# How to numerically verify that principal value?

Mathematica finds

Integrate[Exp[I*s]/(1 + s/(s^2 - 1)^2), {s, -Infinity, Infinity}, PrincipalValue -> True] // ToRadicals (*A huge closed-form expression which is omitted here.*) N[%] (*-1.414 + 0.192275 I*) 

The use of the principal value is grounded by the plots

Plot[{Cos[s]/(1 + s/(s^2 - 1)^2),Sin[s]/(1 + s/(s^2 - 1)^2)},{s,-5,5},WorkingPrecision->30,PlotPoints -> 50] It’s clear that the integrand has its real singularities at the real roots of the denominator, so

sol = Reduce[1 + s/(s^2 - 1)^2 == 0, s, Reals] // ToRadicals;  sol[][] (*-(1/(2 Sqrt[3/(4 + (155/2 - (3 Sqrt)/2)^(1/3) + (1/2 (155 + 3 Sqrt))^(1/3))]))  -  1/2 Sqrt[8/3 - 1/3 (155/2 - (3 Sqrt)/2)^(1/3) -  1/3 (1/2 (155 + 3 Sqrt))^(1/3) +     2 Sqrt[3/( 4 + (155/2 - (3 Sqrt)/2)^(1/3) + (1/2 (155 + 3 Sqrt))^(  1/3))]]*)  N[%] (*-1.49022*)  sol[][] (*-(1/(2 Sqrt[3/(4 + (155/2 - (3 Sqrt)/2)^(1/3) + (1/2 (155 + 3 Sqrt))^(1/3))])) +  1/2 Sqrt[8/3 - 1/3 (155/2 - (3 Sqrt)/2)^(1/3) -   1/3 (1/2 (155 + 3 Sqrt))^(1/3) + 2 Sqrt[3/( 4 + (155/2 - (3 Sqrt)/2)^(1/3)+(1/2 (155 + 3 Sqrt))^( 1/3))]]*) 

However, I have doubts concerning the obtained principal value because the integrand asymptotically equals $$\exp(is)$$ as $$s\to \infty$$ and $$s\to -\infty$$ and $$PV\int_{-\infty}^\infty \exp(is)\,ds$$ does not exist.

In view of it I try to verify it numerically through

NIntegrate[Exp[I*s]/(1+s/(s^2-1)^2),{s,-Infinity, -(1/(2 Sqrt[3/(4+(155/2-(3 Sqrt)/2)^(1/3)+(1/2 (155+3 Sqrt))^(1/3))]))- 1/2 Sqrt[8/3-1/3 (155/2-(3 Sqrt)/2)^(1/3)-1/3 (1/2 (155+3 Sqrt))^(1/3)+ 2 Sqrt[3/(4+(155/2-(3 Sqrt)/2)^(1/3)+(1/2 (155+3 Sqrt))^(1/3))]], -(1/(2 Sqrt[3/(4+(155/2-(3 Sqrt)/2)^(1/3)+(1/2 (155+3 Sqrt))^(1/3))]))+ 1/2 Sqrt[8/3-1/3 (155/2-(3 Sqrt)/2)^(1/3)-1/3 (1/2 (155+3 Sqrt))^(1/3)+ 2 Sqrt[3/(4+(155/2-(3 Sqrt)/2)^(1/3)+(1/2 (155+3 Sqrt))^(1/3))]],Infinity}, Method->"PrincipalValue",AccuracyGoal->3,PrecisionGoal->3,WorkingPrecision->50] 

which results in the error message

NIntegrate::ncvb: NIntegrate failed to converge to prescribed accuracy after 9 recursive bisections in s near {s} = {3.7749613270651398879039428756113970426387939277790*10^28}. NIntegrate obtained 8.8211977939280824575415993952100374290963331174834*10^47 I and 9.194032783290130686998715991388359408878977362628350.*^47 for the integral and error estimates.

and

 (*-2.6098684408162971553635553440779848277629513026488*10^49 +   8.8211977939280824575415993952100374290963331174789*10^47 I*) `

Constructive suggestions are welcome.

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