Integral ( I was too lazy to make a good name... )

Calculus Level 3

\[\large \int _{ 1 }^{ 2 }{ \frac { x }{ { x }^{ 2 }+4x+8 } } dx = \arctan { \frac { a }{ b } } -\arctan { c } +\frac { 1 }{ d } \ln { \frac { e }{ f } } \]

The equation above holds true for some positive integers \(a\), \(b\), \(c\), \(d\), \(e\) and \(f\), with \(\gcd(a,b) = \gcd(e,f) = 1\). What is \( a+b+c+d+e+f \)?

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