I don't know what version of Mathematica you're using, but...
On V9.0.1, I get the same result as you.
On V10.0.1, I get a similar result, but the conditions are embedded in an If
statement and the z
-integral has not been forgotten; however, the If
statement shows that integrand depends on conditions that should have been resolved, given that the z
domain is real:
Integrate[Exp[2/(x^2+y^2+z^2+1)-2]/(x^2+y^2+z^2+1)^4,{x,-∞,∞},{y,-∞,∞},{z, -∞,∞}]

This is certainly a bug, because the user should not expect to see internal local variables such as Integrate`ImproperDump`y$19189
returned to the top level.
Usually, when a ConditionalExpression
is generated, one can try the option GenerateConditions -> False
. On V9, the z
-integral is still dropped, but it remains (unevaluated) in V10:
int= Integrate[
Exp[2/(x^2 + y^2 + z^2 + 1) - 2]/(x^2 + y^2 + z^2 + 1)^4, {x, -∞, ∞}, {y, -∞, ∞}, {z, -∞, ∞},
GenerateConditions -> False]


It returns unevaluated because Mathematica seems to get stuck on it. I'm not sure why. An obvious substitution to try, to any (brave) calc I student, is u == 1/(1+z^2)
or z -> Sqrt[1/u^2 - 1]
. Here is the computation, first showing another way to address a result that includes ConditionalExpression
, that of explicitly including the obvious assumptions (computation in V10.0.1 -- V9 still loses the z
-integral):
int = Assuming[-Infinity < x < Infinity && -Infinity < y < Infinity && -Infinity < z < Infinity,
Integrate[
Exp[2/(x^2 + y^2 + z^2 + 1) - 2]/(x^2 + y^2 + z^2 + 1)^4, {x, -∞, ∞}, {y, -∞, ∞}, {z, -∞, ∞}]
]

Block[{Integrate},
-2 Integrate[First[int] Dt[z] /. z -> Sqrt[1/u^2 - 1] /. Dt[u] -> 1, {u, 0, 1}]
]

Update: I blocked integration while the integrand is extracted from int
, and updated the substitution, which did work before but was typo from what was intended.
Summary: In short, Mathematica losing the outside integral, which it did consistently in V9 under all three conditions, must be considered a bug. The situation is improved in V10.0.1, although not perfect; it is the same in V10.0.2 (thanks @Simon Woods).