Just a small twist

Calculus Level 4

\[\large{\int^{\frac{\pi}{4}}_{0} (\tan^{2016} (x)+\tan^{2014} (x))d(x-\frac{\lfloor x \rfloor}{1!}+\frac{\lfloor x \rfloor^{2}}{2!}-\frac{\lfloor x \rfloor^3}{3!}+\ldots)}\]

If the value of the integral above equals to \(\frac ab\) for coprime positive integers, find the value of \(a+b\).

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