Can't integrate exp(-sqrt(x ** 2 + y ** 2)) over the plane

[nschloe]

Sympy cannot integrate the function

import sympy
from sympy import oo, pi

x, y = sympy.symbols("x, y")
print(sympy.integrate(
    sympy.exp(-sympy.sqrt(x ** 2 + y ** 2)),
    (x, -oo, +oo),
    (y, -oo, +oo)
))

What does work:

• The function in 1D

  print(sympy.integrate(
      sympy.exp(-sympy.sqrt(x ** 2)),
      (x, -oo, +oo),
  ))

  2
  

• the squared exponent

  print(sympy.integrate(
      sympy.exp(-(x ** 2 + y ** 2)),
      (x, -oo, +oo),
      (y, -oo, +oo),
  ))

  pi
  

• Polar coordinates

  r, phi = sympy.symbols("r, phi")
  print(sympy.integrate(
      r * sympy.exp(-r),
      (r, 0, +oo),
      (phi, 0, 2*pi)
  ))

  2 * pi