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$$\frac{2 \left(- 2 x \operatorname{atan}{\left(x \right)} + 1\right)}{\left(x^{2} + 1\right)^{2}} = 0$$
Solve this equationThe roots of this equation
$$x_{1} = -17878.8979748856$$
$$x_{2} = -22114.9272950956$$
$$x_{3} = -42453.4454174628$$
$$x_{4} = -39063.4019253875$$
$$x_{5} = 24788.0714796005$$
$$x_{6} = -40758.4158445424$$
$$x_{7} = -27198.9915740665$$
$$x_{8} = -17031.8133699325$$
$$x_{9} = -29741.2260909927$$
$$x_{10} = -22962.2226401045$$
$$x_{11} = 40042.1249189752$$
$$x_{12} = 19704.3745564126$$
$$x_{13} = 23940.7309886273$$
$$x_{14} = 42584.6652795986$$
$$x_{15} = 25635.4296319425$$
$$x_{16} = -20420.4147645461$$
$$x_{17} = 0.765378926665789$$
$$x_{18} = 40889.6345794937$$
$$x_{19} = 30719.8667071727$$
$$x_{20} = 36652.130400909$$
$$x_{21} = -30588.6591830513$$
$$x_{22} = 34957.164150976$$
$$x_{23} = -35673.4300548609$$
$$x_{24} = -26351.6051691354$$
$$x_{25} = 35804.6443832452$$
$$x_{26} = 38347.1182450815$$
$$x_{27} = -25504.2334265512$$
$$x_{28} = 41737.1481106533$$
$$x_{29} = -39910.9068025809$$
$$x_{30} = 29872.4321264636$$
$$x_{31} = -23809.5402776119$$
$$x_{32} = -33978.4777406911$$
$$x_{33} = 31567.3104913969$$
$$x_{34} = 28177.5939359089$$
$$x_{35} = 33262.2228288295$$
$$x_{36} = -31436.1015983531$$
$$x_{37} = -33131.0115087019$$
$$x_{38} = 22246.1111845123$$
$$x_{39} = -28893.8031518313$$
$$x_{40} = 29025.007564543$$
$$x_{41} = 37499.6218083141$$
$$x_{42} = 18857.1946299244$$
$$x_{43} = -34825.9507519323$$
$$x_{44} = 21398.8367920726$$
$$x_{45} = 18010.055373674$$
$$x_{46} = -21267.6569489348$$
$$x_{47} = 39194.6193822603$$
$$x_{48} = -36520.9152075726$$
$$x_{49} = 27330.1922698542$$
$$x_{50} = -28046.391296672$$
$$x_{51} = 26482.8037290628$$
$$x_{52} = -32283.5525952437$$
$$x_{53} = 23093.4101312038$$
$$x_{54} = -37368.4058086007$$
$$x_{55} = -19573.2044619574$$
$$x_{56} = -41605.9287947859$$
$$x_{57} = -38215.9014923937$$
$$x_{58} = -24656.8778783254$$
$$x_{59} = 32414.7627499852$$
$$x_{60} = 34109.6901394521$$
$$x_{61} = -18726.030446616$$
$$x_{62} = 17162.962929885$$
$$x_{63} = 20551.5900405008$$
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
$$\left(-\infty, 0.765378926665789\right]$$
Convex at the intervals
$$\left[0.765378926665789, \infty\right)$$