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Integrate[(a + b*x + c*x^2 + d*x^3)/(1 + x^4), x] ==
(-2*(Sqrt[2]*a + 2*b + Sqrt[2]*c)* ArcTan[1 - Sqrt[2]*x] + 2*(Sqrt[2]*a - 2*b + Sqrt[2]*c)* ArcTan[1 + Sqrt[2]*x] + Sqrt[2]*(-a + c)* Log[1 - Sqrt[2]*x + x^2] + Sqrt[2]*(a - c)* Log[1 + Sqrt[2]*x + x^2] + 2*d*Log[1 + x^4])/8



                       2      3
          a + b x + c x  + d x
Integrate[---------------------, x] == 
                      4
                 1 + x

(-2 (Sqrt[2] a + 2 b + Sqrt[2] c) 
 
     ArcTan[1 - Sqrt[2] x] + 
 
    2 (Sqrt[2] a - 2 b + Sqrt[2] c) 
 
     ArcTan[1 + Sqrt[2] x] + 
 
                                          2
    Sqrt[2] (-a + c) Log[1 - Sqrt[2] x + x ] + 
 
                                         2
    Sqrt[2] (a - c) Log[1 + Sqrt[2] x + x ] + 
 
                 4
    2 d Log[1 + x ]) / 8

Time to compute: 0.28 second

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