\[f(x)+f \left(1-\dfrac{1}{x}\right)=\tan^{-1} x, \quad \forall x \in \mathbb{R} - \{0\}\]

If \(f:\mathbb{R} \rightarrow \mathbb{R}\) satisfies the functional equation above then \(\displaystyle\int_{0}^{1} f(x) \, dx=\frac{a \pi ^{b}}{c}\), where \(a,b,c\) are positive integers with \(a,c\) coprime.

Find the value of \(a+b+c\).

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