From: samuel.thomas.blake@gmail.com   
      
   I was pleased to see the following concise result from Rubi   
      
   In[8203]:= Int[(2 u + u^2)/(2 + 4 u + 2 u^2 + u^4), u]   
   Out[8203]= ArcTan[u^2/(Sqrt[2] (1 + u))]/Sqrt[2]   
      
   My implementation of Bronstein's Risch algorithm (from Symbolic Integration I)   
   returns two arctangents   
      
   In[8206]:= Risch[(2 u + u^2)/(2 + 4 u + 2 u^2 + u^4), u]   
   Out[8206]= (ArcTan[(-1 + u)/Sqrt[2]] - ArcTan[(2 + 2 u + (-1 + u)   
   u^2)/Sqrt[2]])/Sqrt[2]   
      
   and Mathematica returns the naive form   
      
   In[8207]:= Integrate[(2 u + u^2)/(2 + 4 u + 2 u^2 + u^4), u]   
   Out[8207]= 1/4 RootSum[   
     2 + 4 #1 + 2 #1^2 + #1^4 &, (2 Log[u - #1] #1 + Log[u - #1] #1^2)/(   
      1 + #1 + #1^3) &]   
      
   FriCAS also returns two arctangents   
      
   (8) -> integrate((2*u + u^2)/(2 + 4*u + 2*u^2 + u^4), u)   
      
                        +-+          3    2   
                (u - 1)\|2          u  - u  + 2 u + 2   
           atan(-----------) - atan(-----------------)   
                     2                      +-+   
                                           \|2   
      (8)  -------------------------------------------   
                                +-+   
                               \|2   
                                            Type: Union(Expression(Integer),...)   
      
   Perhaps there is still some room for improvement in the log to arctan   
   conversions in symbolic integrators...   
      
   Sam   
      
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