Bug in Integrate?

I believe the integral in question converges conditionally and can be calculated after substituting $x = \cosh(t)$, giving (on the integration domain) $$t = \text{arccosh}(x), \;\; dt = \frac{dx}{\sqrt{x^2-1}}.$$

Integrate[Sin[x]/Sqrt[x^2 - 1], {x, 1, ∞}]

1/2 π BesselJ[0, 1]

N[%]

1.20197

We can numerically integrate over a large finite range, forcing each zero to be integrated over. This will ensure we're not skipping over any peaks or valleys.

NIntegrate[Sin[Cosh[t]], {t, ##}] & @@ Prepend[ArcCosh[π Range[10000]], 0]

1.20194

This extended syntax of NIntegrate is discussed here.

Using the undocumented function Integrate`InverseIntegrate[]:

Integrate`InverseIntegrate[Sin[Cosh[t]], {t, 0, ∞}]
   1/2 π BesselJ[0, 1]

N[%, 20]
   1.2019697153172064991

which is consistent with Chip's reformulation:

NIntegrate[Sin[x]/Sqrt[x^2 - 1], {x, 1, ∞}, 
           Method -> "DoubleExponentialOscillatory", WorkingPrecision -> 20]
   1.2019697153164287403