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

Optimal. Leaf size=25 \[ \frac{1}{3} \sqrt{1-x^4} x+\frac{2}{3} F\left (\left .\sin ^{-1}(x)\right |-1\right ) \]

[Out]

(x*Sqrt[1 - x^4])/3 + (2*EllipticF[ArcSin[x], -1])/3

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Rubi [A]  time = 0.0113578, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ \frac{1}{3} \sqrt{1-x^4} x+\frac{2}{3} F\left (\left .\sin ^{-1}(x)\right |-1\right ) \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[1 - x^4],x]

[Out]

(x*Sqrt[1 - x^4])/3 + (2*EllipticF[ArcSin[x], -1])/3

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Rubi in Sympy [A]  time = 1.18436, size = 20, normalized size = 0.8 \[ \frac{x \sqrt{- x^{4} + 1}}{3} + \frac{2 F\left (\operatorname{asin}{\left (x \right )}\middle | -1\right )}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x*sqrt(-x**4 + 1)/3 + 2*elliptic_f(asin(x), -1)/3

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Mathematica [A]  time = 0.0306921, size = 39, normalized size = 1.56 \[ \frac{-x^5+2 \sqrt{1-x^4} F\left (\left .\sin ^{-1}(x)\right |-1\right )+x}{3 \sqrt{1-x^4}} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[1 - x^4],x]

[Out]

(x - x^5 + 2*Sqrt[1 - x^4]*EllipticF[ArcSin[x], -1])/(3*Sqrt[1 - x^4])

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Maple [B]  time = 0.008, size = 45, normalized size = 1.8 \[{\frac{x}{3}\sqrt{-{x}^{4}+1}}+{\frac{2\,{\it EllipticF} \left ( x,i \right ) }{3}\sqrt{-{x}^{2}+1}\sqrt{{x}^{2}+1}{\frac{1}{\sqrt{-{x}^{4}+1}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((-x^4+1)^(1/2),x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \sqrt{-x^{4} + 1}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(sqrt(-x^4 + 1), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\sqrt{-x^{4} + 1}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(sqrt(-x^4 + 1), x)

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Sympy [A]  time = 1.76051, size = 31, normalized size = 1.24 \[ \frac{x \Gamma \left (\frac{1}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{1}{4} \\ \frac{5}{4} \end{matrix}\middle |{x^{4} e^{2 i \pi }} \right )}}{4 \Gamma \left (\frac{5}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-x**4+1)**(1/2),x)

[Out]

x*gamma(1/4)*hyper((-1/2, 1/4), (5/4,), x**4*exp_polar(2*I*pi))/(4*gamma(5/4))

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \sqrt{-x^{4} + 1}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(sqrt(-x^4 + 1), x)