In order to find the extrema, we need to solve the equation
$$\frac{d}{d x} f{\left(x \right)} = 0$$
(the derivative equals zero),
and the roots of this equation are the extrema of this function:
$$\frac{d}{d x} f{\left(x \right)} = $$
the first derivative$$\frac{-1 + \frac{1}{x^{2} + 1}}{x^{3}} - \frac{3 \left(- x + \operatorname{atan}{\left(x \right)}\right)}{x^{4}} = 0$$
Solve this equationThe roots of this equation
$$x_{1} = -38411.7535678211$$
$$x_{2} = 20746.0147149245$$
$$x_{3} = -42649.461827958$$
$$x_{4} = -25699.2605579154$$
$$x_{5} = 24983.0265404262$$
$$x_{6} = -32479.1013813967$$
$$x_{7} = 22440.7726333195$$
$$x_{8} = 17356.7709091699$$
$$x_{9} = -27394.1738017033$$
$$x_{10} = 32610.3194185706$$
$$x_{11} = 19051.3391839221$$
$$x_{12} = -22309.5718142799$$
$$x_{13} = -15531.1805503588$$
$$x_{14} = 23288.1767401818$$
$$x_{15} = -39259.289744171$$
$$x_{16} = 27525.3856373513$$
$$x_{17} = 25830.4694498413$$
$$x_{18} = 36000.3853896247$$
$$x_{19} = -36716.6907745185$$
$$x_{20} = 34305.3436017062$$
$$x_{21} = -24851.8193555671$$
$$x_{22} = 40238.0520918502$$
$$x_{23} = 36847.9121154636$$
$$x_{24} = -34174.124093174$$
$$x_{25} = -28241.6439581724$$
$$x_{26} = 24135.5952065001$$
$$x_{27} = -18072.8562532625$$
$$x_{28} = 33457.8291397138$$
$$x_{29} = -14684.0459720068$$
$$x_{30} = 41085.5943179336$$
$$x_{31} = 26677.922810015$$
$$x_{32} = 16509.5376458429$$
$$x_{33} = 42780.6862304154$$
$$x_{34} = -18920.1512789125$$
$$x_{35} = -13836.9711337571$$
$$x_{36} = 28372.8570679257$$
$$x_{37} = -29089.1220673667$$
$$x_{38} = -29936.6074421376$$
$$x_{39} = 37695.4423747551$$
$$x_{40} = -23156.9735604547$$
$$x_{41} = 30067.8227785119$$
$$x_{42} = 18204.0397130023$$
$$x_{43} = -30784.0994722808$$
$$x_{44} = -21462.1864505397$$
$$x_{45} = -19767.4735221121$$
$$x_{46} = -41801.9150374336$$
$$x_{47} = 12274.1733072055$$
$$x_{48} = 30915.3157846703$$
$$x_{49} = -31631.5976140599$$
$$x_{50} = 31762.8148239848$$
$$x_{51} = 38542.9759312704$$
$$x_{52} = -40954.3706546333$$
$$x_{53} = 35152.8624565848$$
$$x_{54} = 29220.3363398227$$
$$x_{55} = 19898.6653005008$$
$$x_{56} = -26546.7123745741$$
$$x_{57} = -20614.8195410024$$
$$x_{58} = -35021.6422921722$$
$$x_{59} = 39390.5125693788$$
$$x_{60} = 15662.345886645$$
$$x_{61} = 14815.2030082688$$
$$x_{62} = 13121.1033142041$$
$$x_{63} = -40106.8288341669$$
$$x_{64} = 41933.1390816712$$
$$x_{65} = 21593.3846176302$$
$$x_{66} = -17225.5925849612$$
$$x_{67} = -24004.389916208$$
$$x_{68} = -37564.2205050519$$
$$x_{69} = -16378.3652980817$$
$$x_{70} = -35869.1646158396$$
$$x_{71} = 13968.1182452817$$
$$x_{72} = -12143.0529026344$$
$$x_{73} = -33326.6103384257$$
$$x_{74} = -12989.9682054958$$
The values of the extrema at the points:
(-38411.753567821135, -6.77725727007463e-10)
(20746.014714924495, -2.32325967128191e-9)
(-42649.46182795796, -5.49739437736566e-10)
(-25699.260557915426, -1.51402204783306e-9)
(24983.02654042619, -1.60207408398598e-9)
(-32479.101381396697, -9.47918471742499e-10)
(22440.772633319477, -1.98561017295073e-9)
(17356.77090916988, -3.31911907519973e-9)
(-27394.17380170328, -1.33247381592315e-9)
(32610.31941857058, -9.40305497051386e-10)
(19051.339183922148, -2.75494652621263e-9)
(-22309.571814279894, -2.00903244998659e-9)
(-15531.180550358766, -4.1452157438356e-9)
(23288.17674018178, -1.84374022001946e-9)
(-39259.289744170994, -6.4878043831623e-10)
(27525.385637351348, -1.31980081249435e-9)
(25830.469449841286, -1.49868027442496e-9)
(36000.38538962473, -7.71554752122418e-10)
(-36716.69077451845, -7.41744536841238e-10)
(34305.34360170624, -8.49682291355882e-10)
(-24851.8193555671, -1.61903476140945e-9)
(40238.0520918502, -6.17602647537309e-10)
(36847.91211546358, -7.36471112583106e-10)
(-34174.124093174025, -8.56219774242942e-10)
(-28241.643958172423, -1.25370655711109e-9)
(24135.595206500064, -1.71654703584026e-9)
(-18072.85625326246, -3.0613195463161e-9)
(33457.829139713795, -8.93272763815866e-10)
(-14684.04597200684, -4.6372666663652e-9)
(41085.59431793356, -5.92385266867228e-10)
(26677.92281001498, -1.40498085169395e-9)
(16509.53764584295, -3.66850281768929e-9)
(42780.68623041544, -5.4637215795632e-10)
(-18920.151278912526, -2.79328168924977e-9)
(-13836.9711337571, -5.2223819027556e-9)
(28372.85706792574, -1.24213790730051e-9)
(-29089.122067366698, -1.18172199785001e-9)
(-29936.60744213758, -1.11576323518388e-9)
(37695.442374755075, -7.03726999829967e-10)
(-23156.973560454706, -1.86469128439758e-9)
(30067.8227785119, -1.10604640353812e-9)
(18204.03971300229, -3.01735894283166e-9)
(-30784.099472280774, -1.05517606679392e-9)
(-21462.186450539717, -2.17080237132013e-9)
(-19767.47352211211, -2.55895799826681e-9)
(-41801.9150374336, -5.72257276955845e-10)
(12274.173307205472, -6.63681814917335e-9)
(30915.31578467033, -1.04623816816246e-9)
(-31631.597614059938, -9.99392713138864e-10)
(31762.81482398482, -9.91152673962698e-10)
(38542.97593127045, -6.73118943310333e-10)
(-40954.37065463332, -5.96187449881745e-10)
(35152.86245658479, -8.09206204108099e-10)
(29220.33633982265, -1.17113303715945e-9)
(19898.665300500757, -2.5253281641932e-9)
(-26546.712374574054, -1.41890334782319e-9)
(-20614.81954100241, -2.35292363828522e-9)
(-35021.64229217217, -8.15281345135363e-10)
(39390.51256937884, -6.44465119800202e-10)
(15662.345886645018, -4.07608115537644e-9)
(14815.203008268796, -4.55552819924657e-9)
(13121.103314204072, -5.80774176311859e-9)
(-40106.82883416693, -6.21650577669032e-10)
(41933.13908167115, -5.68681345571628e-10)
(21593.384617630203, -2.14450452279427e-9)
(-17225.59258496116, -3.36986151711332e-9)
(-24004.389916207994, -1.73536260726775e-9)
(-37564.22050505195, -7.08652097474588e-10)
(-16378.365298081748, -3.72749647410585e-9)
(-35869.16461583956, -7.77210137690153e-10)
(13968.118245281725, -5.12478151527373e-9)
(-12143.05290263435, -6.78091102495798e-9)
(-33326.61033842568, -9.0032071522487e-10)
(-12989.968205495828, -5.92558601529429e-9)
Intervals of increase and decrease of the function:Let's find intervals where the function increases and decreases, as well as minima and maxima of the function, for this let's look how the function behaves itself in the extremas and at the slightest deviation from:
The function has no minima
The function has no maxima
Decreasing at the entire real axis