If range of function `f(x)=sin^(-1)x+2tan^(-1)x+x^2+4x+1` is `[p , q],` then the value of `(p+q)` is_______

Explanation

`f(x) = sin^-1x+2tan^-1x+x^2+4x+1`

`f(x) = sin^-1x+2tan^-1x+(x+2)^2-3`

Now, `f(x)` will be minimum when `x = -1` and maximum when `x = 1`.

`:. f(x)_min = p = sin^-1(-1)+2tan^-1(-1)+(-1+2)^2-3`

`=>p = -pi/2+2(-pi/4)+1-3`

`=>p = -pi-2`

`:. f(x)_max = p = sin^-1(1)+2tan^-1(1)+(1+2)^2-3`

`=>q = pi/2+2(pi/4)+9-3`

...