clearvars,clc

syms x

F= (1/(x^2+1));

for i=1:20

  i

  F=diff(F)

  x=sym(1);

  eval(F)

end

finds that for integral n, the evaluation of the 4n+1 derivative of 1/(x^2+1) is always zero. For n midway between integers, or quarters of the way, I can't find a pattern. Below, the ith derivative appears below it, followed by that derivative's evaluation. When i = 4*n+3, with n an integer, the evaluation is zero, starting with i = 3, meaning n = 0:

i =

     1

F =

-(2*x)/(x^2 + 1)^2

ans =

-1/2

i =

     2

F =

(8*x^2)/(x^2 + 1)^3 - 2/(x^2 + 1)^2

ans =

1/2

i =

     3

F =

(24*x)/(x^2 + 1)^3 - (48*x^3)/(x^2 + 1)^4

ans =

0

i =

     4

F =

24/(x^2 + 1)^3 - (288*x^2)/(x^2 + 1)^4 + (384*x^4)/(x^2 + 1)^5

ans =

-3

i =

     5

F =

(3840*x^3)/(x^2 + 1)^5 - (720*x)/(x^2 + 1)^4 - (3840*x^5)/(x^2 + 1)^6

ans =

15

i =

     6

F =

(17280*x^2)/(x^2 + 1)^5 - 720/(x^2 + 1)^4 - (57600*x^4)/(x^2 + 1)^6 + (46080*x^6)/(x^2 + 1)^7

ans =

-45

i =

     7

F =

(40320*x)/(x^2 + 1)^5 - (403200*x^3)/(x^2 + 1)^6 + (967680*x^5)/(x^2 + 1)^7 - (645120*x^7)/(x^2 + 1)^8

ans =

0

i =

     8

F =

40320/(x^2 + 1)^5 - (1612800*x^2)/(x^2 + 1)^6 + (9676800*x^4)/(x^2 + 1)^7 - (18063360*x^6)/(x^2 + 1)^8 + (10321920*x^8)/(x^2 + 1)^9

ans =

1260

i =

     9

F =

(58060800*x^3)/(x^2 + 1)^7 - (3628800*x)/(x^2 + 1)^6 - (243855360*x^5)/(x^2 + 1)^8 + (371589120*x^7)/(x^2 + 1)^9 - (185794560*x^9)/(x^2 + 1)^10

ans =

-11340

i =

    10

F =

(217728000*x^2)/(x^2 + 1)^7 - 3628800/(x^2 + 1)^6 - (2032128000*x^4)/(x^2 + 1)^8 + (6502809600*x^6)/(x^2 + 1)^9 - (8360755200*x^8)/(x^2 + 1)^10 + (3715891200*x^10)/(x^2 + 1)^11

ans =

56700

i =

    11

F =

(479001600*x)/(x^2 + 1)^7 - (11176704000*x^3)/(x^2 + 1)^8 + (71530905600*x^5)/(x^2 + 1)^9 - (183936614400*x^7)/(x^2 + 1)^10 + (204374016000*x^9)/(x^2 + 1)^11 - (81749606400*x^11)/(x^2 + 1)^12

ans =

0

i =

    12

F =

479001600/(x^2 + 1)^7 - (40236134400*x^2)/(x^2 + 1)^8 + (536481792000*x^4)/(x^2 + 1)^9 - (2575112601600*x^6)/(x^2 + 1)^10 + (5518098432000*x^8)/(x^2 + 1)^11 - (5395474022400*x^10)/(x^2 + 1)^12 + (1961990553600*x^12)/(x^2 + 1)^13

ans =

-3742200

i =

    13

F =

(2789705318400*x^3)/(x^2 + 1)^9 - (87178291200*x)/(x^2 + 1)^8 - (25107347865600*x^5)/(x^2 + 1)^10 + (95647039488000*x^7)/(x^2 + 1)^11 - (175352905728000*x^9)/(x^2 + 1)^12 + (153035263180800*x^11)/(x^2 + 1)^13 - (51011754393600*x^13)/(x^2 + 1)^14

ans =

48648600

i =

    14

F =

(9763968614400*x^2)/(x^2 + 1)^9 - 87178291200/(x^2 + 1)^8 - (175751435059200*x^4)/(x^2 + 1)^10 + (1171676233728000*x^6)/(x^2 + 1)^11 - (3682411020288000*x^8)/(x^2 + 1)^12 + (5891857632460800*x^10)/(x^2 + 1)^13 - (4642069649817600*x^12)/(x^2 + 1)^14 + (1428329123020800*x^14)/(x^2 + 1)^15

ans =

-340540200

i =

    15

F =

(20922789888000*x)/(x^2 + 1)^9 - (878757175296000*x^3)/(x^2 + 1)^10 + (10545086103552000*x^5)/(x^2 + 1)^11 - (55236165304320000*x^7)/(x^2 + 1)^12 + (147296440811520000*x^9)/(x^2 + 1)^13 - (208893134241792000*x^11)/(x^2 + 1)^14 + (149974557917184000*x^13)/(x^2 + 1)^15 - (42849873690624000*x^15)/(x^2 + 1)^16

ans =

0

i =

    16

F =

20922789888000/(x^2 + 1)^9 - (3012881743872000*x^2)/(x^2 + 1)^10 + (70300574023680000*x^4)/(x^2 + 1)^11 - (618645051408384000*x^6)/(x^2 + 1)^12 + (2651335934607360000*x^8)/(x^2 + 1)^13 - (6127531937759232000*x^10)/(x^2 + 1)^14 + (7798677011693568000*x^12)/(x^2 + 1)^15 - (5141984842874880000*x^14)/(x^2 + 1)^16 + (1371195958099968000*x^16)/(x^2 + 1)^17

ans =

40864824000

i =

    17

F =

(341459930972160000*x^3)/(x^2 + 1)^11 - (6402373705728000*x)/(x^2 + 1)^10 - (5258482936971264000*x^5)/(x^2 + 1)^12 + (36058168710660096000*x^7)/(x^2 + 1)^13 - (130210053677383680000*x^9)/(x^2 + 1)^14 + (265155018397581312000*x^11)/(x^2 + 1)^15 - (305948098151055360000*x^13)/(x^2 + 1)^16 + (186482650301595648000*x^15)/(x^2 + 1)^17 - (46620662575398912000*x^17)/(x^2 + 1)^18

ans =

-694702008000

i =

    18

F =

(1152427267031040000*x^2)/(x^2 + 1)^11 - 6402373705728000/(x^2 + 1)^10 - (33804533166243840000*x^4)/(x^2 + 1)^12 + (378610771461931008000*x^6)/(x^2 + 1)^13 - (2109402869573615616000*x^8)/(x^2 + 1)^14 + (6562586705340137472000*x^10)/(x^2 + 1)^15 - (11931975827891159040000*x^12)/(x^2 + 1)^16 + (12587578895357706240000*x^14)/(x^2 + 1)^17 - (7132961374036033536000*x^16)/(x^2 + 1)^18 + (1678343852714360832000*x^18)/(x^2 + 1)^19

ans =

6252318072000

i =

    19

F =

(2432902008176640000*x)/(x^2 + 1)^11 - (160571532539658240000*x^3)/(x^2 + 1)^12 + (3082973424761438208000*x^5)/(x^2 + 1)^13 - (26719103014599131136000*x^7)/(x^2 + 1)^14 + (124689147401462611968000*x^9)/(x^2 + 1)^15 - (340061311094898032640000*x^11)/(x^2 + 1)^16 + (558049331027524976640000*x^13)/(x^2 + 1)^17 - (542105064426738548736000*x^15)/(x^2 + 1)^18 + (286996798814155702272000*x^17)/(x^2 + 1)^19 - (63777066403145711616000*x^19)/(x^2 + 1)^20

ans =

0

i =

    20

F =

2432902008176640000/(x^2 + 1)^11 - (535238441798860800000*x^2)/(x^2 + 1)^12 + (19268583904758988800000*x^4)/(x^2 + 1)^13 - (267191030145991311360000*x^6)/(x^2 + 1)^14 + (1870337211021939179520000*x^8)/(x^2 + 1)^15 - (7481348844087756718080000*x^10)/(x^2 + 1)^16 + (18136603258394561740800000*x^12)/(x^2 + 1)^17 - (27105253221336927436800000*x^14)/(x^2 + 1)^18 + (24394727899203234693120000*x^16)/(x^2 + 1)^19 - (12117642616597685207040000*x^18)/(x^2 + 1)^20 + (2551082656125828464640000*x^20)/(x^2 + 1)^21

ans =

-1187940433682048