\[\frac{1}{2}\zeta\left(\zeta^{2}+\alpha^{2}\right)^{1/2}+\frac{1}{2}\alpha^{2}%
\operatorname{arcsinh}\left(\frac{\zeta}{\alpha}\right)=\left(1+a^{2}\right)^{%
1/2}\operatorname{arctanh}\left(x\left(\frac{1+a^{2}}{x^{2}+a^{2}}\right)^{1/2%
}\right)-\operatorname{arcsinh}\left(\frac{x}{a}\right),\]