The number of integral values in the range of the function $f\left(x\right)={\sin }^{-1}x-{\cot }^{-1}x+x^2+2x+6$ is

$f\left(x\right)={\sin }^{-1}x-(\frac{\pi }{2}-{\tan }^{-1}x)+{(x+1)}^2+5$

$={\sin }^{-1}x+{\tan }^{-1}x-\frac{\pi }{2}+{(x+1)}^2+5$

Domain of $f\left(x\right)is[-1,1]$ and  are all increasing in $[-1,1]$

$\therefore f\left(x\right)$ is increasing $f(-1)=\frac{-5\pi }{4}+5andf\left(1\right)=\frac{\pi }{4}+9$

$\therefore$ Range of $f\left(x\right)$ is $\left[\frac{-5\pi }{4}+5,\frac{\pi }{4}+9\right]$.