# Thou shalt generalize it first!

**Calculus**Level 5

\[ \large \displaystyle \int_{0}^{\infty}e^{-x^{2}}\cos^{3}{(x)}\mathbb{d}x =\dfrac{\pi^{\frac{1}{A}}}{B} \left[\eta e^{-\frac{C}{D}}+e^{-\frac{E}{F}}\right] \]

If the above integral is true for positive integers \(A,B,C,D,E,F,\eta\), where \(\gcd(C,D) = \gcd(E,F) = 1\), find the value of \(A+B+C+D+E+F-\eta \).