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Evaluating complicated integral numerically


I am trying to evaluate a complicated integral numerically, but when my limits are > +/-5, then Mathematica fails:


sigma = 1;
gaussian[x_, y_, z_] := 
  1/(Sqrt[2 \[Pi]] \[Sigma])^3 Exp[-((x^2 + y^2 + z^2)/(2 sigma^2))];

gamma = 10^4;
lorentzian[f_, f0_] := gamma/(2 \[Pi]) 1/((f - f0)^2 + gamma^2/4);

f = 1000;
 lorentzian[ f, Sqrt[x^2 + y^2 + 4 z^2]]*gaussian[x, y, z] Sqrt[
  x^2 + y^2]/Sqrt[x^2 + y^2 + 4 z^2],
 {x, -500, 500},
 {y, -500, 500},
 {z, -500, 500}]

I don't see what goes wrong here? Is it really true that NIntegrate can't handle this?!

Thanks in advance.


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