An integral with fractional part function

Calculus Level 5

01{1x}1{1x}3dx1x \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 aπkcd,\dfrac{a\pi^k}{c\sqrt{d}}, where aa and cc are coprime and dd is square-free, find a+k+c+da+k+c+d.

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

01{1x}1{1x}ndx1x\int_0^1 \sqrt[n]{\frac{\big\{\frac1x\big\}}{1-\big\{\frac1x\big\}}}\frac{dx}{1-x}


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