f = Integrate[ 1/(1 + x^2 y^2), {x, 0, [Infinity]}, {y, 0, [Infinity]}]

(* Out[1407]= !(
*SubsuperscriptBox[([Integral]), (0), ([Infinity])](
*FractionBox[([Pi]), (2 x)] [DifferentialD]x)) *)

f // N

(* Out[1418]= 366.404 *)

fi = Integrate[ 1/(1 + x^2 y^2), {x, 0, t}, {y, 0, t}, Assumptions -> t > 0]

(* Out[1411]= 1/2 I (PolyLog[2, -I t^2] - PolyLog[2, I t^2]) *)

Series[fi, {t, [Infinity], 2}] // Normal

(* Out[1416]= 1/t^2 - [Pi] Log[1/t] *)

Limit[fi, t -> [Infinity]]

(* Out[1400]= [Infinity] *)