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$$\frac{1 - \frac{1}{x^{2} + 1}}{x^{4}} - \frac{4 \left(x - \operatorname{atan}{\left(x \right)}\right)}{x^{5}} = 0$$
Solve this equationThe roots of this equation
$$x_{1} = -9635.16086967068$$
$$x_{2} = -7672.62881789392$$
$$x_{3} = -6582.35921146526$$
$$x_{4} = -5492.1222907781$$
$$x_{5} = -5710.16615325996$$
$$x_{6} = -2657.95348889226$$
$$x_{7} = 9014.74634499065$$
$$x_{8} = -1786.39023902939$$
$$x_{9} = -8326.80089666195$$
$$x_{10} = 7924.45270508001$$
$$x_{11} = -8544.85957816161$$
$$x_{12} = 1384.66529443179$$
$$x_{13} = 4653.73530019236$$
$$x_{14} = -8108.74284034062$$
$$x_{15} = -6800.41102478015$$
$$x_{16} = -2004.22107401941$$
$$x_{17} = -5274.08063017358$$
$$x_{18} = 6398.07480973781$$
$$x_{19} = 6834.1775567283$$
$$x_{20} = 2255.84284777311$$
$$x_{21} = -7018.46399104355$$
$$x_{22} = 7488.33997130571$$
$$x_{23} = 9232.80671223416$$
$$x_{24} = 4435.70530122338$$
$$x_{25} = 2037.95787900293$$
$$x_{26} = -10943.5355971552$$
$$x_{27} = -3311.87763474688$$
$$x_{28} = -3093.88923759967$$
$$x_{29} = -10071.2844248865$$
$$x_{30} = 8796.68646966634$$
$$x_{31} = -9853.22245920811$$
$$x_{32} = -8980.97863348592$$
$$x_{33} = 1820.11848049166$$
$$x_{34} = 6180.02544917054$$
$$x_{35} = -9417.09968254908$$
$$x_{36} = 4217.67951297655$$
$$x_{37} = 1602.34377731744$$
$$x_{38} = 9450.86753715877$$
$$x_{39} = 10105.0524601947$$
$$x_{40} = 8578.6271240014$$
$$x_{41} = 8360.56834969763$$
$$x_{42} = -6146.25953865561$$
$$x_{43} = -7236.5180052863$$
$$x_{44} = -7454.57297490512$$
$$x_{45} = -5928.2119725539$$
$$x_{46} = 10977.303824696$$
$$x_{47} = 2909.66713500245$$
$$x_{48} = 6616.12555682936$$
$$x_{49} = 10323.1148306644$$
$$x_{50} = -4619.97201179297$$
$$x_{51} = -9199.03892662363$$
$$x_{52} = 10759.2405320859$$
$$x_{53} = -2440.01361836832$$
$$x_{54} = -6364.30867056898$$
$$x_{55} = 5743.93152489527$$
$$x_{56} = -3529.87628681345$$
$$x_{57} = -4838.00511805066$$
$$x_{58} = 3781.64352081394$$
$$x_{59} = -4401.94262668798$$
$$x_{60} = 2691.70445475524$$
$$x_{61} = 10541.1775279508$$
$$x_{62} = -8762.91883789065$$
$$x_{63} = -10507.4093905257$$
$$x_{64} = 7706.39594324275$$
$$x_{65} = 7052.23069237461$$
$$x_{66} = -2875.91346755289$$
$$x_{67} = 3345.63518005307$$
$$x_{68} = 8142.51019295217$$
$$x_{69} = 5307.84532286983$$
$$x_{70} = 3127.64505064611$$
$$x_{71} = 5961.97762857279$$
$$x_{72} = 3563.63525073288$$
$$x_{73} = 2473.76111677168$$
$$x_{74} = 5525.88734327328$$
$$x_{75} = -2222.09990170263$$
$$x_{76} = -10725.4723482245$$
$$x_{77} = -10289.3467426694$$
$$x_{78} = -4183.91755155239$$
$$x_{79} = -3965.89750630056$$
$$x_{80} = 9886.99043828033$$
$$x_{81} = -5056.04145920651$$
$$x_{82} = 9668.92878863151$$
$$x_{83} = -3747.8833805644$$
$$x_{84} = -1568.62798192089$$
$$x_{85} = 5089.8057442349$$
$$x_{86} = 3999.65863276751$$
$$x_{87} = 7270.28486085012$$
$$x_{88} = -7890.68546137018$$
$$x_{89} = 4871.76893866043$$
$$x_{90} = -1350.96857451446$$
The values of the extrema at the points:
(-9635.160869670677, -1.11776958145545e-12)
(-7672.628817893916, -2.21349487302131e-12)
(-6582.359211465264, -3.50551332472551e-12)
(-5492.122290778105, -6.03469308883257e-12)
(-5710.16615325996, -5.369505499223e-12)
(-2657.9534888922576, -5.3223226078453e-11)
(9014.746344990652, 1.36478358583227e-12)
(-1786.3902390293924, -1.75262529830896e-10)
(-8326.80089666195, -1.73174335887348e-12)
(7924.452705080006, 2.00912093288887e-12)
(-8544.859578161615, -1.60252706153691e-12)
(1384.6652944317864, 3.76246832259478e-10)
(4653.735300192361, 9.9185542641628e-12)
(-8108.742840340623, -1.87523321172825e-12)
(-6800.411024780154, -3.17902683397301e-12)
(-2004.2210740194118, -1.24114558285075e-10)
(-5274.080630173582, -6.8144428561162e-12)
(6398.074809737811, 3.81720454449933e-12)
(6834.1775567282975, 3.13214171811575e-12)
(2255.842847773106, 8.70504551674902e-11)
(-7018.463991043547, -2.89185538223141e-12)
(7488.339971305707, 2.38096076334153e-12)
(9232.806712234158, 1.27035435320526e-12)
(4435.705301223378, 1.14540254982792e-11)
(2037.9578790029336, 1.18053705543013e-10)
(-10943.535597155234, -7.62894874802996e-13)
(-3311.8776347468806, -2.75151041710324e-11)
(-3093.889237599666, -3.37493341432853e-11)
(-10071.28442488646, -9.78763306996241e-13)
(8796.686469666336, 1.46880826811012e-12)
(-9853.222459208115, -1.04519157327288e-12)
(-8980.97863348592, -1.38023504438876e-12)
(1820.1184804916636, 1.65701403634991e-10)
(6180.0254491705355, 4.23563785298216e-12)
(-9417.099682549078, -1.19722576549115e-12)
(4217.6795129765505, 1.33234752979403e-11)
(1602.343777317439, 2.42832673982205e-10)
(9450.867537158769, 1.18443924224715e-12)
(10105.052460194725, 9.68984370258066e-13)
(8578.627124001401, 1.58367879911771e-12)
(8360.56834969763, 1.71084631052185e-12)
(-6146.259538655606, -4.30582444498396e-12)
(-7236.518005286297, -2.63825510263823e-12)
(-7454.572974905121, -2.41346038082211e-12)
(-5928.211972553896, -4.79859116263575e-12)
(10977.303824695986, 7.55876424564143e-13)
(2909.6671350024462, 4.05728584898987e-11)
(6616.125556829361, 3.45211831990863e-12)
(10323.114830664428, 9.08869732367151e-13)
(-4619.972011792972, -1.01375801500692e-11)
(-9199.038926623629, -1.28439459797409e-12)
(10759.240532085852, 8.02771127626173e-13)
(-2440.0136183683203, -6.87929048602917e-11)
(-6364.308670568983, -3.87827948720368e-12)
(5743.931524895269, 5.27537659773035e-12)
(-3529.87628681345, -2.27262759870871e-11)
(-4838.005118050664, -8.82795313995201e-12)
(3781.6435208139396, 1.84832286538086e-11)
(-4401.942626687983, -1.17195756557215e-11)
(2691.7044547552405, 5.12465254472768e-11)
(10541.177527950755, 8.53626521964498e-13)
(-8762.918837890646, -1.48585278088761e-12)
(-10507.40939052567, -8.61882601343844e-13)
(7706.395943242754, 2.18452751894021e-12)
(7052.23069237461, 2.85051763938332e-12)
(-2875.913467552889, -4.20179961154258e-11)
(3345.6351800530674, 2.66907231397105e-11)
(8142.510192952174, 1.85200128943106e-12)
(5307.845322869832, 6.68523527959022e-12)
(3127.6450506461083, 3.26685225521898e-11)
(5961.977628572791, 4.71752866551541e-12)
(3563.6352507328766, 2.20865978104899e-11)
(2473.7611167716786, 6.60162631030637e-11)
(5525.887343273277, 5.92475591907681e-12)
(-2222.0999017026343, -9.10756411810749e-11)
(-10725.472348224523, -8.10377010365927e-13)
(-10289.346742669373, -9.17846994344074e-13)
(-4183.917551552389, -1.36485838112269e-11)
(-3965.8975063005573, -1.60252023237651e-11)
(9886.990438280332, 1.03451944451266e-12)
(-5056.041459206507, -7.73451687364544e-12)
(9668.928788631507, 1.10609991563322e-12)
(-3747.883380564403, -1.89871493516154e-11)
(-1568.6279819208949, -2.58824384730569e-10)
(5089.805744234899, 7.5816257601581e-12)
(3999.658632767512, 1.56228640389328e-11)
(7270.284860850115, 2.60166804735459e-12)
(-7890.685461370175, -2.03502312117089e-12)
(4871.768938660432, 8.64569519810368e-12)
(-1350.9685745144648, -4.05097200287e-10)
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