1 The increasing interval of the function $f(x)=\frac{(x+1)^{2}}{x^{2}+1}$ is
$1[-1,1]$. Let $g(x)=\frac{2 x}{x^{2}+1}$,
then
$$
f(x)=\frac{x^{2}+2 x+1}{x^{2}+1}=1+g(x) .
$$

When $x \neq 0$, $g(x)=\frac{2}{x+\frac{1}{x}}$.
It is easy to see that $x+\frac{1}{x}$ is a decreasing function on $(0,1]$, and $x+\frac{1}{x}>0$, so $g(x)$ is an increasing function on $(0,1]$. Since $g(x)$ is an odd function, $g(x)$ is also an increasing function on $[-1,0)$.
When $x=0$, $g(0)=0$.
Thus, $g(x)$ is an increasing function on $[-1,1]$.
Therefore, the increasing interval of $f(x)$ is $[-1,1]$.