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

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

[Out]

1/(2*Sqrt[1 - x^4])

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Rubi [A]  time = 0.00808117, 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/(1 - x^4)^(3/2),x]

[Out]

1/(2*Sqrt[1 - x^4])

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Rubi in Sympy [A]  time = 1.92273, size = 10, normalized size = 0.67 \[ \frac{1}{2 \sqrt{- x^{4} + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/(2*sqrt(-x**4 + 1))

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Mathematica [A]  time = 0.00484006, size = 15, normalized size = 1. \[ \frac{1}{2 \sqrt{1-x^4}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^3/(1 - x^4)^(3/2),x]

[Out]

1/(2*Sqrt[1 - x^4])

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Maple [A]  time = 0.006, size = 23, normalized size = 1.5 \[ -{\frac{ \left ( -1+x \right ) \left ( 1+x \right ) \left ({x}^{2}+1 \right ) }{2} \left ( -{x}^{4}+1 \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^3/(-x^4+1)^(3/2),x)

[Out]

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

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Maxima [A]  time = 1.43415, 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/(-x^4 + 1)^(3/2),x, algorithm="maxima")

[Out]

1/2/sqrt(-x^4 + 1)

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Fricas [A]  time = 0.279419, 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/(-x^4 + 1)^(3/2),x, algorithm="fricas")

[Out]

1/2/sqrt(-x^4 + 1)

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Sympy [A]  time = 1.36861, size = 10, normalized size = 0.67 \[ \frac{1}{2 \sqrt{- x^{4} + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/(2*sqrt(-x**4 + 1))

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GIAC/XCAS [A]  time = 0.213182, 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/(-x^4 + 1)^(3/2),x, algorithm="giac")

[Out]

1/2/sqrt(-x^4 + 1)