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

Optimal. Leaf size=15 \[ -\frac{1}{2} \sqrt{1-x^4} \]

[Out]

-Sqrt[1 - x^4]/2

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Rubi [A]  time = 0.00806581, antiderivative size = 15, 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 \[ -\frac{1}{2} \sqrt{1-x^4} \]

Antiderivative was successfully verified.

[In]  Int[x^3/Sqrt[1 - x^4],x]

[Out]

-Sqrt[1 - x^4]/2

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Rubi in Sympy [A]  time = 1.93221, size = 10, normalized size = 0.67 \[ - \frac{\sqrt{- x^{4} + 1}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(-x**4 + 1)/2

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Mathematica [A]  time = 0.0052842, size = 15, normalized size = 1. \[ -\frac{1}{2} \sqrt{1-x^4} \]

Antiderivative was successfully verified.

[In]  Integrate[x^3/Sqrt[1 - x^4],x]

[Out]

-Sqrt[1 - x^4]/2

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Maple [A]  time = 0.007, size = 23, normalized size = 1.5 \[{\frac{ \left ( -1+x \right ) \left ( 1+x \right ) \left ({x}^{2}+1 \right ) }{2}{\frac{1}{\sqrt{-{x}^{4}+1}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^3/(-x^4+1)^(1/2),x)

[Out]

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

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Maxima [A]  time = 1.44102, size = 15, normalized size = 1. \[ -\frac{1}{2} \, \sqrt{-x^{4} + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*sqrt(-x^4 + 1)

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Fricas [A]  time = 0.273306, size = 15, normalized size = 1. \[ -\frac{1}{2} \, \sqrt{-x^{4} + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*sqrt(-x^4 + 1)

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Sympy [A]  time = 0.388837, size = 10, normalized size = 0.67 \[ - \frac{\sqrt{- x^{4} + 1}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(-x**4 + 1)/2

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GIAC/XCAS [A]  time = 0.210913, size = 15, normalized size = 1. \[ -\frac{1}{2} \, \sqrt{-x^{4} + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*sqrt(-x^4 + 1)