Party time integral!

Calculus Level 5

\[ \large \int_0^{\frac\pi4} \sqrt[3]{\tan(x)} \, dx \]

If the value of the integral above is equals to \( \dfrac{\sqrt a }b \pi - \dfrac1c \ln(2) \) for positive integers \(a,b\) and \(c\) with \(a\) square-free, find the value of \(a+b+c\).