# How can I achieve this transformation?

I calculate a integration which results in the below. $$\log{\left|\frac{x+\sqrt{x^2+1}-1}{x+\sqrt{x^2+1}+1}\right|}$$

I've checked the integration with Wolfram Alpha and it give me this result. $$\log{\left|\frac{\sqrt{x^2+1}-1}{x}\right|}$$

These expressions seems to have the equal value, but I cannot find the way to transform one to the other.

Is it possible to perform the transformation?
If possible, how can I do that?

• The argument of the first log above is 1. You're missing a sign. Jul 29, 2017 at 12:59
• Apparently you have miscopied something. $\frac{x+ \sqrt{x^2+ 1}- 1}{x+ \sqrt{x^2+ 1}- 1}= 1$ so that first logarithm is 0 for all x, Jul 29, 2017 at 13:02
• I've missed the sign indeed. Thanks! Jul 29, 2017 at 13:04

\begin{align} \frac{x-1+\sqrt{x^2+1}}{x+1+\sqrt{x^2+1}}\cdot\frac{x+1-\sqrt{x^2+1}}{x+1-\sqrt{x^2+1}} &=\frac{(x^2-1)+2\sqrt{x^2+1}-(x^2+1)}{(x+1)^2-(x^2+1)} \\ &= \frac{2\sqrt{x^2+1}-2}{2x} \\ &= \frac{\sqrt{x^2+1}-1}{x} \\ \end{align}