Twelve? Ain't Nobody got time for that! Part 2

Calculus Level 5

\[ \large \displaystyle \int_0^\infty \frac {1}{x^{12} + 1} \mathrm{d}x = \frac{ \pi (\sqrt{a} + \sqrt{b} )}{c} \]

If the equation above is satisfied for positive integers \(a,b\) and \(c\), what is the smallest value of \(a+b+c\)?