Integrate? Great!

Calculus Level 4

\[ \large \int_0^1 \dfrac{dx}{x^2+2x\cos\alpha+1}\]

Let \(\alpha\) be a constant real number such that the integral above can be expressed as \(\dfrac{a\alpha}{b\sin(c\alpha)} \), where \(a,b\) and \(c\) are positive integers with \(a\) and \(b\) being coprime . Find \(a+2b+3c\).

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