Let's find the inflection points, we'll need to solve the equation for this
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$
(the second derivative equals zero),
the roots of this equation will be the inflection points for the specified function graph:
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = $$
the second derivative$$\frac{- \frac{2 x^{2}}{x^{2} + 1} - \frac{x^{2}}{\left(x^{2} + 1\right) \log{\left(x^{2} + 1 \right)}} + 1}{\left(x^{2} + 1\right) \sqrt{\log{\left(x^{2} + 1 \right)}}} = 0$$
Solve this equationSolutions are not found,
maybe, the function has no inflections