How to get the asymptotic form of this oscilatting integral?

This is more of a comment, but by mimicking the procedure in this answer near the endpoint $x=1$ we can show that

$$ \int_1^\infty \frac{\cos xt}{(x^2-1)\left[\left(\ln\frac{x-1}{x+1}\right)^2+\pi^2\right]}\,dx \approx \operatorname{Re} \left\{ \frac{e^{it}}{2} \int_0^\infty \frac{e^{-yt}}{y \left[\left(\log \frac{iy}{2}\right)^2 + \pi^2\right]}\,dy\right\} $$

to first order as $t \to \infty$. Numerically these agree very well; below is a plot of the left integral in red and the approximation in blue.

This image was created with the following code in Mathematica. The program complains about the singularities in the integrands but ultimately it probably handles them alright.

pointSpacing = 1/5; (* decrease this for a higher resolution plot *)
Show[
 ListPlot[
  Table[{t, 
    Re[E^(I t)/2 NIntegrate[
        E^(-y t)/(y (Log[I y/2]^2 + Pi^2)), {y, 0, \[Infinity]}] // 
      Quiet]},
   {t, 30, 50, pointSpacing}],
  Joined -> True, PlotRange -> All],
 ListPlot[
  Table[{t, 
    NIntegrate[
      Cos[x t]/((x^2 - 1) (Log[(x - 1)/(x + 1)]^2 + Pi^2)), {x, 
       1, \[Infinity]}, MaxRecursion -> 20] // Quiet},
   {t, 30, 50, pointSpacing}],
  PlotStyle -> Red, Joined -> True]
 ]

Because of the slow $y \to 0$ logarithmic singularity in the denominator we lose a lot of numerical accuracy if we replace the denominator by a simpler approximation. To first order the denominator is

$$ \sim \frac{1}{y(\log y)^2} $$

near $y=0$, so if we replace the denominator with this in the integrand (and replace the integration interval $(0,\infty)$ with $(0,c)$ for some $0 < c < 1$) then we should still obtain the correct leading order behavior (but, again, it will be a worse approximation numerically). Then, since the integral is real, we can use $\operatorname{Re} e^{it} = \cos t$ to obtain the approximation

$$ \int_1^\infty \frac{\cos xt}{(x^2-1)\left[\left(\ln\frac{x-1}{x+1}\right)^2+\pi^2\right]}\,dx \approx \frac{\cos t}{2} \int_0^c \frac{e^{-yt}}{y (\log y)^2}\,dy $$

as $t \to \infty$. Now we can use a result of Erdélyi (see this answer) to conclude that the leading order behavior should be

$$ \int_1^\infty \frac{\cos xt}{(x^2-1)\left[\left(\ln\frac{x-1}{x+1}\right)^2+\pi^2\right]}\,dx \approx \frac{\cos t}{2\log t} $$

as $t \to \infty$. It is important to note here that by ignoring complex contributions from the approximating integral we no longer accurately approximate the phase of the integral. If higher-order corrections to this approximation were computed, terms of the form $o(1/\log t)\sin t$ will appear compensate for this.

Below is a plot of this $\cos t/2\log t$ approximation in blue versus the original integral in red. As mentioned earlier, this is a worse approximation than before, but at least it captures the correct logarithmic decrease of the amplitude of the oscillation.

This image was created with the following code.

pointSpacing = 1/2; (*decrease this for a higher res plot*)
Show[
 Plot[Cos[t]/(2 Log[t]),
  {t, 570, 600},
  PlotRange -> All],
 ListPlot[
  Table[{t, 
    NIntegrate[
      Cos[x t]/((x^2 - 1) (Log[(x - 1)/(x + 1)]^2 + Pi^2)), {x, 
       1, \[Infinity]}, MaxRecursion -> 20] // Quiet},
   {t, 570, 600, pointSpacing}],
  PlotStyle -> Red, Joined -> True]
 ]