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(-1)^(1/4)*(EllipticF[I*ArcSinh[(-1)^(1/4)*x], -1] - 2*EllipticPi[I, ArcSin[(-1)^(3/4)*x], -1])

Integrate[(1 + x^2)/((1 - x^2)*Sqrt[1 + x^4]), x] ==
(-1)^(1/4)*(EllipticF[I*ArcSinh[(-1)^(1/4)*x], -1] - 2*EllipticPi[I, ArcSin[(-1)^(3/4)*x], -1]) 

                      2
                 1 + x
Integrate[---------------------, x] == 
                2            4
          (1 - x ) Sqrt[1 + x ]

    1/4                          1/4
(-1)    (EllipticF[I ArcSinh[(-1)    x], -1] - 
 
                               3/4
    2 EllipticPi[I, ArcSin[(-1)    x], -1])
Time to compute: 0.10 second
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