# Why does the mix of Exclusions and SymbolicProcessing->0 sets my integral to $0$?

I am trying to compute a $$5$$-dimensional integral, which may or may not be zero. I do believe it is finite though, at least for the values $$\epsilon$$, $$\sigma$$ used here. The integral is given in the script below and the integrand depends on the variables $$r, \theta_1, \theta_2, \phi, \tau_4$$. I set "SymbolicProcessing" -> 0 and Exclusions->r==0, since this point is an indeterminate one for some values of $$\epsilon$$ and $$\sigma$$. However, in the case I present here there should be nothing special about this point.

I have observed the following behavior: having both SymbolicProcessing and Exclusions as above results in the integral being evaluated to $$0$$ in about $$5$$ seconds. There is no error estimate from the IntegrationMonitor, i.e. the variable errors remains empty! Something is fishy about that. When I comment out the SymbolicProcessing but leave Exclusions, I get $$-1.3945964 \cdot 10^7$$ in $$\sim 100$$ seconds. When I comment out Exclusions and keep SymbolicProcessing, I obtain $$-9.7620595 \cdot 10^6$$ in $$\sim 90$$ seconds. When I comment both out, the result appears to be $$-9.7620595 \cdot 10^6$$ in $$\sim 80$$ seconds.

Surely the result with both options cannot be trusted. If I put Exclusions->r==13 instead of Exclusions->r==0, I also get $$0$$ in $$5$$ seconds and no error estimate. The other results seem in phase with one another. So why does the integral behave that way when both options are activated?

My code:

Clear[ϵ, σ, r, θ1, θ2, ϕ, x, y, z, \
τ, τ4, d, x15, x24, x25, x45, R, S, a, f, Φ, \
Y245, integrand, errors]

ϵ = 2;
σ = -2;

x = r*Cos[θ1];
y = r*Sin[θ1]*Sin[θ2]*Cos[ϕ];
z = r*Sin[θ1]*Sin[θ2]*Sin[ϕ];
τ = r*Sin[θ1]*Cos[θ2];

d = Sqrt[x^2 + y^2 + z^2] // FullSimplify;
x15 = (1 - x)^2 + y^2 + z^2 + τ^2 // FullSimplify;
x24 = ϵ^2 + σ^2 + τ4^2 // FullSimplify;
x25 = (ϵ - x)^2 + (σ - y)^2 + z^2 + τ^2 //
FullSimplify;
x45 = x^2 + y^2 + z^2 + (τ4 - τ)^2 // FullSimplify;
R = x24/x25;
S = x45/x25;
a = 1/4 Sqrt[4*R*S - (1 - R - S)^2];
f = I Sqrt[-((1 - R - S - 4*I*a)/(1 - R - S + 4*I*a))];
Φ =
1/a Im[PolyLog[2, f*Sqrt[R/S]] +
Log[Sqrt[R/S]]*Log[1 - f*Sqrt[R/S]]];
Y245 = 1/x25 Φ;
integrand = 1/(d*x15^2) (τ^2/x15 - 1) Y245;

NIntegrate[
integrand, {r, 0, ∞}, {θ1, 0, π}, {θ2,
0, π}, {ϕ, 0,
2 π}, {τ4, -∞, ∞},
Exclusions -> {r == 0},
Method -> {"GlobalAdaptive", "SymbolicProcessing" -> 0,
"SingularityHandler" -> None}, PrecisionGoal -> 2,
AccuracyGoal -> 2, WorkingPrecision -> 20,
IntegrationMonitor :> ((errors = Through[#1@"Error"]) &)] // Timing
Length@errors
Total@errors
• Which Mathematica version are you using? (I assume 12.0 or later, but let's be sure.) – Anton Antonov Apr 27 '20 at 11:32
• @AntonAntonov I have 12.0.0.0. – Jxx Apr 27 '20 at 12:08
• Is it coming from mathematica.stackexchange.com/questions/218725/… ? – Alex Trounev Apr 27 '20 at 12:46
• @AlexTrounev It is related to that integral, yes (it is different though). – Jxx Apr 27 '20 at 15:37
• @Jxx Then see my answer. – Alex Trounev Apr 27 '20 at 16:55

There is a missing Jacobian $$r^3 \sin\theta _1^2\sin \theta _2$$ in integrand, so after inserting it in a code we have

NIntegrate[
integrand r^3 Sin[\[Theta]1]^2 Sin[\[Theta]2], {r,
0, \[Infinity]}, {\[Theta]1, 0, \[Pi]}, {\[Theta]2,
0, \[Pi]}, {\[Phi], 0,
2 \[Pi]}, {\[Tau]4, -\[Infinity], \[Infinity]},