\[\large \int^{\pi /3}_{\pi /4} \dfrac{(\sin^3\theta-\cos^2\theta-\cos^3\theta)(\sin\theta+\cos\theta+\cos^2\theta )^{2007}}{(\sin\theta)^{2009}(\cos\theta)^{2009}} \,d\theta \]

If the above integral is of the form \[ \dfrac{(a+\sqrt{b})^n-(1+\sqrt{c})^n}{d}, \]

where \(a,b,c\) and \(d\) are positive integers, find \(a+b+c+d\).

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