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$$x^{\operatorname{acot}{\left(x \right)}} \left(\frac{2 x \log{\left(x \right)}}{\left(x^{2} + 1\right)^{2}} + \left(\frac{\log{\left(x \right)}}{x^{2} + 1} - \frac{\operatorname{acot}{\left(x \right)}}{x}\right)^{2} - \frac{2}{x \left(x^{2} + 1\right)} - \frac{\operatorname{acot}{\left(x \right)}}{x^{2}}\right) = 0$$
Solve this equationThe roots of this equation
$$x_{1} = 39191.441387957$$
$$x_{2} = 29146.7959912156$$
$$x_{3} = 31387.2342774224$$
$$x_{4} = 56851.5698030362$$
$$x_{5} = 35853.5549721892$$
$$x_{6} = 45839.9203538363$$
$$x_{7} = 36967.2655875309$$
$$x_{8} = 40301.9615395353$$
$$x_{9} = 41411.4720654588$$
$$x_{10} = 34738.7250266363$$
$$x_{11} = 33622.7472181323$$
$$x_{12} = 53556.237548898$$
$$x_{13} = 48048.7173456137$$
$$x_{14} = 57948.5847324975$$
$$x_{15} = 51355.6131554866$$
$$x_{16} = 30267.6437856181$$
$$x_{17} = 46944.7525283045$$
$$x_{18} = 4.72804164681674$$
$$x_{19} = 38079.8851304912$$
$$x_{20} = 49151.8357308118$$
$$x_{21} = 52456.3103687384$$
$$x_{22} = 28024.6680422269$$
$$x_{23} = 54655.4120979409$$
$$x_{24} = 42519.9986873266$$
$$x_{25} = 32505.5930158416$$
$$x_{26} = 55753.850791162$$
$$x_{27} = 26901.2412138305$$
$$x_{28} = 43627.56633321$$
$$x_{29} = 50254.1278542409$$
$$x_{30} = 24650.4494520772$$
$$x_{31} = 25776.5029056205$$
$$x_{32} = 44734.1991207973$$
Сonvexity and concavity intervals:Let’s find the intervals where the function is convex or concave, for this look at the behaviour of the function at the inflection points:
Concave at the intervals
$$\left[4.72804164681674, \infty\right)$$
Convex at the intervals
$$\left(-\infty, 4.72804164681674\right]$$