Absolute value of sum of all integral values of x in the domain of $f\left(x\right)={\cot }^{-1}\sqrt{x\left(x+3\right)}+{\cos }^{-1}\sqrt{x^2+3x+1}$ is

$$\begin{gathered} We have f\left(x\right)=co{t}^{-1}\sqrt{x\left(x+3\right)}+{\cos }^{-1}\sqrt{x^2+3x+1} \\ f\left(x\right) is defined if x\left(x+3\right)\ge 0 and x^2+3x+1\le 1 \\ \Rightarrow x\left(x+3\right)\ge 0 and x\left(x+3\right)\le 0 \\ \Rightarrow x\left(x+3\right)=0 \\ \Rightarrow x=0,-3 \\ \therefore sum of values of x is -3. \\ \therefore absolute value of sum of values of x =3 \end{gathered}$$