93.13 Problem number 21

\[ \int \frac {1}{(b \text {sech}(c+d x))^{5/2}} \, dx \]

Optimal antiderivative \[ \frac {2 \sinh \left (d x +c \right )}{5 b d \left (b \,\mathrm {sech}\left (d x +c \right )\right )^{\frac {3}{2}}}-\frac {6 i \sqrt {\frac {\cosh \left (d x +c \right )}{2}+\frac {1}{2}}\, \EllipticE \left (i \sinh \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )}{5 \cosh \left (\frac {d x}{2}+\frac {c}{2}\right ) b^{2} d \sqrt {\cosh \left (d x +c \right )}\, \sqrt {b \,\mathrm {sech}\left (d x +c \right )}} \]

command

integrate(1/(b*sech(d*x+c))^(5/2),x, algorithm="fricas")

Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output

\[ -\frac {24 \, \sqrt {2} {\left (\cosh \left (d x + c\right )^{3} + 3 \, \cosh \left (d x + c\right )^{2} \sinh \left (d x + c\right ) + 3 \, \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{2} + \sinh \left (d x + c\right )^{3}\right )} \sqrt {b} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cosh \left (d x + c\right ) + \sinh \left (d x + c\right )\right )\right ) - \sqrt {2} {\left (\cosh \left (d x + c\right )^{6} + 6 \, \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{5} + \sinh \left (d x + c\right )^{6} + {\left (15 \, \cosh \left (d x + c\right )^{2} - 11\right )} \sinh \left (d x + c\right )^{4} - 11 \, \cosh \left (d x + c\right )^{4} + 4 \, {\left (5 \, \cosh \left (d x + c\right )^{3} - 11 \, \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{3} + {\left (15 \, \cosh \left (d x + c\right )^{4} - 66 \, \cosh \left (d x + c\right )^{2} - 13\right )} \sinh \left (d x + c\right )^{2} - 13 \, \cosh \left (d x + c\right )^{2} + 2 \, {\left (3 \, \cosh \left (d x + c\right )^{5} - 22 \, \cosh \left (d x + c\right )^{3} - 13 \, \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right ) - 1\right )} \sqrt {\frac {b \cosh \left (d x + c\right ) + b \sinh \left (d x + c\right )}{\cosh \left (d x + c\right )^{2} + 2 \, \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + \sinh \left (d x + c\right )^{2} + 1}}}{20 \, {\left (b^{3} d \cosh \left (d x + c\right )^{3} + 3 \, b^{3} d \cosh \left (d x + c\right )^{2} \sinh \left (d x + c\right ) + 3 \, b^{3} d \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{2} + b^{3} d \sinh \left (d x + c\right )^{3}\right )}} \]

Fricas 1.3.7 via sagemath 9.3 output

\[ {\rm integral}\left (\frac {\sqrt {b \operatorname {sech}\left (d x + c\right )}}{b^{3} \operatorname {sech}\left (d x + c\right )^{3}}, x\right ) \]