Look familiar? (My seventeenth integral problem)

Calculus Level 5

\[\large \displaystyle \int_{0}^{1} \dfrac{1-x^{3}}{1+x^{3}} \, dx \]

If the above integral can be expressed in the form

\[ -a + \dfrac{b \pi \sqrt{c}}{d} + \dfrac{f\ln 2}{g}\]

where \( a, b, c, d, f, g\) are positive integers and \(\gcd(b, d)=\gcd(f, g)= 1 \), find \(a+b+c+d+f+g\).

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