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]] (*-(1/(2 Sqrt[3/(4 + (155/2 - (3 Sqrt[849])/2)^(1/3) + (1/2 (155 + 3 Sqrt[849]))^(1/3))])) - 1/2 Sqrt[8/3 - 1/3 (155/2 - (3 Sqrt[849])/2)^(1/3) - 1/3 (1/2 (155 + 3 Sqrt[849]))^(1/3) + 2 Sqrt[3/( 4 + (155/2 - (3 Sqrt[849])/2)^(1/3) + (1/2 (155 + 3 Sqrt[849]))^( 1/3))]]*) N[%] (*-1.49022*) sol[[2]][[2]] (*-(1/(2 Sqrt[3/(4 + (155/2 - (3 Sqrt[849])/2)^(1/3) + (1/2 (155 + 3 Sqrt[849]))^(1/3))])) + 1/2 Sqrt[8/3 - 1/3 (155/2 - (3 Sqrt[849])/2)^(1/3) - 1/3 (1/2 (155 + 3 Sqrt[849]))^(1/3) + 2 Sqrt[3/( 4 + (155/2 - (3 Sqrt[849])/2)^(1/3)+(1/2 (155 + 3 Sqrt[849]))^( 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[849])/2)^(1/3)+(1/2 (155+3 Sqrt[849]))^(1/3))]))- 1/2 Sqrt[8/3-1/3 (155/2-(3 Sqrt[849])/2)^(1/3)-1/3 (1/2 (155+3 Sqrt[849]))^(1/3)+ 2 Sqrt[3/(4+(155/2-(3 Sqrt[849])/2)^(1/3)+(1/2 (155+3 Sqrt[849]))^(1/3))]], -(1/(2 Sqrt[3/(4+(155/2-(3 Sqrt[849])/2)^(1/3)+(1/2 (155+3 Sqrt[849]))^(1/3))]))+ 1/2 Sqrt[8/3-1/3 (155/2-(3 Sqrt[849])/2)^(1/3)-1/3 (1/2 (155+3 Sqrt[849]))^(1/3)+ 2 Sqrt[3/(4+(155/2-(3 Sqrt[849])/2)^(1/3)+(1/2 (155+3 Sqrt[849]))^(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 9.1940327832901306869987159913883594088789773626283`50.*^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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