\[ \int \frac {\cos ^6(c+d x)}{(b \cos (c+d x))^{3/2}} \, dx \]
Optimal antiderivative \[ \frac {14 \left (b \cos \left (d x +c \right )\right )^{\frac {3}{2}} \sin \left (d x +c \right )}{45 b^{3} d}+\frac {2 \left (b \cos \left (d x +c \right )\right )^{\frac {7}{2}} \sin \left (d x +c \right )}{9 b^{5} d}+\frac {14 \sqrt {\frac {\cos \left (d x +c \right )}{2}+\frac {1}{2}}\, \EllipticE \left (\sin \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \sqrt {b \cos \left (d x +c \right )}}{15 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) b^{2} d \sqrt {\cos \left (d x +c \right )}} \]
command
integrate(cos(d*x+c)^6/(b*cos(d*x+c))^(3/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {2 \, {\left (5 \, \cos \left (d x + c\right )^{3} + 7 \, \cos \left (d x + c\right )\right )} \sqrt {b \cos \left (d x + c\right )} \sin \left (d x + c\right ) + 21 i \, \sqrt {2} \sqrt {b} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) - 21 i \, \sqrt {2} \sqrt {b} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right )}{45 \, b^{2} d} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {\sqrt {b \cos \left (d x + c\right )} \cos \left (d x + c\right )^{4}}{b^{2}}, x\right ) \]