Integration of modulus involving function

Calculus Level 4

Let \[\displaystyle{I=\int _{ -\frac { \pi }{ 3 } }^{ \frac { \pi }{ 3 } }{ \frac { \pi +{ 4x }^{ 3 } }{ 2-\cos { \left( |x|+\frac { \pi }{ 3 } \right) } } } dx}\]

So, I can be written as \[\displaystyle{\frac { a\pi }{ \sqrt { b } } \left( \tan ^{ -1 }{ c } -\frac { \pi }{ d } \right) }\] where a,b,c and d are positive integers and b is square free. Find the value of \(\frac { a }{ d } -\frac { b }{ c } \)

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