Show that `f(x) = 2x + cot^-1 x + log(sqrt(1+x^2)-x)` is increasing in `R`

Explanation

`f(x) = 2x + cot^-1 x + log(sqrt(1+x^2)-x)`

`f'(x)=2-frac{1}{1+x^2}+frac{1}{sqrt(1+x^2)-1}(frac{x}{sqrt(1+x^2)}-1)`

`=2-frac{1}{1+x^2}-frac{1}{sqrt(1+x^2)}`

Now `1+x^2, sqrt(1+x^2)>-1` for all real x

`f'(x)>0` for all real `x`.

`f(x)` is increasing on `R`