\[ \large \int_{-\pi}^{\pi} \dfrac{2x (1+\sin x)}{1+ \cos^2 x} \,dx \]

If the integral can be expressed as \( \dfrac{a\pi^b}c \), where \(a,b\) and \(c\) are positive integers and with \(a,c\) coprime, find \(a+b+c\).

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