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