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

Calculus Level 5

01x12+1dx=π(a+b)c \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,ba,b and cc, what is the smallest value of a+b+ca+b+c?

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