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