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

Optimal. Leaf size=8 \[ \frac{1}{2} \sin ^{-1}\left (x^2\right ) \]

[Out]

ArcSin[x^2]/2

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Rubi [A]  time = 0.0126796, antiderivative size = 8, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ \frac{1}{2} \sin ^{-1}\left (x^2\right ) \]

Antiderivative was successfully verified.

[In]  Int[x/Sqrt[1 - x^4],x]

[Out]

ArcSin[x^2]/2

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Rubi in Sympy [A]  time = 2.49598, size = 5, normalized size = 0.62 \[ \frac{\operatorname{asin}{\left (x^{2} \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

asin(x**2)/2

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Mathematica [A]  time = 0.00867826, size = 8, normalized size = 1. \[ \frac{1}{2} \sin ^{-1}\left (x^2\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[x/Sqrt[1 - x^4],x]

[Out]

ArcSin[x^2]/2

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Maple [A]  time = 0.01, size = 7, normalized size = 0.9 \[{\frac{\arcsin \left ({x}^{2} \right ) }{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x/(-x^4+1)^(1/2),x)

[Out]

1/2*arcsin(x^2)

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Maxima [A]  time = 1.58955, size = 22, normalized size = 2.75 \[ -\frac{1}{2} \, \arctan \left (\frac{\sqrt{-x^{4} + 1}}{x^{2}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*arctan(sqrt(-x^4 + 1)/x^2)

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Fricas [A]  time = 0.245405, size = 24, normalized size = 3. \[ -\arctan \left (\frac{\sqrt{-x^{4} + 1} - 1}{x^{2}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-arctan((sqrt(-x^4 + 1) - 1)/x^2)

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Sympy [A]  time = 3.23801, size = 19, normalized size = 2.38 \[ \begin{cases} - \frac{i \operatorname{acosh}{\left (x^{2} \right )}}{2} & \text{for}\: \left |{x^{4}}\right | > 1 \\\frac{\operatorname{asin}{\left (x^{2} \right )}}{2} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/(-x**4+1)**(1/2),x)

[Out]

Piecewise((-I*acosh(x**2)/2, Abs(x**4) > 1), (asin(x**2)/2, True))

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GIAC/XCAS [A]  time = 0.218459, size = 8, normalized size = 1. \[ \frac{1}{2} \, \arcsin \left (x^{2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*arcsin(x^2)