A nasty integral of a rational function

Calculus Level 4

\[\large \int_0^{\infty} \dfrac{x^8 - 4x^6 + 9x^4 - 5x^2 + 1}{x^{12} - 10 x^{10} + 37x^8 - 42x^6 + 26x^4 - 8x^2 + 1} \, dx = \, ? \]

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