How to solve the following equation? $\left(\sqrt{u^2-1}+u\right)^{1/u}=\pi ^{1/\pi }$

Question

I have no clue:

$$\left(\sqrt{u^2-1}+u\right)^{1/u}=\pi ^{1/\pi }$$

Answer 1

Letting $u=\cosh t$ we are left with solving the transcendental equation $~\dfrac t{\cosh t}~=~\dfrac{\ln\pi}\pi$ ,

whose two solutions, unfortunately, cannot even be expressed in terms of the special Lambert

W function. The only way forward is with the help of numerical methods, yielding

$$t_1~=~0.3928615102385430405833389297736055465796116466357\ldots \\ t_2~=~2.6862059675777010426011674393973417566869073839410\ldots$$

corresponding to

$$u_1~=~1.0781677404428052383937449983538496606471995047941\ldots \\ u_2~=~7.3720141651664644041169505229630376609985273104503\ldots$$