\[ \int \frac {\cos ^5(c+d x)}{(b \cos (c+d x))^{3/2}} \, dx \]
Optimal antiderivative \[ \frac {2 \left (b \cos \left (d x +c \right )\right )^{\frac {5}{2}} \sin \left (d x +c \right )}{7 b^{4} d}+\frac {10 \sqrt {\frac {\cos \left (d x +c \right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sin \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \left (\sqrt {\cos }\left (d x +c \right )\right )}{21 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) b d \sqrt {b \cos \left (d x +c \right )}}+\frac {10 \sin \left (d x +c \right ) \sqrt {b \cos \left (d x +c \right )}}{21 b^{2} d} \]
command
integrate(cos(d*x+c)^5/(b*cos(d*x+c))^(3/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {2 \, \sqrt {b \cos \left (d x + c\right )} {\left (3 \, \cos \left (d x + c\right )^{2} + 5\right )} \sin \left (d x + c\right ) - 5 i \, \sqrt {2} \sqrt {b} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) + 5 i \, \sqrt {2} \sqrt {b} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )}{21 \, b^{2} d} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {\sqrt {b \cos \left (d x + c\right )} \cos \left (d x + c\right )^{3}}{b^{2}}, x\right ) \]