An integral with fractional part function

Calculus Level 5

\[ \int_0^1 \sqrt[3]{\frac{\big\{\frac1x\big\}}{1-\big\{\frac1x\big\}}}\frac{dx}{1-x} \]

If the closed form of the value of the integral above can be expressed as \(\dfrac{a\pi^k}{c\sqrt{d}},\) where \(a\) and \(c\) are coprime and \(d\) is square-free, find \(a+k+c+d\).


Also, is it possible to find the following in a closed form?

\[\int_0^1 \sqrt[n]{\frac{\big\{\frac1x\big\}}{1-\big\{\frac1x\big\}}}\frac{dx}{1-x} \]

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