Strange result for divergent double integral $\int _0^{\infty }\int _0^{\infty }\frac{1}{x^2 y^2+1}dydx$

Question

Mathematica 10.1.0 returns a strange result for the following double integral

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

(* Out[1407]= \!\(
\*SubsuperscriptBox[\(∫\), \(0\), \(∞\)]\(
\*FractionBox[\(π\), \(2\ x\)] \[DifferentialD]x\)\) *)

in Latex:

$$\int_0^{\infty } \frac{\pi }{2 x} \, dx$$

There is no warning that the result might be infinite or not existent. Applying Simplify[] or FullSimplify[] to $f$ does not help. Numerically, however, the divergence is revealed

f // N

During evaluation of In[1418]:= NIntegrate::slwcon: Numerical integration converging too slowly; suspect one of the following: singularity, value of the integration is 0, highly oscillatory integrand, or WorkingPrecision too small. >>

and further error Messages which I have dropped here.

(* Out[1418]= 366.404 *)

I suggest to consider this behaviour a bug.

Taking finite integration regions reveals the logarithmic divergnce of the double integral:

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, ∞, 2}] // Normal

(* Out[1416]= 1/t^2 - π Log[1/t] *)

Limit[fi, t -> ∞]

(* Out[1400]= ∞ *)

Answer 1

One may use the definition of the double improper integral (see Multivariable improper integrals in https://en.wikipedia.org/wiki/Improper_integral ). Let us calculate

r = Integrate[1/(1 + x^2*y^2), {x, 0, a}, {y, 0, b},Assumptions -> a > 0 && b > 0]

1/2 I (PolyLog[2, -I a b] - PolyLog[2, I a b]

If the improper integral under consideration exists, the one equals $$\lim\limits_{a\to \infty,\,b\to \infty} r $$ and this limit is finite and is equal to the iterated limit $$\lim_{b\to\infty}(\lim_{a\to\infty} r).$$ But

Limit[ r, a -> Infinity, Assumptions -> b > 0] 

Infinity

Answer 2

In:

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

Out:

Infinity