You can make your definition of f[t] slightly simpler
f[t_]=Piecewise[{{2 Sqrt[1-t^2]/Pi,-1<t<1},{0,True}}]
which will speed it up slightly, but that doesn't make enough of a difference.
You can look at the output from this
Plot[f[t],{t,-2,2}]
which seems to indicate that your integrand will be zero when t<=-1 or when t>=1.
Can you think for a few minutes to see whether this information might let you integrate over a far smaller range than -infinity to infinity and and get the same result while perhaps speeding up your calculation?
