numerical integration - How to improve accuracy of NIntegrate over ImplicitRegion

I'm trying to compute the area of a implicit region given as

R = ImplicitRegion[
       0 < Sinh[u]/Cosh[v] < 1 && 0 < Sinh[v]/Cosh[u] < 1,
       {{u, 0, Infinity}, {v, 0, Infinity}}
    ]

The region looks like this:

RegionPlot[

  0 < Sinh[u]/Cosh[v] < 1 && 0 < Sinh[v]/Cosh[u] < 1, 
  {u, 0, 2}, {v, 0, 2}
]

I used NIntegrate (Mathematica 10.4) like this:

NIntegrate[1, Element[{u, v}, R]]
(* Out: 0.884886 *)

However I happen to know that the correct answer should be $\pi^2/8 = 1.2337$.

I'm reading the documentation for NIntegrate, but I'm quite overwhelmed by the amount of possible options. How can I improve the result of NIntegrate in this case?

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