\[ \large \int_{1}^{\frac{1+\sqrt{5}}{2}} \dfrac{x^2+1}{x^4-x^2+1} \ln\left( 1+x-\dfrac{1}{x} \right) \, dx \]

The above expression can be expressed as \(\dfrac{\pi}{a}\ln(b) \), where \(a\) and \(b\) are positive integers, with \(b\) minimized. What is the value of \(a+b\)?

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