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Integrate[(Sin[x]*Sqrt[1 - Cos[x]])/ (5 - Cos[x] - 2*Sqrt[1 - Cos[x]]), x] ==
(Sqrt[1 - Cos[x]]*Csc[x/2]^2* (48 + 4*ArcTan[(3 - Cos[x])/(2*Sqrt[3])]* Sqrt[3 - 3*Cos[x]] - 48*Cos[x] - 96*Sqrt[1 - Cos[x]]*Log[Sec[x/4]^2] + 6*Sqrt[1 - Cos[x]]*RootSum[ 4 - 8*#1 + 2*#1^2 + 2*#1^3 + #1^4 & , (-16*Log[1 - #1 + Tan[x/4]^2] - 2*Log[1 - #1 + Tan[x/4]^2]*#1 + 17*Log[1 - #1 + Tan[x/4]^2]*#1^2 + 8*Log[1 - #1 + Tan[x/4]^2]*#1^3)/ (-4 + 2*#1 + 3*#1^2 + 2*#1^3) & ] - 3*RootSum[1 - 12*#1^2 + 86*#1^4 - 12*#1^6 + #1^8 & , (2*ArcTan[Sin[x/2]/(Cos[x/2] - #1)] - I*Log[1 - 2*Cos[x/2]*#1 + #1^2] - 18*ArcTan[Sin[x/2]/(Cos[x/2] - #1)]*#1^2 + (9*I)*Log[1 - 2*Cos[x/2]*#1 + #1^2]*#1^2 - 18*ArcTan[Sin[x/2]/(Cos[x/2] - #1)]*#1^4 + (9*I)*Log[1 - 2*Cos[x/2]*#1 + #1^2]*#1^4 + 2*ArcTan[Sin[x/2]/(Cos[x/2] - #1)]*#1^6 - I*Log[1 - 2*Cos[x/2]*#1 + #1^2]*#1^6)/ (-3*#1 + 43*#1^3 - 9*#1^5 + #1^7) & ]* Sin[x/2] + 3*RootSum[1 - 12*#1^2 + 86*#1^4 - 12*#1^6 + #1^8 & , (2*ArcTan[Sin[x/2]/(Cos[x/2] - #1)] - I*Log[1 - 2*Cos[x/2]*#1 + #1^2] - 146*ArcTan[Sin[x/2]/(Cos[x/2] - #1)]*#1^2 + (73*I)*Log[1 - 2*Cos[x/2]*#1 + #1^2]*#1^2 - 146*ArcTan[Sin[x/2]/(Cos[x/2] - #1)]*#1^4 + (73*I)*Log[1 - 2*Cos[x/2]*#1 + #1^2]*#1^4 + 2*ArcTan[Sin[x/2]/(Cos[x/2] - #1)]*#1^6 - I*Log[1 - 2*Cos[x/2]*#1 + #1^2]*#1^6)/ (-3*#1 + 43*#1^3 - 9*#1^5 + #1^7) & ]* Sin[x/2]))/48



              Sin[x] Sqrt[1 - Cos[x]]
Integrate[-------------------------------, x] == 
          5 - Cos[x] - 2 Sqrt[1 - Cos[x]]

                      x 2
(Sqrt[1 - Cos[x]] Csc[-]  
                      2
 
                   3 - Cos[x]
    (48 + 4 ArcTan[----------] Sqrt[3 - 3 Cos[x]] - 
                   2 Sqrt[3]
 
                                              x 2
      48 Cos[x] - 96 Sqrt[1 - Cos[x]] Log[Sec[-] ] + 
                                              4
 
      6 Sqrt[1 - Cos[x]] 
 
                              2       3     4
       RootSum[4 - 8 #1 + 2 #1  + 2 #1  + #1  & , 
 
                              x 2
        (-16 Log[1 - #1 + Tan[-] ] - 
                              4
 
                               x 2
            2 Log[1 - #1 + Tan[-] ] #1 + 
                               4
 
                                x 2    2
            17 Log[1 - #1 + Tan[-] ] #1  + 
                                4
 
                               x 2    3
            8 Log[1 - #1 + Tan[-] ] #1 ) / 
                               4
 
                           2       3
          (-4 + 2 #1 + 3 #1  + 2 #1 ) & ] - 
 
                         2        4        6
      3 RootSum[1 - 12 #1  + 86 #1  - 12 #1  + 
 
                                  x
                              Sin[-]
            8                     2
          #1  & , (2 ArcTan[-----------] - 
                                x
                            Cos[-] - #1
                                2
 
                            x         2
            I Log[1 - 2 Cos[-] #1 + #1 ] - 
                            2
 
                            x
                        Sin[-]
                            2        2
            18 ArcTan[-----------] #1  + 
                          x
                      Cos[-] - #1
                          2
 
                                x         2    2
            (9 I) Log[1 - 2 Cos[-] #1 + #1 ] #1  - 
                                2
 
                            x
                        Sin[-]
                            2        4
            18 ArcTan[-----------] #1  + 
                          x
                      Cos[-] - #1
                          2
 
                                x         2    4
            (9 I) Log[1 - 2 Cos[-] #1 + #1 ] #1  + 
                                2
 
                           x
                       Sin[-]
                           2        6
            2 ArcTan[-----------] #1  - 
                         x
                     Cos[-] - #1
                         2
 
                            x         2    6
            I Log[1 - 2 Cos[-] #1 + #1 ] #1 ) / 
                            2
 
                        3       5     7          x
          (-3 #1 + 43 #1  - 9 #1  + #1 ) & ] Sin[-] + 
                                                 2
 
                         2        4        6
      3 RootSum[1 - 12 #1  + 86 #1  - 12 #1  + 
 
                                  x
                              Sin[-]
            8                     2
          #1  & , (2 ArcTan[-----------] - 
                                x
                            Cos[-] - #1
                                2
 
                            x         2
            I Log[1 - 2 Cos[-] #1 + #1 ] - 
                            2
 
                             x
                         Sin[-]
                             2        2
            146 ArcTan[-----------] #1  + 
                           x
                       Cos[-] - #1
                           2
 
                                 x         2    2
            (73 I) Log[1 - 2 Cos[-] #1 + #1 ] #1  - 
                                 2
 
                             x
                         Sin[-]
                             2        4
            146 ArcTan[-----------] #1  + 
                           x
                       Cos[-] - #1
                           2
 
                                 x         2    4
            (73 I) Log[1 - 2 Cos[-] #1 + #1 ] #1  + 
                                 2
 
                           x
                       Sin[-]
                           2        6
            2 ArcTan[-----------] #1  - 
                         x
                     Cos[-] - #1
                         2
 
                            x         2    6
            I Log[1 - 2 Cos[-] #1 + #1 ] #1 ) / 
                            2
 
                        3       5     7          x
          (-3 #1 + 43 #1  - 9 #1  + #1 ) & ] Sin[-]))\
                                                 2
 
   / 48

Time to compute: 2.20 second

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