Day 14: Christmas is the Solution

Calculus Level 3

\[C = \int_0^{3\pi} \bigg [ \frac{1 - \cos^2{\frac{x}{12}}} {1 - \sin^2{\frac{x}{12}}} + 1 \bigg ] \, \mathrm{d}x\] If the value of \(2 + \frac{C}{144}\) is in the form \(a/b\) where \(a\) and \(b\) are positive coprime integers and enter \(a+b\) as your answer.

This problem is part of the set Advent Calendar 2014.
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