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