Thanks for your reply. Omega and epsilon are real. Due to the complexity of the integral adding in the assumptions didn't change anything (the kernel just crashes after a while).

I've already done some attempts at verifying the authors' results. In fact, the lack of agreement was why I'm trying to re-do their solving of this integral to begin with. Their formula for $G_1$ only works in one situation. To explain more I need to write some more definitions:

$$\epsilon=(\omega-\epsilon_t)(\omega-\epsilon_p)-4$$ $$\Delta p=1-(\omega-\epsilon_p)\Delta\epsilon_t$$ $$\Omega=\frac{\Delta p^2 + 1}{2\Delta p} + \frac{\epsilon}{2}$$

Here, the true fundamental parameters are $\epsilon_t$, the energy of a certain orbital (diagonal Hamiltonian matrix element), $\epsilon_p$ is another orbital energy, and $\Delta\epsilon_t$ is a perturbation of a surface ion's energy due to the change in atomic environment. The formula for $G_1$ only works when $\Delta\epsilon_t=0$, and only gives the correct value for the imaginary part (contributing to the density of states). The real part disagrees with what is plotted in the paper. Similarly though, solving the integral for G1 numerically is pretty unstable and I'm unable to get agreement with the paper except in the $\Delta\epsilon_t=0$ case for the imaginary part.

Here's some code showing some more playing around with this system:

\[Epsilon]p = -9.0;
\[Epsilon]t = -6.0;
\[CapitalDelta]\[Epsilon]t = 0.0;
\[Epsilon][w_] := (w - \[Epsilon]p) (w - \[Epsilon]t) - 4
\[CapitalDelta]p[w_] := 
 1 - (w + \[Epsilon]p) \[CapitalDelta]\[Epsilon]t
\[CapitalOmega][w_] := ((\[CapitalDelta]p[w])^2 + 1)/(
  2 \[CapitalDelta]p[w]) + \[Epsilon][w]/2
Plot[\[Epsilon][w], {w, -12, -3}]
Plot[\[CapitalDelta]p[w], {w, -12, -3}]
Plot[\[CapitalOmega][w], {w, -12, -3}]

Manipulate[
 Plot[{Re[(1/(x + \[CapitalOmega][
          w])) ((1 - (x - \[Epsilon][w]/2)^2)/(1 - x^2))^(1/2)], 
   Im[(1/(x + \[CapitalOmega][w])) ((1 - (x - \[Epsilon][w]/2)^2)/(1 -
           x^2))^(1/2)]}, {x, -1, 1}], {w, -12.5, -3}]

\[Alpha][w_] := (\[CapitalDelta]p[w] + 1)^2/(\[CapitalOmega][w] + 1)
\[Beta][w_] := (\[CapitalDelta]p[w] - 1)^2/(\[CapitalOmega][w] + 1)
\[Gamma][w_] := ((\[CapitalDelta]p[w])^2 - 1)/\[CapitalDelta]p[w]
\[Eta][w_] := 1/4 \[Epsilon][w]
\[Eta]1[w_] := 1 - (\[Eta][w])^2
\[Xi][w_] := 1/\[Eta][w]
G2[w_] := ((\[CapitalDelta]p[w])^2 - 1)/(
 4 (\[CapitalDelta]p[w])^2 Sqrt[(\[CapitalOmega][w])^2 - 1])

t = -4.7; N[
 Piecewise[{{1/(
     2 \[CapitalDelta]p[t]) (1 + 
       2/Pi (1 + \[Xi][
           t]) (\[Alpha][
            t] EllipticPi[-\[Beta][t] \[Xi][t], (Abs[\[Xi][t]])^2] - 
          EllipticPi[-\[Xi][t], (Abs[\[Xi][t]])^2])), 
    Abs[\[Epsilon][t]] > 
     4}, {(1 + \[Eta][t])/(Pi \[CapitalDelta]p[t])
       Sign[\[Epsilon][
        t]] (\[Alpha][
          t] EllipticPi[-\[Beta][t] \[Eta][t], (Abs[\[Eta][t]])^2] - 
        EllipticPi[-\[Eta][t], (Abs[\[Eta][t]])^2]) + 
     Sign[\[Gamma][t]] G2[t] - ((I \[Eta][t])/(
       Pi \[CapitalDelta]p[
         t])) (\[Beta][
          t] EllipticPi[-\[Alpha][t] (1 + \[Eta][t]), (Abs[\[Eta]1[
             t]])^2] - 
        EllipticPi[(1 + \[Eta][t]), (Abs[\[Eta]1[t]])^2]), 
    Abs[\[Epsilon][t]] < 4}}]]
1/(2 \[CapitalDelta]p[t]) (1 - 
   I/Pi NIntegrate[(1/(x + \[CapitalOmega][
           t])) ((1 - (x - \[Epsilon][t]/2)^2)/(1 - x^2))^(1/
         2), {x, -1, 1}])

Plot[-2/Pi Abs[w + 9] Re[
   1/(2 \[CapitalDelta]p[w]) (1 - 
      I/Pi NIntegrate[(1/(x + \[CapitalOmega][
              w])) ((1 - (x - \[Epsilon][w]/2)^2)/(1 - x^2))^(1/
            2), {x, -1, 1}])], {w, -12.5, -3}, 
 PlotRange -> {{-3, -12}, {-2, 2}}]
Plot[-1/Pi (w + 9) Re[
   Piecewise[{{1/(
       2 \[CapitalDelta]p[w]) (1 + 
         2/Pi (1 + \[Xi][
             w]) (\[Alpha][
              w] EllipticPi[-\[Beta][w] \[Xi][w], (Abs[\[Xi][w]])^2] -
             EllipticPi[-\[Xi][w], (Abs[\[Xi][w]])^2])), 
      Abs[\[Epsilon][w]] > 
       4}, {(1 + \[Eta][w])/(Pi \[CapitalDelta]p[w])
         Sign[\[Epsilon][
          w]] (\[Alpha][
            w] EllipticPi[-\[Beta][w] \[Eta][w], (Abs[\[Eta][w]])^2] -
           EllipticPi[-\[Eta][w], (Abs[\[Eta][w]])^2]) + 
       Sign[\[Gamma][w]] G2[w] - ((I \[Eta][w])/(
         Pi \[CapitalDelta]p[
           w])) (\[Beta][
            w] EllipticPi[-\[Alpha][w] (1 + \[Eta][w]), (Abs[\[Eta]1[
               w]])^2] - 
          EllipticPi[(1 + \[Eta][w]), (Abs[\[Eta]1[w]])^2]), 
      Abs[\[Epsilon][w]] < 4}}]], {w, -12.5, -3}, 
 PlotRange -> {{-3, -12.5}, {-2, 2}}]
Plot[-2/Pi Abs[w + 9] Im[
   1/(2 \[CapitalDelta]p[w]) (1 - 
      I/Pi NIntegrate[(1/(x + \[CapitalOmega][
              w])) ((1 - (x - \[Epsilon][w]/2)^2)/(1 - x^2))^(1/
            2), {x, -1, 1}])], {w, -12.5, -3}, 
 PlotRange -> {{-3, -12}, {0, 3}}]
Plot[-2/Pi Abs[w + 9] Im[
   Piecewise[{{1/(
       2 \[CapitalDelta]p[w]) (1 + 
         2/Pi (1 + \[Xi][
             w]) (\[Alpha][
              w] EllipticPi[-\[Beta][w] \[Xi][w], (Abs[\[Xi][w]])^2] -
             EllipticPi[-\[Xi][w], (Abs[\[Xi][w]])^2])), 
      Abs[\[Epsilon][w]] > 
       4}, {(1 + \[Eta][w])/(Pi \[CapitalDelta]p[w])
         Sign[\[Epsilon][
          w]] (\[Alpha][
            w] EllipticPi[-\[Beta][w] \[Eta][w], (Abs[\[Eta][w]])^2] -
           EllipticPi[-\[Eta][w], (Abs[\[Eta][w]])^2]) + 
       Sign[\[Gamma][w]] G2[w] - ((I \[Eta][w])/(
         Pi \[CapitalDelta]p[
           w])) (\[Beta][
            w] EllipticPi[-\[Alpha][w] (1 + \[Eta][w]), (Abs[\[Eta]1[
               w]])^2] - 
          EllipticPi[(1 + \[Eta][w]), (Abs[\[Eta]1[w]])^2]), 
      Abs[\[Epsilon][w]] < 4}}]], {w, -12.5, -3}, 
 PlotRange -> {{-3, -12.5}, {0, 3}}]

Here are some things I've tried:

Integrate[(1/(x + \[CapitalOmega])) ((1 - (x + \[Epsilon]/2)^2)/(
   1 - x^2))^(1/2), {x, -1, 1}]

Which doesn't go anywhere. By doing some manipulations by hand, getting the radical completely in the denominator, then doing polynomial division on the term outside, I can get three separate integrals that are listed in a table of integrals. The simplest of these is the following:

Integrate[1/((1 - x) (-1 - x) (1 - \[Epsilon]/2 - x) (-1 - \[Epsilon]/
     2 - x))^(1/2), {x, -1, 1}]

Which also doesn't result in an answer. However, replacing the variables with explicit values gives an actual answer:

In[1]:= Integrate[1/((1-x)(-1-x)(3-x)(-3-x))^(1/2),{x,-1,1}]
Out[1]= 2/3 EllipticK[1/9]

As soon as the roots aren't from difference of squares terms though, it doesn't work again:

Integrate[1/((1 - x) (-1 - x) (2 - x) (-3 - x))^(1/2), {x, -1, 1}]

I think I'm just going to have to simplify the entire thing by hand to get it in terms of EllipticK and EllipticPi, then evaluate it at that point.