Party time integral!

Calculus Level 5

0π4tan(x)3dx \large \int_0^{\frac\pi4} \sqrt[3]{\tan(x)} \, dx

If the value of the integral above is equals to abπ1cln(2) \dfrac{\sqrt a }b \pi - \dfrac1c \ln(2) for positive integers a,ba,b and cc with aa square-free, find the value of a+b+ca+b+c.

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