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