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

Explanation

To show that the function \( f(x) = 2x + \cot^{-1}(x) + \log(\sqrt{1+x^2} - x) \) is increasing in \( \mathbb{R} \), we need to find the derivative \( f'(x) \) and analyze its sign.

### Step 1: Differentiate \( f(x) \)

The derivative of \( f(x) \) can be calculated as follows:

1. The derivative of \( 2x \) is:
\[
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