\[ \int (b \cos (c+d x))^{5/2} \sec ^8(c+d x) \, dx \]
Optimal antiderivative \[ \frac {2 b^{7} \sin \left (d x +c \right )}{9 d \left (b \cos \left (d x +c \right )\right )^{\frac {9}{2}}}+\frac {14 b^{5} \sin \left (d x +c \right )}{45 d \left (b \cos \left (d x +c \right )\right )^{\frac {5}{2}}}+\frac {14 b^{3} \sin \left (d x +c \right )}{15 d \sqrt {b \cos \left (d x +c \right )}}-\frac {14 b^{2} \sqrt {\frac {\cos \left (d x +c \right )}{2}+\frac {1}{2}}\, \EllipticE \left (\sin \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \sqrt {b \cos \left (d x +c \right )}}{15 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) d \sqrt {\cos \left (d x +c \right )}} \]
command
integrate((b*cos(d*x+c))^(5/2)*sec(d*x+c)^8,x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {-21 i \, \sqrt {2} b^{\frac {5}{2}} \cos \left (d x + c\right )^{5} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) + 21 i \, \sqrt {2} b^{\frac {5}{2}} \cos \left (d x + c\right )^{5} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) + 2 \, {\left (21 \, b^{2} \cos \left (d x + c\right )^{4} + 7 \, b^{2} \cos \left (d x + c\right )^{2} + 5 \, b^{2}\right )} \sqrt {b \cos \left (d x + c\right )} \sin \left (d x + c\right )}{45 \, d \cos \left (d x + c\right )^{5}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\sqrt {b \cos \left (d x + c\right )} b^{2} \cos \left (d x + c\right )^{2} \sec \left (d x + c\right )^{8}, x\right ) \]