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ReplaceAll[(Sqrt[x y]/(1 + (x + y)^2)^2) (1/2), {x -> (u + v)/2, y -> (-u + v)/2}]
ReplaceAll[(Sqrt[x y]/(1 + (x + y)^2)^2) (1/2), {x -> (u + v)/2, y -> (-u + v)/2}]
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