One thing I thought was promising was a polynomial integration with
\int_{-\infty}^\infty \frac{1}{t^4 +c_1t^3+c_2t^2+c_3t+c_4} dt
where c1 = -2x, c2 = 2x^2, c3 = -x, c4 = x^2 + 1
but I couldn't find anything useful from there.
I'm interested in seeing this what would your strategies...