\[\zeta\sqrt{\zeta^{2}+\alpha^{2}}+\alpha^{2}\operatorname{arcsinh}\left(\frac{%
\zeta}{\alpha}\right)=\frac{X}{\mu}-\frac{2\kappa}{\mu}\ln\left(\frac{X+x-2%
\kappa}{2\sqrt{\mu^{2}-\kappa^{2}}}\right)-2\ln\left(\frac{\mu X+2\mu^{2}-%
\kappa x}{x\sqrt{\mu^{2}-\kappa^{2}}}\right).\]