I do not know if improvement, but I would have made it otherwise, calculating
by residues:

with(IntegrationTools):
with(plots):
J := Int(1/(y^4-2*y^3+2*I*r*y^2-2*I*r*y-r^2),y=-1..1);
f:=GetIntegrand(J);
s:=singular(f,y);

Numerically:

L:=map(x->op([1,2],x),[s]);
L1:=eval(L,r=0.1);

L1 := [.2236067978-.2236067978*I, -.2236067978+.2236067978*I,
-.1246114e-2+.4993777184e-1*I, 2.001246114-.4993777184e-1*I]

And ploted:

complexplot(L1,view=[-3..3,-0.3..0.3],style=point,symbol=solidbox,color=red,axes=normal);

I choose as path on the complex plane the segment on the real axis (-1,1), the
straight lines from its end points down to +/1-I*infinity, and closed at
infinity. It encloses the first pole above, and its residue times (2*I*Pi) can
be calculated as:

f1:=factor(f);
(2*I*Pi)*residue(f1,op(s[1]));

With a minus sign gives the value of the integral along this path made in
clockwise sense.

Now, the contribution at infinity and the half infinity stright lines is
finite and can be expanded in powers of r (easy to do).

Hence the leading contribution to J as r->0 is

That probably gives it away...

The thing that really blows my mind is that this phenomenon, namely that for parabolic Julia sets, the escape rate in a neighbourhood of the landing point of a ray that goes to the Julia set at the (forward orbit of) the critical value shows up in my PhD thesis!  Basically, what I show there is that there is a relation between the speed of escape of such points (in this case, polynomial in the number of iterations) and the ``geometry'' of the Julia set (basically what kind of 'wedges' one can stick in along landing rays).  Linear wedges (in the natural coordinates) correspond to any case where there is a hyperbolic metric, corresponding to exponential escapes.  When one gets to polynomial escapes, the wedges are asymptotically narrower (think x^2 around 0), which is what one observes in the parabolic and (conjecturally) in some of the Siegel cases too [ie in the exterior of the Siegel domains, the interior parts in those cases are simple].  These Julia sets (conjecturally in my thesis, but I believe that has been proven since) correspond exactly to those where the exterior conformal map extends to a H"older continuous map on the circle.