3.899 \(\int \frac{x}{\left (1-x^4\right )^{3/2}} \, dx\)

Optimal. Leaf size=18 \[ \frac{x^2}{2 \sqrt{1-x^4}} \]

[Out]

x^2/(2*Sqrt[1 - x^4])

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Rubi [A]  time = 0.0127446, antiderivative size = 18, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077 \[ \frac{x^2}{2 \sqrt{1-x^4}} \]

Antiderivative was successfully verified.

[In]  Int[x/(1 - x^4)^(3/2),x]

[Out]

x^2/(2*Sqrt[1 - x^4])

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Rubi in Sympy [A]  time = 2.26679, size = 12, normalized size = 0.67 \[ \frac{x^{2}}{2 \sqrt{- x^{4} + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x/(-x**4+1)**(3/2),x)

[Out]

x**2/(2*sqrt(-x**4 + 1))

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Mathematica [A]  time = 0.0112336, size = 18, normalized size = 1. \[ \frac{x^2}{2 \sqrt{1-x^4}} \]

Antiderivative was successfully verified.

[In]  Integrate[x/(1 - x^4)^(3/2),x]

[Out]

x^2/(2*Sqrt[1 - x^4])

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Maple [A]  time = 0.005, size = 26, normalized size = 1.4 \[ -{\frac{ \left ( -1+x \right ) \left ( 1+x \right ) \left ({x}^{2}+1 \right ){x}^{2}}{2} \left ( -{x}^{4}+1 \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x/(-x^4+1)^(3/2),x)

[Out]

-1/2*(-1+x)*(1+x)*(x^2+1)*x^2/(-x^4+1)^(3/2)

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Maxima [A]  time = 1.43034, size = 19, normalized size = 1.06 \[ \frac{x^{2}}{2 \, \sqrt{-x^{4} + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/(-x^4 + 1)^(3/2),x, algorithm="maxima")

[Out]

1/2*x^2/sqrt(-x^4 + 1)

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Fricas [A]  time = 0.26828, size = 50, normalized size = 2.78 \[ -\frac{\sqrt{-x^{4} + 1} x^{2} - x^{2}}{2 \,{\left (x^{4} + \sqrt{-x^{4} + 1} - 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/(-x^4 + 1)^(3/2),x, algorithm="fricas")

[Out]

-1/2*(sqrt(-x^4 + 1)*x^2 - x^2)/(x^4 + sqrt(-x^4 + 1) - 1)

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Sympy [A]  time = 1.66813, size = 32, normalized size = 1.78 \[ \begin{cases} - \frac{i x^{2}}{2 \sqrt{x^{4} - 1}} & \text{for}\: \left |{x^{4}}\right | > 1 \\\frac{x^{2}}{2 \sqrt{- x^{4} + 1}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/(-x**4+1)**(3/2),x)

[Out]

Piecewise((-I*x**2/(2*sqrt(x**4 - 1)), Abs(x**4) > 1), (x**2/(2*sqrt(-x**4 + 1))
, True))

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GIAC/XCAS [A]  time = 0.218817, size = 28, normalized size = 1.56 \[ -\frac{\sqrt{-x^{4} + 1} x^{2}}{2 \,{\left (x^{4} - 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/(-x^4 + 1)^(3/2),x, algorithm="giac")

[Out]

-1/2*sqrt(-x^4 + 1)*x^2/(x^4 - 1)