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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