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I am new to Mathematica. So I just run a basic procedure but Mathematica didn't calculate this integral. Maybe it is too complicated to be done.

Integrate[(a E^(I g x - 1/2 (d + a ArcSinh[(-c + x)/b])^2))/(b Sqrt[ 2 π] 
          Sqrt[1 + (-c + x)^2/b^2]), {x, -∞, ∞}]

$a,b,c,d$ and $g$ are constants. Could you suggest what I should do?

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One way to find where the problem is, is to keep throwing away terms until you find where it is stuck. one problem is this Integrate[E^x E^ArcSinh[-c + x], x] can't do it in closed form. But Integrate[E^ArcSinh[-c + x], x] can. So that is where to start working at. Then start adding back stuff that was thrown out, until it gets stuck again. –  Nasser Jan 30 '14 at 7:53

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