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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