Calculation of the double integral, over D, of (1+sin(xy)+y)/(1+x^2+y^2) dxdy

[wlw]

The first one is the region of integration, the second one is the double integral, and when I did it in polar coordinates, I didn't know what to do with sin of xy, and when I did it in rectangular coordinates, I got so complicated that I got stuck with the calculation. Please understand the big man teach me how to do.

 

[Gatineau]

This is unsolvable analytically. Numerically, Wolfram Alpha gave me around 1.08879.
That's because this requires solving the integral sinx/x, which is only possible using non-elementary functions. I attached my method, until I hit that roadblock. Switching the order of integration also doesn't help, since you'd have to solve the integral of sin(sin(x)) which I don't even know of a non-elementary function that does that.