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$$- \frac{\sin{\left(x \right)}}{x + 1} - \frac{\cos{\left(x \right)}}{\left(x + 1\right)^{2}} = 0$$
Solve this equationThe roots of this equation
$$x_{1} = 9.32825706323943$$
$$x_{2} = 43.9600588531378$$
$$x_{3} = -94.2370546693974$$
$$x_{4} = 2.88996969767843$$
$$x_{5} = 72.242978694986$$
$$x_{6} = 18.7990914357831$$
$$x_{7} = 69.1007741687956$$
$$x_{8} = -31.3830252979972$$
$$x_{9} = -25.0912562079058$$
$$x_{10} = -91.0950880256329$$
$$x_{11} = -28.2376364595748$$
$$x_{12} = -637.741738184573$$
$$x_{13} = -53.3879890840753$$
$$x_{14} = 40.8167952172419$$
$$x_{15} = 1313.18496827279$$
$$x_{16} = -81.6690132946536$$
$$x_{17} = 15.6479679638982$$
$$x_{18} = -72.24259540785$$
$$x_{19} = -2.57625015820118$$
$$x_{20} = 6.14411351301787$$
$$x_{21} = -12.4794779911025$$
$$x_{22} = 94.2372799036618$$
$$x_{23} = 84.8113487041494$$
$$x_{24} = 81.6693131963402$$
$$x_{25} = 56.5312876685112$$
$$x_{26} = -47.1022022669651$$
$$x_{27} = 100.521115065812$$
$$x_{28} = -100.520917114109$$
$$x_{29} = -62.815677356778$$
$$x_{30} = 37.673259943911$$
$$x_{31} = -97.378996929011$$
$$x_{32} = 47.1031041186137$$
$$x_{33} = 65.9585122146304$$
$$x_{34} = 78.5272426949571$$
$$x_{35} = -87.9530943542027$$
$$x_{36} = -15.6397620877646$$
$$x_{37} = 53.388691007263$$
$$x_{38} = -21.9434371567881$$
$$x_{39} = -59.6732185170696$$
$$x_{40} = 12.492390025579$$
$$x_{41} = 172.781841669816$$
$$x_{42} = 50.2459712046114$$
$$x_{43} = 31.38505790634$$
$$x_{44} = -75.3847808857452$$
$$x_{45} = 34.5293808983144$$
$$x_{46} = -37.6718497263809$$
$$x_{47} = -34.527701946778$$
$$x_{48} = 75.3851328811964$$
$$x_{49} = -6.08916120309943$$
$$x_{50} = -43.9590233567938$$
$$x_{51} = 59.6737803264459$$
$$x_{52} = -69.1003552230555$$
$$x_{53} = -40.8155939881502$$
$$x_{54} = 28.2401476526276$$
$$x_{55} = -9.30494468339504$$
$$x_{56} = -78.5269183093816$$
$$x_{57} = 21.9475985837942$$
$$x_{58} = -18.7934144113698$$
$$x_{59} = -56.5306616416093$$
$$x_{60} = 91.0953290668266$$
$$x_{61} = 87.9533529268738$$
$$x_{62} = 197.915309953386$$
$$x_{63} = -84.8110706151124$$
$$x_{64} = 25.0944376288815$$
$$x_{65} = 62.8161843480611$$
$$x_{66} = -50.2451786914948$$
$$x_{67} = -65.9580523911179$$
$$x_{68} = 97.379207861883$$
The values of the extrema at the points:
(9.328257063239425, -0.0963710979823201)
(43.960058853137774, 0.022236464203186)
(-94.23705466939735, -0.010724732692878)
(2.8899696976784344, -0.248976134877405)
(72.242978694986, -0.0136519134817116)
(18.79909143578314, 0.0504430691319447)
(69.10077416879557, 0.0142637264671467)
(-31.38302529799723, -0.0328953023371544)
(-25.091256207905772, -0.0414731225016059)
(-91.09508802563293, 0.0110987005999837)
(-28.237636459574798, 0.0366891865463047)
(-637.7417381845734, 0.00157049350907233)
(-53.387989084075315, 0.0190848682073296)
(40.81679521724192, -0.023907001519389)
(1313.1849682727866, 0.000760927673528925)
(-81.66901329465364, -0.0123953812433342)
(15.647967963898166, -0.0599593189797558)
(-72.24259540785, 0.0140351638863266)
(-2.5762501582011796, 0.535705052303484)
(6.1441135130178655, 0.138623930394573)
(-12.479477991102517, -0.0867833198945747)
(94.23727990366179, 0.0104995111118831)
(84.81134870414938, -0.0116526790492257)
(81.66931319634023, 0.0120955020439642)
(56.53128766851124, 0.0173792211238612)
(-47.10220226696507, 0.0216858368023364)
(100.52111506581193, 0.00984968979094353)
(-100.52091711410945, -0.0100476316966419)
(-62.815677356778, -0.0161750096209984)
(37.673259943911006, 0.0258490197028825)
(-97.37899692901101, 0.0103751461271118)
(47.10310411861372, -0.0207841885412821)
(65.95851221463039, -0.0149329557083856)
(78.52724269495707, -0.0125733134820883)
(-87.95309435420273, -0.0114996928375307)
(-15.63976208776456, 0.0681483206400774)
(53.388691007263, -0.0183830682189117)
(-21.94343715678808, 0.0476933188520339)
(-59.673218517069586, 0.0170410762454831)
(12.492390025578958, 0.0739131230459364)
(172.781841669816, -0.0057542458670116)
(50.24597120461141, 0.0195100148956696)
(31.385057906339963, 0.0308637274812354)
(-75.38478088574516, -0.0134423955413013)
(34.5293808983144, -0.0281345781753277)
(-37.67184972638089, -0.0272587398500595)
(-34.52770194677802, 0.0298128246468963)
(75.38513288119637, 0.0130904310684593)
(-6.089161203099427, -0.192809042427521)
(-43.95902335679378, -0.0232716924030311)
(59.67378032644585, -0.0164793457895915)
(-69.1003552230555, -0.0146826283229769)
(-40.81559398815024, 0.0251078697468112)
(28.240147652627645, -0.0341795711715136)
(-9.304944683395044, 0.119546681963348)
(-78.5269183093816, 0.0128976727485698)
(21.947598583794207, -0.0435362264748061)
(-18.793414411369753, -0.0561120230339157)
(-56.53066164160934, -0.0180051500304447)
(91.09532906682657, -0.0108576739325778)
(87.9533529268738, 0.0112411368826843)
(197.91530995338616, -0.00502720159537522)
(-84.81107061511238, 0.0119307487512748)
(25.094437628881476, 0.0382942342355763)
(62.81618434806106, 0.0156680826074814)
(-50.245178691494786, -0.0203023709567303)
(-65.9580523911179, 0.0153927263543733)
(97.37920786188297, -0.0101642243790071)
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:
$$x_{1} = 9.32825706323943$$
$$x_{2} = -94.2370546693974$$
$$x_{3} = 2.88996969767843$$
$$x_{4} = 72.242978694986$$
$$x_{5} = -31.3830252979972$$
$$x_{6} = -25.0912562079058$$
$$x_{7} = 40.8167952172419$$
$$x_{8} = -81.6690132946536$$
$$x_{9} = 15.6479679638982$$
$$x_{10} = -12.4794779911025$$
$$x_{11} = 84.8113487041494$$
$$x_{12} = -100.520917114109$$
$$x_{13} = -62.815677356778$$
$$x_{14} = 47.1031041186137$$
$$x_{15} = 65.9585122146304$$
$$x_{16} = 78.5272426949571$$
$$x_{17} = -87.9530943542027$$
$$x_{18} = 53.388691007263$$
$$x_{19} = 172.781841669816$$
$$x_{20} = -75.3847808857452$$
$$x_{21} = 34.5293808983144$$
$$x_{22} = -37.6718497263809$$
$$x_{23} = -6.08916120309943$$
$$x_{24} = -43.9590233567938$$
$$x_{25} = 59.6737803264459$$
$$x_{26} = -69.1003552230555$$
$$x_{27} = 28.2401476526276$$
$$x_{28} = 21.9475985837942$$
$$x_{29} = -18.7934144113698$$
$$x_{30} = -56.5306616416093$$
$$x_{31} = 91.0953290668266$$
$$x_{32} = 197.915309953386$$
$$x_{33} = -50.2451786914948$$
$$x_{34} = 97.379207861883$$
Maxima of the function at points:
$$x_{34} = 43.9600588531378$$
$$x_{34} = 18.7990914357831$$
$$x_{34} = 69.1007741687956$$
$$x_{34} = -91.0950880256329$$
$$x_{34} = -28.2376364595748$$
$$x_{34} = -637.741738184573$$
$$x_{34} = -53.3879890840753$$
$$x_{34} = 1313.18496827279$$
$$x_{34} = -72.24259540785$$
$$x_{34} = -2.57625015820118$$
$$x_{34} = 6.14411351301787$$
$$x_{34} = 94.2372799036618$$
$$x_{34} = 81.6693131963402$$
$$x_{34} = 56.5312876685112$$
$$x_{34} = -47.1022022669651$$
$$x_{34} = 100.521115065812$$
$$x_{34} = 37.673259943911$$
$$x_{34} = -97.378996929011$$
$$x_{34} = -15.6397620877646$$
$$x_{34} = -21.9434371567881$$
$$x_{34} = -59.6732185170696$$
$$x_{34} = 12.492390025579$$
$$x_{34} = 50.2459712046114$$
$$x_{34} = 31.38505790634$$
$$x_{34} = -34.527701946778$$
$$x_{34} = 75.3851328811964$$
$$x_{34} = -40.8155939881502$$
$$x_{34} = -9.30494468339504$$
$$x_{34} = -78.5269183093816$$
$$x_{34} = 87.9533529268738$$
$$x_{34} = -84.8110706151124$$
$$x_{34} = 25.0944376288815$$
$$x_{34} = 62.8161843480611$$
$$x_{34} = -65.9580523911179$$
Decreasing at intervals
$$\left[197.915309953386, \infty\right)$$
Increasing at intervals
$$\left(-\infty, -100.520917114109\right]$$