<s>[INST] 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?
 [/INST] $$
\begin{align}
\frac{x-1+\sqrt{x^2+1}}{x+1+\sqrt{x^2+1}}\cdot\frac{x+1-\sqrt{x^2+1}}{x+1-\sqrt{x^2+1}}
&=\frac{(x^2-1)+2\sqrt{x^2+1}-(x^2+1)}{(x+1)^2-(x^2+1)} \\
&= \frac{2\sqrt{x^2+1}-2}{2x} \\
&= \frac{\sqrt{x^2+1}-1}{x} \\
\end{align}
$$
</s>