We first factor the denominator of the integrand. First, by sum of cubes, we obtain
[math]x^6 + 1 = (x^2 + 1)(x^4 - x^2 + 1). \tag*{}[/math]
Next, we factor the quartic factor by a fairly unorthodox completing the square:
[math]x^4 - x^2 + 1 = (x^4 + 1)^2 - 3x^2 = (x^2 + x \sqrt{3} + 1)(x^2 - x\sqrt{3} + 1). \tag*{}[/math]
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