Double integral of e^(x^2)+e^(y^2)

Hey all, I'm having trouble solving the double integral [0,1] to [0,1] from e^(x^2)+e^(y^2). Splitting them up does no good, and it seems whether you switch the bounds, and thus dxdy to dydx, the function is still not integrable in an elementary fashion.

Any help would be greatly appreciated.

There is an excellent candidate for numerical methods if you insist on a solution. It's all so clean and well-behaved.

Note: Did you notice the symmetry across x = y? This helps it look a little cleaner.