Differentiating unusual

Calculus Level 5

\[ \large f(x)=\frac{1}{1+x+x^2}\]

For the function \(f(x) \) as described above, then \({\dfrac{d^{6} f(x)}{dx^6}}\) can be represented as

\[ \frac ab \sin^c \left( \text{arctan} \left( \frac {\sqrt d}{ex+f} \right) \right) \sin \left( g \cdot \text{arctan} \left( \frac {\sqrt d}{ex+f} \right) \right) \]

where \(a,b,c,d,e,f,g\) are integers independent of \(x\). for \(\gcd(a,b) = 1 \) and \(d\) square free.

Evaluate \(a+b+c+d+e+f+g\).

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