1. The problem statement, all variables and given/known data

Verify that $\displaystyle \int_0^{\infty}\frac{x^8-4x^6+9x^4-5x^2+1}{x^{12}-10x^{10}+37x^8-42x^6+26x^4-8x^2+1}dx=\frac{\pi}{2}$

2. Relevant equations

Newton Leibniz formula.

3. The attempt at a solution

This is a particulary interesting integral in that one cannot directly use the partial fractions decomposition nor the methods of residues due to the irreducibility of both numerator and denominator. Advanced methods such as Feynman and Schwinger parametrizations also fail. On the other hand, Mathematica has no difficulty in evaluating this integral so I must be missing on something.

Any help would be greatly appreciated.

Thanks!