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

Time to compute: 0.09 second

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