\[ \large \int_1^{(1+\sqrt5)/2} \dfrac{x^2+1}{x^4-x^2+1} \ln \left( 1 + x - \dfrac1x\right) \, dx \]

If the value of the integral above is equal to \( \dfrac {a\pi^b}c \ln d \), where \(a,b,c\) and \(d\) are positive integers with \(d\) minimized, find \(a+b+2c+d\).

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