\[ \int \frac {1}{(b \text {sech}(c+d x))^{7/2}} \, dx \]
Optimal antiderivative \[ \frac {2 \sinh \left (d x +c \right )}{7 b d \left (b \,\mathrm {sech}\left (d x +c \right )\right )^{\frac {5}{2}}}+\frac {10 \sinh \left (d x +c \right )}{21 b^{3} d \sqrt {b \,\mathrm {sech}\left (d x +c \right )}}-\frac {10 i \sqrt {\frac {\cosh \left (d x +c \right )}{2}+\frac {1}{2}}\, \EllipticF \left (i \sinh \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \left (\sqrt {\cosh }\left (d x +c \right )\right ) \sqrt {b \,\mathrm {sech}\left (d x +c \right )}}{21 \cosh \left (\frac {d x}{2}+\frac {c}{2}\right ) b^{4} d} \]
command
integrate(1/(b*sech(d*x+c))^(7/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {80 \, \sqrt {2} {\left (\cosh \left (d x + c\right )^{4} + 4 \, \cosh \left (d x + c\right )^{3} \sinh \left (d x + c\right ) + 6 \, \cosh \left (d x + c\right )^{2} \sinh \left (d x + c\right )^{2} + 4 \, \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} + \sinh \left (d x + c\right )^{4}\right )} \sqrt {b} {\rm weierstrassPInverse}\left (-4, 0, \cosh \left (d x + c\right ) + \sinh \left (d x + c\right )\right ) + \sqrt {2} {\left (3 \, \cosh \left (d x + c\right )^{8} + 24 \, \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{7} + 3 \, \sinh \left (d x + c\right )^{8} + 2 \, {\left (42 \, \cosh \left (d x + c\right )^{2} + 13\right )} \sinh \left (d x + c\right )^{6} + 26 \, \cosh \left (d x + c\right )^{6} + 12 \, {\left (14 \, \cosh \left (d x + c\right )^{3} + 13 \, \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{5} + 30 \, {\left (7 \, \cosh \left (d x + c\right )^{4} + 13 \, \cosh \left (d x + c\right )^{2}\right )} \sinh \left (d x + c\right )^{4} + 8 \, {\left (21 \, \cosh \left (d x + c\right )^{5} + 65 \, \cosh \left (d x + c\right )^{3}\right )} \sinh \left (d x + c\right )^{3} + 2 \, {\left (42 \, \cosh \left (d x + c\right )^{6} + 195 \, \cosh \left (d x + c\right )^{4} - 13\right )} \sinh \left (d x + c\right )^{2} - 26 \, \cosh \left (d x + c\right )^{2} + 4 \, {\left (6 \, \cosh \left (d x + c\right )^{7} + 39 \, \cosh \left (d x + c\right )^{5} - 13 \, \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right ) - 3\right )} \sqrt {\frac {b \cosh \left (d x + c\right ) + b \sinh \left (d x + c\right )}{\cosh \left (d x + c\right )^{2} + 2 \, \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + \sinh \left (d x + c\right )^{2} + 1}}}{168 \, {\left (b^{4} d \cosh \left (d x + c\right )^{4} + 4 \, b^{4} d \cosh \left (d x + c\right )^{3} \sinh \left (d x + c\right ) + 6 \, b^{4} d \cosh \left (d x + c\right )^{2} \sinh \left (d x + c\right )^{2} + 4 \, b^{4} d \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} + b^{4} d \sinh \left (d x + c\right )^{4}\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {\sqrt {b \operatorname {sech}\left (d x + c\right )}}{b^{4} \operatorname {sech}\left (d x + c\right )^{4}}, x\right ) \]