I’m attempting to compute a $5$-dimensional integral, which can or might not be zero. I do imagine it’s finite although, not less than for the values $epsilon$, $sigma$ used right here. The integral is given within the script beneath and the integrand is determined by the variables $r, theta_1, theta_2, phi, tau_4$. I set `"SymbolicProcessing" -> 0`

and `Exclusions->r==0`

, since this level is an indeterminate one for some values of $epsilon$ and $sigma$. Nonetheless, within the case I current right here there must be nothing particular about this level.

I’ve noticed the next conduct: having each `SymbolicProcessing`

and `Exclusions`

as above leads to the integral being evaluated to $0$ in about $5$ seconds. There isn’t a error estimate from the `IntegrationMonitor`

, i.e. the variable `errors`

stays empty! One thing is fishy about that. After I remark out the `SymbolicProcessing`

however go away `Exclusions`

, I get $-1.3945964 cdot 10^7$ in $sim 100$ seconds. After I remark out `Exclusions`

and preserve `SymbolicProcessing`

, I acquire $-9.7620595 cdot 10^6$ in $sim 90$ seconds. After I remark each out, the consequence seems to be $-9.7620595 cdot 10^6$ in $sim 80$ seconds.

Absolutely the consequence with each choices can’t be trusted. If I put `Exclusions->r==13`

as an alternative of `Exclusions->r==0`

, I additionally get $0$ in $5$ seconds and no error estimate. The opposite outcomes appear in section with each other. So why does the integral behave that means when each choices 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/four 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
Size@errors
Whole@errors
```