\[ \int (b \text {sech}(c+d x))^{7/2} \, dx \]
Optimal antiderivative \[ \frac {2 b \left (b \,\mathrm {sech}\left (d x +c \right )\right )^{\frac {5}{2}} \sinh \left (d x +c \right )}{5 d}+\frac {6 i b^{4} \sqrt {\frac {\cosh \left (d x +c \right )}{2}+\frac {1}{2}}\, \EllipticE \left (i \sinh \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )}{5 \cosh \left (\frac {d x}{2}+\frac {c}{2}\right ) d \sqrt {\cosh \left (d x +c \right )}\, \sqrt {b \,\mathrm {sech}\left (d x +c \right )}}+\frac {6 b^{3} \sinh \left (d x +c \right ) \sqrt {b \,\mathrm {sech}\left (d x +c \right )}}{5 d} \]
command
integrate((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 {2 \, {\left (3 \, \sqrt {2} {\left (b^{3} \cosh \left (d x + c\right )^{4} + 4 \, b^{3} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} + b^{3} \sinh \left (d x + c\right )^{4} + 2 \, b^{3} \cosh \left (d x + c\right )^{2} + b^{3} + 2 \, {\left (3 \, b^{3} \cosh \left (d x + c\right )^{2} + b^{3}\right )} \sinh \left (d x + c\right )^{2} + 4 \, {\left (b^{3} \cosh \left (d x + c\right )^{3} + b^{3} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )\right )} \sqrt {b} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cosh \left (d x + c\right ) + \sinh \left (d x + c\right )\right )\right ) + \sqrt {2} {\left (3 \, b^{3} \cosh \left (d x + c\right )^{5} + 15 \, b^{3} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{4} + 3 \, b^{3} \sinh \left (d x + c\right )^{5} + 8 \, b^{3} \cosh \left (d x + c\right )^{3} + b^{3} \cosh \left (d x + c\right ) + 2 \, {\left (15 \, b^{3} \cosh \left (d x + c\right )^{2} + 4 \, b^{3}\right )} \sinh \left (d x + c\right )^{3} + 6 \, {\left (5 \, b^{3} \cosh \left (d x + c\right )^{3} + 4 \, b^{3} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{2} + {\left (15 \, b^{3} \cosh \left (d x + c\right )^{4} + 24 \, b^{3} \cosh \left (d x + c\right )^{2} + b^{3}\right )} \sinh \left (d x + c\right )\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}}\right )}}{5 \, {\left (d \cosh \left (d x + c\right )^{4} + 4 \, d \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} + d \sinh \left (d x + c\right )^{4} + 2 \, d \cosh \left (d x + c\right )^{2} + 2 \, {\left (3 \, d \cosh \left (d x + c\right )^{2} + d\right )} \sinh \left (d x + c\right )^{2} + 4 \, {\left (d \cosh \left (d x + c\right )^{3} + d \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right ) + d\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\sqrt {b \operatorname {sech}\left (d x + c\right )} b^{3} \operatorname {sech}\left (d x + c\right )^{3}, x\right ) \]