\[ \int_0^1 \int_0^1 \int_0^1\! \dfrac{3}{x+y+z} \: \mathrm{d}x \: \mathrm{d}y \: \mathrm{d}z = \frac{A}{B} \ln \frac{C}{D} \]

The above equation holds true for positive integers \(A\), \(B\), \(C\), and \(D\) such that \( \gcd(A,B) = \gcd(C,D) = 1\) and that the values of \(C\) and \(D\) are minimized.

Determine \(A+B+C+D\).

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