Excluding singular curves in NIntegrate

Question

I would like to calculate

NIntegrate[1/(x2 x3 + (1 - x2 - x3) (x2 + x3)), {x2, 0, 1}, {x3, 0, 1}, Exclusions -> {x2 == 1/2 (1 - x3 + Sqrt[1 + 2 x3 - 3 x3^2])}]

I have tried excluding the singular curve. Any ideas about how to calculate this? Most answers here suggest Exclusions, but alas, it doesn't work. I also tried

NIntegrate[1/(x2 x3 + (1 - x2 - x3) (x2 + x3)), {x2, 0, 1}, {x3, 0, 1},Method -> {"GlobalAdaptive", "SingularityHandler" -> None, MaxRecursion -> 120]

Any ideas will be helpful.

Akku14 · Answer

Do first integration analytically. int1 = Integrate[1/(x2 x3 + (1 - x2 - x3) (x2 + x3)), {x2, 0, 1}, PrincipalValue -> True, Assumptions -> 0 < x3 < 1] (* Log[(1 - 2 x3^3 + Sqrt[1 + (2 - 3 x3) x3] + x3 (3 + 2 Sqrt[1 + (2 - 3 x3) x3]))/( 2 x3^3)]/Sqrt[1 + (2 - 3 x3) x3] *) NIntegrate[int1, {x3, 0, 1}] (* 3.51586 *) Edit Numerical test yield the same result. Do a two -step numerical integration, regarding x3 as a parameter for the first NIntegrate. nint2[x3_?NumericQ] := NIntegrate[1/(x2 x3 + (1 - x2 - x3) (x2 + x3)), {x2, 0, 1}, Method -> "PrincipalValue", Exclusions -> {(x2 x3 + (1 - x2 - x3) (x2 + x3)) == 0}, MaxRecursion -> 100, WorkingPrecision -> 30] pl = Plot[nint2[Rationalize[x3, 0]], {x3, 0, 1}, WorkingPrecision -> 25, PlotRange -> {0, 30}] NIntegrate[nint2[x3], {x3, 0, 1}, WorkingPrecision -> 25] // Timing (* {48.016, 3.515860858034190463653627} *)

Excluding singular curves in NIntegrate

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