3.879 \(\int \frac{1}{x^3 \sqrt{1-x^4}} \, dx\)

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

[Out]

-Sqrt[1 - x^4]/(2*x^2)

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Rubi [A]  time = 0.0151032, antiderivative size = 18, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067 \[ -\frac{\sqrt{1-x^4}}{2 x^2} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^3*Sqrt[1 - x^4]),x]

[Out]

-Sqrt[1 - x^4]/(2*x^2)

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Rubi in Sympy [A]  time = 2.53008, size = 14, normalized size = 0.78 \[ - \frac{\sqrt{- x^{4} + 1}}{2 x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(-x**4 + 1)/(2*x**2)

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Mathematica [A]  time = 0.0104622, size = 18, normalized size = 1. \[ -\frac{\sqrt{1-x^4}}{2 x^2} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^3*Sqrt[1 - x^4]),x]

[Out]

-Sqrt[1 - x^4]/(2*x^2)

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Maple [A]  time = 0.007, size = 26, normalized size = 1.4 \[{\frac{ \left ( -1+x \right ) \left ( 1+x \right ) \left ({x}^{2}+1 \right ) }{2\,{x}^{2}}{\frac{1}{\sqrt{-{x}^{4}+1}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^3/(-x^4+1)^(1/2),x)

[Out]

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

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Maxima [A]  time = 1.42646, size = 19, normalized size = 1.06 \[ -\frac{\sqrt{-x^{4} + 1}}{2 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*sqrt(-x^4 + 1)/x^2

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Fricas [A]  time = 0.237728, size = 50, normalized size = 2.78 \[ \frac{x^{4} + \sqrt{-x^{4} + 1} - 1}{2 \,{\left (\sqrt{-x^{4} + 1} x^{2} - x^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 1.87038, size = 34, normalized size = 1.89 \[ \begin{cases} - \frac{i \sqrt{x^{4} - 1}}{2 x^{2}} & \text{for}\: \left |{x^{4}}\right | > 1 \\- \frac{\sqrt{- x^{4} + 1}}{2 x^{2}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.213929, size = 12, normalized size = 0.67 \[ -\frac{1}{2} \, \sqrt{\frac{1}{x^{4}} - 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*sqrt(1/x^4 - 1)