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g[s_, p_] := NIntegrate[ 1/2*r*Log[r^2 + s^2 - 2*r*s*Cos[t - p]], {r, 0, 1/( 2 + Cos[t])}, {t, 0, 2*Pi}] ContourPlot[g[Sqrt[x^2 + y^2], ArcTan[y/x]], {x, -2, 2}, {y, -2, 2}]
