Why all integration problems have integrand?

Calculus Level 5

\[\displaystyle\int_{-\pi/3}^{\pi/3} \dfrac{\pi+4x^3}{2-\cos\left(|x|+\frac{\pi}{3}\right)}\ dx\]

If the above integral can be expressed as \(\dfrac{a\pi}{\sqrt{b}} \tan^{-1} \left(\frac{1}{c}\right)\) for positive integers \(a,b,c\) with \(a,b\) coprime and prime \(c\). Find \(a+b+c\).

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