Dig the Range

Geometry Level 5

\[ \large \displaystyle f(x)=\left(\cos^{-1}\dfrac{x}{2}\right)^2+\pi \sin^{-1}\dfrac{x}{2}-\left(\sin^{-1}\dfrac{x}{2}\right)^2+\dfrac{\pi^2}{12}(x^2+6 x+8) \]

If the range of \(f(x)\) above is \([a\pi^2,b \pi^2]\), what is the value of \(2(a+b)\)?