\[ \int \frac {A+C \cos ^2(c+d x)}{(b \sec (c+d x))^{3/2}} \, dx \]
Optimal antiderivative \[ \frac {2 \left (7 A +5 C \right ) \sin \left (d x +c \right )}{21 b d \sqrt {b \sec \left (d x +c \right )}}+\frac {2 \left (7 A +5 C \right ) \sqrt {\frac {\cos \left (d x +c \right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sin \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \left (\sqrt {\cos }\left (d x +c \right )\right ) \sqrt {b \sec \left (d x +c \right )}}{21 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) b^{2} d}+\frac {2 b^{2} C \tan \left (d x +c \right )}{7 d \left (b \sec \left (d x +c \right )\right )^{\frac {7}{2}}} \]
command
integrate((A+C*cos(d*x+c)^2)/(b*sec(d*x+c))^(3/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {\sqrt {2} {\left (-7 i \, A - 5 i \, C\right )} \sqrt {b} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) + \sqrt {2} {\left (7 i \, A + 5 i \, C\right )} \sqrt {b} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) + 2 \, {\left (3 \, C \cos \left (d x + c\right )^{3} + {\left (7 \, A + 5 \, C\right )} \cos \left (d x + c\right )\right )} \sqrt {\frac {b}{\cos \left (d x + c\right )}} \sin \left (d x + c\right )}{21 \, b^{2} d} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {{\left (C \cos \left (d x + c\right )^{2} + A\right )} \sqrt {b \sec \left (d x + c\right )}}{b^{2} \sec \left (d x + c\right )^{2}}, x\right ) \]