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