Evaluate the integral \( \int_a^b {\sin(\tan^{-1}(\cot|x|)) \times arg(z) } d(x - [|x|])\) where \(z=3x + i4x^3\) and \([.]\) denotes the greatest integer part of \(x\) . Given lower limit \(a = -2010\) and upper limit \(b = 2013\) .

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