If range of the function $f\left(x\right)={\sin }^{-1}x+2{\tan }^{-1}x+x^2+4x+1$ is  $[p,q]$then the value of $p+q$ is

$$\begin{gathered} The domain of f\left(x\right)={\sin }^{-1}x+2{\tan }^{-1}x+x^2+4x+1 is \left[-1,1\right]. \\ Also f\left(x\right) is increa\sin g in \left[-1,1\right] \\ \therefore Range of f\left(x\right)=\left[f\left(-1\right),f\left(1\right)\right] \\ =\left[-\mathrm{\pi }-2,\mathrm{\pi }+6\right] \\ =\left[p,q\right] \\ \therefore p+q=4 \end{gathered}$$