An integral with fractional part function

Calculus Level 5

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

If the closed form of the value of the integral above can be expressed 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?

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


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