\left (d x +c \right )\right )}{195 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) d}+\frac {8 a b \left (11 a^{2} \left (7 A +5 C \right )+5 b^{2} \left (11 A +9 C \right )\right ) \sqrt {\frac {\cos \left (d x +c \right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sin \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \left (\sqrt {\cos }\left (d x +c \right )\right ) \left (\sqrt {\sec }\left (d x +c \right )\right )}{231 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) d} \]
command
integrate((a+b*cos(d*x+c))^4*(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
\[ -\frac {780 \, \sqrt {2} {\left (11 i \, {\left (7 \, A + 5 \, C\right )} a^{3} b + 5 i \, {\left (11 \, A + 9 \, C\right )} a b^{3}\right )} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) + 780 \, \sqrt {2} {\left (-11 i \, {\left (7 \, A + 5 \, C\right )} a^{3} b - 5 i \, {\left (11 \, A + 9 \, C\right )} a b^{3}\right )} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) + 231 \, \sqrt {2} {\left (-39 i \, {\left (5 \, A + 3 \, C\right )} a^{4} - 78 i \, {\left (9 \, A + 7 \, C\right )} a^{2} b^{2} - 7 i \, {\left (13 \, A + 11 \, C\right )} b^{4}\right )} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) + 231 \, \sqrt {2} {\left (39 i \, {\left (5 \, A + 3 \, C\right )} a^{4} + 78 i \, {\left (9 \, A + 7 \, C\right )} a^{2} b^{2} + 7 i \, {\left (13 \, A + 11 \, C\right )} b^{4}\right )} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) - \frac {2 \, {\left (3465 \, C b^{4} \cos \left (d x + c\right )^{6} + 16380 \, C a b^{3} \cos \left (d x + c\right )^{5} + 385 \, {\left (78 \, C a^{2} b^{2} + {\left (13 \, A + 11 \, C\right )} b^{4}\right )} \cos \left (d x + c\right )^{4} + 2340 \, {\left (11 \, C a^{3} b + {\left (11 \, A + 9 \, C\right )} a b^{3}\right )} \cos \left (d x + c\right )^{3} + 77 \, {\left (117 \, C a^{4} + 78 \, {\left (9 \, A + 7 \, C\right )} a^{2} b^{2} + 7 \, {\left (13 \, A + 11 \, C\right )} b^{4}\right )} \cos \left (d x + c\right )^{2} + 780 \, {\left (11 \, {\left (7 \, A + 5 \, C\right )} a^{3} b + 5 \, {\left (11 \, A + 9 \, C\right )} a b^{3}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{\sqrt {\cos \left (d x + c\right )}}}{45045 \, d} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {C b^{4} \cos \left (d x + c\right )^{6} + 4 \, C a b^{3} \cos \left (d x + c\right )^{5} + 4 \, A a^{3} b \cos \left (d x + c\right ) + A a^{4} + {\left (6 \, C a^{2} b^{2} + A b^{4}\right )} \cos \left (d x + c\right )^{4} + 4 \, {\left (C a^{3} b + A a b^{3}\right )} \cos \left (d x + c\right )^{3} + {\left (C a^{4} + 6 \, A a^{2} b^{2}\right )} \cos \left (d x + c\right )^{2}}{\sqrt {\sec \left (d x + c\right )}}, x\right ) \]