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 $a+b+c$ ?