The difference between F and G is of the order of .2 or less. It is too small to perceive it in the two plots you have made. Try plotting the difference:

With[{n = E, y = 0}, 
 Plot[NIntegrate[
    Sqrt[(x + Cos[\[CurlyPhi]])^2 + (y + 
         Sin[\[CurlyPhi]])^2], {\[CurlyPhi], 0, 2 \[Pi]}] - 
   Power[(2 \[Pi])^E + (2 \[Pi] Sqrt[x^2 + y^2])^E, (E)^-1], {x, 0, 
   2 Pi}]]

I tested your formula for F numerically, and it doesn't seem true:

With[{x = 1, 
   y = 2}, {Integrate[
    Sqrt[(x + Cos[\[CurlyPhi]])^2 + (y + 
         Sin[\[CurlyPhi]])^2], {\[CurlyPhi], 0, 2 \[Pi]}],
   Power[(2 \[Pi])^E + (2 \[Pi] Sqrt[x^2 + y^2])^E, (E)^-1]}] // N
With[{y = 2}, 
 Plot[NIntegrate[
    Sqrt[(x + Cos[\[CurlyPhi]])^2 + (y + 
         Sin[\[CurlyPhi]])^2], {\[CurlyPhi], 0, 2 \[Pi]}] -
   Power[(2 \[Pi])^E + (2 \[Pi] Sqrt[x^2 + y^2])^E, (E)^-1], {x, -20, 
   20}]]