$6\left[-\frac{1}{4}, \frac{1}{4}\right]$ Hint: $f(x)=\frac{x}{1+x^{2}} \cdot \frac{1-x^{2}}{1+x^{2}}$, let $x=\tan \alpha$, then
$$
f=\frac{1}{2} \sin 2 \alpha \cdot \cos 2 \alpha=\frac{1}{4} \sin 4 \alpha,
$$

Thus, $-\frac{1}{4} \leqslant f \leqslant \frac{1}{4}$, and the equality holds when $x=-\tan \frac{\pi}{8}, \tan \frac{\pi}{8}$.
6. The range of the function $f(x)=\frac{x-x^{3}}{\left(1+x^{2}\right)^{2}}$ is $\qquad$