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

Explanation

We have, `f(x) = 2x+cot^(-1)x+log(sqrt(1+x^(2))-x)`
`therefore f^(')(x) = 2+(-1/(1+x^(2))) + (1/(sqrt(1+x^(2))-x))(1/(2sqrt(1+x^(2)).2x-1))`
`=2-1/(1+x^(2))+1/(sqrt(1+x^(2))-x).(x-sqrt(1+x^(2)))/(sqrt(1+x^(2)))`
`=2-(1/(1+x^(2)))-1/(sqrt(1+x^(2)))`
`=((2+2x^(2)-1-sqrt(1+x^(2)))/(1+x^(2)))=(1+2x^(2)-sqrt(1+x^(2)))/(1+x^(2))`
For increasing function, `f^(')(x) gt- 0`
`rArr (1+2x^(2)-sqrt(1-x^(2)))/(1+x^(2)) ge0`
`rArr 1+2x^(2)gesqrt(1+x^(2))`
`rArr 1+4x^(2)+4x^(2) ge1 + x^(2)`
`rArr 4x^(4)+3x^(2) ge0`
`rArr x^(2)(4x^(2)+3) ge0`
Which is true for any real value of x.
Hence, f(x) is increasing in R.