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I'm trying to do the following integral, but Mathematica won't evaluate it - but it returned unevaluated.

W[βr_, βi_, ϵr_, ϵi_, α_] :=Expand[1/(π (Abs[α]^4 + 3 Abs[α]^2 + 1)) 2*
    E^(-2 Abs[α - (0.87*(βr + I*βi) - 0.5*(ϵr + I*ϵi))]^2) (-Abs[α]^2 +
    Abs[-2*Conjugate[α]*(0.87*(βr + I*βi) - 0.5*(ϵr + I*ϵi)) - (2*(0.87*(βr + I*βi) - 0.5*(ϵr + I*ϵi)))/(2*(0.87*(βr + I*βi) - 0.5*(ϵr + 
 I*ϵi)) - α) + α/(2*(0.87*(βr + I*βi) - 
 0.5*(ϵr + I*ϵi)) - α) + α*Conjugate[α]]^2)*(2*E^(-2 Abs[(0.5*(βr + I*βi) + 0.87*(ϵr + I*ϵi))]^2))/π];

K[βr_, ϵr_, α_] = Integrate[W[βr, βi, ϵr, ϵi, α], {ϵi, -∞, +∞}, {βi, -∞, +∞}]

Mathematica returns the same expression. I really need to know how to do it? can anyone help?

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This is what I get on 11.1.1

r = Simplify@
  ComplexExpand[
   Rationalize@
    W[\[Beta]r, \[Beta]i, \[Epsilon]r, \[Epsilon]i, \[Alpha]], \
\[Alpha]]

Mathematica graphics

 Integrate[%, {\[Epsilon]i, -\[Infinity], +\[Infinity]}]
 Integrate[%, {\[Beta]i, -\[Infinity], +\[Infinity]}]

Mathematica graphics

(8*E^(-((10069*(\[Beta]r^2 + \[Epsilon]r^2))/5000) + ((87*\[Beta]r)/25 - 2*\[Epsilon]r)*Re[\[Alpha]] - 2*Re[\[Alpha]]^2)*
   (2500*Im[\[Alpha]]^4 + (50 + (87*\[Beta]r - 50*\[Epsilon]r)*Re[\[Alpha]] - 50*Re[\[Alpha]]^2)^2 + 
    Im[\[Alpha]]^2*(7569*\[Beta]r^2 - 8700*\[Beta]r*\[Epsilon]r + 2500*(2 + \[Epsilon]r^2) - 100*(87*\[Beta]r - 50*\[Epsilon]r)*Re[\[Alpha]] + 5000*Re[\[Alpha]]^2)))/
  (10069*Pi*(1 + Im[\[Alpha]]^4 + 3*Re[\[Alpha]]^2 + Re[\[Alpha]]^4 + Im[\[Alpha]]^2*(3 + 2*Re[\[Alpha]]^2)))
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  • $\begingroup$ I am very grateful for your help. Thank you. $\endgroup$ – Nabil Khossossi Aug 26 '17 at 23:55
  • $\begingroup$ The use of ComplexExpand without any specified complex variables assumes that all variables are real -- including α. However, the presence of Conjugate[α] implies that α is complex. The validity of your result depends on α being real. $\endgroup$ – Bob Hanlon Aug 27 '17 at 1:59
  • $\begingroup$ @BobHanlon thanks. I actually did not notice the conjugate, saw abs only. corrected. $\endgroup$ – Nasser Aug 27 '17 at 3:22

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