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$$\cos{\left(x \right)} \operatorname{atan}{\left(x \right)} + \frac{\sin{\left(x \right)}}{x^{2} + 1} = 0$$
Solve this equationThe roots of this equation
$$x_{1} = -48.694958052076$$
$$x_{2} = -11.0011109024118$$
$$x_{3} = -7.8649958747173$$
$$x_{4} = 26.7044507808002$$
$$x_{5} = 29.8458596412272$$
$$x_{6} = 23.5631212086715$$
$$x_{7} = -89.5354705986824$$
$$x_{8} = 0$$
$$x_{9} = -26.7044507808002$$
$$x_{10} = -92.6770579048997$$
$$x_{11} = -42.4118599365484$$
$$x_{12} = 95.8186457300403$$
$$x_{13} = 51.8365185642119$$
$$x_{14} = 58.1196545885159$$
$$x_{15} = 73.827545153735$$
$$x_{16} = 7.8649958747173$$
$$x_{17} = -51.8365185642119$$
$$x_{18} = 11.0011109024118$$
$$x_{19} = -70.6859632509207$$
$$x_{20} = -95.8186457300403$$
$$x_{21} = 42.4118599365484$$
$$x_{22} = 64.4028043800929$$
$$x_{23} = -54.9780844551589$$
$$x_{24} = 89.5354705986824$$
$$x_{25} = -1.79103397776014$$
$$x_{26} = -23.5631212086715$$
$$x_{27} = 67.5443828898684$$
$$x_{28} = -45.5534044627264$$
$$x_{29} = -4.74359618362999$$
$$x_{30} = 39.2703275097022$$
$$x_{31} = 20.4219240188353$$
$$x_{32} = 48.694958052076$$
$$x_{33} = 92.6770579048997$$
$$x_{34} = 61.2612281128569$$
$$x_{35} = 70.6859632509207$$
$$x_{36} = 83.2522978663322$$
$$x_{37} = 36.1288116046131$$
$$x_{38} = -29.8458596412272$$
$$x_{39} = -14.1404840184881$$
$$x_{40} = 80.1107126426188$$
$$x_{41} = -61.2612281128569$$
$$x_{42} = -80.1107126426188$$
$$x_{43} = -17.2809654364073$$
$$x_{44} = 54.9780844551589$$
$$x_{45} = -58.1196545885159$$
$$x_{46} = 98.9602340090719$$
$$x_{47} = -86.3938838884988$$
$$x_{48} = -64.4028043800929$$
$$x_{49} = 17.2809654364073$$
$$x_{50} = 32.987318864864$$
$$x_{51} = -76.9691283508747$$
$$x_{52} = 14.1404840184881$$
$$x_{53} = -39.2703275097022$$
$$x_{54} = -32.987318864864$$
$$x_{55} = -73.827545153735$$
$$x_{56} = -98.9602340090719$$
$$x_{57} = -83.2522978663322$$
$$x_{58} = 1.79103397776014$$
$$x_{59} = 86.3938838884988$$
$$x_{60} = -36.1288116046131$$
$$x_{61} = -20.4219240188353$$
$$x_{62} = -67.5443828898684$$
$$x_{63} = 45.5534044627264$$
$$x_{64} = 76.9691283508747$$
$$x_{65} = 4.74359618362999$$
The values of the extrema at the points:
(-48.694958052076046, -1.55026314860543)
(-11.001110902411808, -1.4801228582299)
(-7.864995874717303, 1.44424164730901)
(26.704450780800197, 1.53336623503182)
(29.84585964122719, -1.53730296241164)
(23.56312120867149, -1.52838152134177)
(-89.53547059868244, 1.55962802821361)
(0, 0)
(-26.704450780800197, 1.53336623503182)
(-92.6770579048997, -1.5600065842339)
(-42.41185993654844, -1.54722228447912)
(95.8186457300403, 1.56036031976415)
(51.83651856421186, 1.55150725579965)
(58.119654588515886, 1.55359211279627)
(73.82754515373497, -1.5572520643401)
(7.864995874717303, 1.44424164730901)
(-51.83651856421186, 1.55150725579965)
(11.001110902411808, -1.4801228582299)
(-70.68596325092066, 1.55665017729342)
(-95.8186457300403, 1.56036031976415)
(42.41185993654844, -1.54722228447912)
(64.4028043800929, 1.5552702816364)
(-54.97808445515894, -1.55260923119595)
(89.53547059868244, 1.55962802821361)
(-1.7910339777601358, 1.03593336473382)
(-23.56312120867149, -1.52838152134177)
(67.54438288986843, -1.55599231281641)
(-45.553404462726384, 1.54884752107669)
(-4.7435961836299905, -1.36236430357899)
(39.270327509702184, 1.54533717389503)
(20.42192401883534, 1.52186654514573)
(48.694958052076046, -1.55026314860543)
(92.6770579048997, -1.5600065842339)
(61.26122811285691, -1.5544742153389)
(70.68596325092066, 1.55665017729342)
(83.25229786633224, 1.55878521728754)
(36.128811604613105, -1.54312446153498)
(-29.84585964122719, -1.53730296241164)
(-14.140484018488104, 1.50018667629801)
(80.11071264261882, -1.55831424223611)
(-61.26122811285691, -1.5544742153389)
(-80.11071264261882, -1.55831424223611)
(-17.280965436407328, -1.51298997049872)
(54.97808445515894, -1.55260923119595)
(-58.119654588515886, 1.55359211279627)
(98.96023400907185, -1.56069159831347)
(-86.39388388849885, -1.55922194448498)
(-64.4028043800929, 1.5552702816364)
(17.280965436407328, -1.51298997049872)
(32.98731886486398, 1.54049065473337)
(-76.96912835087466, 1.55780482663327)
(14.140484018488104, 1.50018667629801)
(-39.270327509702184, 1.54533717389503)
(-32.98731886486398, 1.54049065473337)
(-73.82754515373497, -1.5572520643401)
(-98.96023400907185, -1.56069159831347)
(-83.25229786633224, 1.55878521728754)
(1.7910339777601358, 1.03593336473382)
(86.39388388849885, -1.55922194448498)
(-36.128811604613105, -1.54312446153498)
(-20.42192401883534, 1.52186654514573)
(-67.54438288986843, -1.55599231281641)
(45.553404462726384, 1.54884752107669)
(76.96912835087466, 1.55780482663327)
(4.7435961836299905, -1.36236430357899)
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:
Minima of the function at points:
$$x_{1} = -48.694958052076$$
$$x_{2} = -11.0011109024118$$
$$x_{3} = 29.8458596412272$$
$$x_{4} = 23.5631212086715$$
$$x_{5} = 0$$
$$x_{6} = -92.6770579048997$$
$$x_{7} = -42.4118599365484$$
$$x_{8} = 73.827545153735$$
$$x_{9} = 11.0011109024118$$
$$x_{10} = 42.4118599365484$$
$$x_{11} = -54.9780844551589$$
$$x_{12} = -23.5631212086715$$
$$x_{13} = 67.5443828898684$$
$$x_{14} = -4.74359618362999$$
$$x_{15} = 48.694958052076$$
$$x_{16} = 92.6770579048997$$
$$x_{17} = 61.2612281128569$$
$$x_{18} = 36.1288116046131$$
$$x_{19} = -29.8458596412272$$
$$x_{20} = 80.1107126426188$$
$$x_{21} = -61.2612281128569$$
$$x_{22} = -80.1107126426188$$
$$x_{23} = -17.2809654364073$$
$$x_{24} = 54.9780844551589$$
$$x_{25} = 98.9602340090719$$
$$x_{26} = -86.3938838884988$$
$$x_{27} = 17.2809654364073$$
$$x_{28} = -73.827545153735$$
$$x_{29} = -98.9602340090719$$
$$x_{30} = 86.3938838884988$$
$$x_{31} = -36.1288116046131$$
$$x_{32} = -67.5443828898684$$
$$x_{33} = 4.74359618362999$$
Maxima of the function at points:
$$x_{33} = -7.8649958747173$$
$$x_{33} = 26.7044507808002$$
$$x_{33} = -89.5354705986824$$
$$x_{33} = -26.7044507808002$$
$$x_{33} = 95.8186457300403$$
$$x_{33} = 51.8365185642119$$
$$x_{33} = 58.1196545885159$$
$$x_{33} = 7.8649958747173$$
$$x_{33} = -51.8365185642119$$
$$x_{33} = -70.6859632509207$$
$$x_{33} = -95.8186457300403$$
$$x_{33} = 64.4028043800929$$
$$x_{33} = 89.5354705986824$$
$$x_{33} = -1.79103397776014$$
$$x_{33} = -45.5534044627264$$
$$x_{33} = 39.2703275097022$$
$$x_{33} = 20.4219240188353$$
$$x_{33} = 70.6859632509207$$
$$x_{33} = 83.2522978663322$$
$$x_{33} = -14.1404840184881$$
$$x_{33} = -58.1196545885159$$
$$x_{33} = -64.4028043800929$$
$$x_{33} = 32.987318864864$$
$$x_{33} = -76.9691283508747$$
$$x_{33} = 14.1404840184881$$
$$x_{33} = -39.2703275097022$$
$$x_{33} = -32.987318864864$$
$$x_{33} = -83.2522978663322$$
$$x_{33} = 1.79103397776014$$
$$x_{33} = -20.4219240188353$$
$$x_{33} = 45.5534044627264$$
$$x_{33} = 76.9691283508747$$
Decreasing at intervals
$$\left[98.9602340090719, \infty\right)$$
Increasing at intervals
$$\left(-\infty, -98.9602340090719\right]$$