How can the same Mathematica version give different symbolic results on different machines?

Question

I'm testing the following code:

$Version
fRe[x_, σ_] = 1/(Sqrt[2 π] σ) Exp[-(x^2/(2σ^2))];
Integrate[fRe[x, σ]/(x - X), {x, -∞, ∞}, 
  PrincipalValue -> True, Assumptions -> σ > 0 && X < 0]

On one machine (Core i7-4765T) I get this as output:

"9.0 for Linux x86 (32-bit) (November 20, 2012)"
-((Sqrt[2] DawsonF[X/(Sqrt[2] σ)])/σ)

On another machine (EEE PC 1015PN with Intel Atom N570) the result is different:

"9.0 for Linux x86 (32-bit) (November 20, 2012)"
E^(X^2/(2*σ^2))/(2*Sqrt[2*Pi]*σ)
  (-EulerGamma + CoshIntegral[X^2/(2*σ^2)] - 2*Pi*Erfi[X/(Sqrt[2]*σ)] + Log[2] - 
     2*Log[-X] + 2*Log[σ] + SinhIntegral[X^2/(2*σ^2)] + 
       E^(X^2/(2*σ^2))*Derivative[1, 0, 0][Hypergeometric1F1][1, 1, -(X^2/(2*σ^2))])

On both machines Help->About Mathematica gives version number 9.0.0.0.

How is this possible? I'd expect something similar for numerical calculations due to CPU-specific optimized code paths which might lead to different precision for different CPUs, but this is purely symbolic integration, isn't it?

Answer 1

THIS IS AN EXTENDED COMMENT RATHER THAN AN ANSWER.

$Version

(*  "10.4.1 for Mac OS X x86 (64-bit) (April 11, 2016)"  *)

fRe[x_, σ_] = 1/(Sqrt[2 π] σ) Exp[-(x^2/(2 σ^2))];

Integrate[fRe[x, σ]/(x - X), {x, -∞, ∞}, 
 PrincipalValue -> True, Assumptions -> σ > 0]

(*  ConditionalExpression[
   -((Sqrt[2]*DawsonF[X/(Sqrt[2]*σ)])/σ), 
   Re[X] > 0 && Im[X] == 0]  *)

The condition requires that X be real, positive

Integrate[fRe[x, σ]/(x - X), {x, -∞, ∞}, 
 PrincipalValue -> True, Assumptions -> σ > 0 && Element[X, Reals]]

(*  ConditionalExpression[
   -((Sqrt[2]*DawsonF[X/(Sqrt[2]*σ)])/σ), X > 0]  *)

The condition requires that X be positive

Integrate[fRe[x, σ]/(x - X), {x, -∞, ∞}, 
 PrincipalValue -> True, Assumptions -> σ > 0 && X > 0]

(*  -((Sqrt[2]*DawsonF[X/(Sqrt[2]*σ)])/σ)  *)

Satisfying the conditions returned the expected result. However, explicitly violating the condition (as in the OP's question) returns the same result

Integrate[fRe[x, σ]/(x - X), {x, -∞, ∞}, 
 PrincipalValue -> True, Assumptions -> σ > 0 && X < 0]

(*  -((Sqrt[2]*DawsonF[X/(Sqrt[2]*σ)])/σ)  *)

% == %%

(*  True  *)

It appears that either the conditions are wrong or the integral for X < 0 is wrong.

Note: I get identical results with version 10.1 and 9.0