\[ \large \int_0^1 \dfrac{\tan^{-1}x}{x^{2/3}} \, dx \]

Given the integral above equals to

\[ \pi \left(\dfrac ab - \dfrac{\sqrt c}d \right) + \dfrac ef \ \ln(g) \]

where \(a,b,c,d,e\) and \(f\) are positive integers with prime \(g\) and \(\gcd(a,b) = \gcd(c,d) =\gcd(e,f) = 1 \).

Find the value of the 7-digit integer, \( \overline{abcdefg} \).

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