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  restart; Digits:=14;     # using Maple 12.02
  J:=Int(arctan(z)^2/(1+4*z^2),z=0..infinity);


  evalf(J);
                           0.40325004753375

  value(J); evalf(%); Re(%);

                           0.87721779610207


This seems not to depend on a possibly to low value
for the Digits used, more likely the symbolic result
by value(J) is not correct (and it would not change
if assuming 0 < z because of symmetry in z=0).
infinity

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