Computing an interesting definite unbound integral

Computing an interesting definite unbound integral

I try to compute (or at least to say if it converge or not) the unbounded
integral in 3 variables (x,y and z over the reels) of the following
function:
(((1/(1+x^2))^2)*
((1/(1+y^2))^2)*((1/(1+z^2))^2))/(1-(((1/(1+x^2))+(1/(1+y^2))+(1/(1+z^2)))^2)/144)
Sorry I don't know how to write it in a more readable form.
I have tried in Wolfram without success.
Maybe somebody has an idea?
Thanks a lot for your help G.OLDANI