Take a look

here, especially the comments and the links within.

The very first things to try is to turn off symbolic processing using Method -> {Automatic, "SymbolicProcessing" -> 0} For me this speeds up your integral by an order of magnitude. If that's not enough, the next step is to Compile[] the integrand. I think you'll need to rewrite the piecewise as an If[] to make sure compilation gives you optimal performance.

EDIT:

Here's how to compile to integrand for more efficient evaluation:

cf = Compile[{{x, _Real}, {y, _Real}},

Module[{e1, e2},

e2 = R^2 - (b/2 + x)^2;

e1 = e2 (R^2 - (b/2 - x)^2);

(79 2 x^2 E^(y/2)

Sinh[y] If[e2 > 0, Sqrt[e2], 0] If[e1 > 0, e1, 0])/(

1/3 (137 (4 3.14 R) (4 Sinh[4.197/2])) ((1/100 (t + 2) Sinh[y])^2 +

x^2)^(3/2))],

CompilationOptions -> {"InlineExternalDefinitions" -> True}]

fun[x_?NumericQ, y_?NumericQ] := cf[x, y] (* avoid errors about non-numeric arguments *)

This gives another 2-3 times speedup on my machine compared to just turning off symbolic processing. If you reduce the accuracy goal, as you did, it can be sped up even more significantly.