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Integrate[((x/s^2)*Exp[-(x^2/(2*s^2))]*Cos[y])/ (Pi*Sqrt[x^2 - Sin[y]^2]), x] ==
(Cos[y]*Sqrt[-1 + 2*x^2 + Cos[2*y]]* (-1 + Erf[Sqrt[(-1 + 2*x^2 + Cos[2*y])/s^2]/2]))/ (E^(Sin[y]^2/(2*s^2))*Sqrt[2*Pi]*s^2* Sqrt[(-1 + 2*x^2 + Cos[2*y])/s^2])



                     2
          x         x
          -- Exp[-(----)] Cos[y]
           2          2
          s        2 s
Integrate[----------------------, x] == 
                   2         2
          Pi Sqrt[x  - Sin[y] ]

                     2
(Cos[y] Sqrt[-1 + 2 x  + Cos[2 y]] 
 
                           2
                   -1 + 2 x  + Cos[2 y]
              Sqrt[--------------------]
                             2
                            s
    (-1 + Erf[--------------------------])) / 
                          2
 
          2     2
    Sin[y] /(2 s )             2
  (E               Sqrt[2 Pi] s  
 
                 2
         -1 + 2 x  + Cos[2 y]
    Sqrt[--------------------])
                   2
                  s

Time to compute: 0.30 second

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