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 $a+b+c$ .