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

Optimal. Leaf size=4 \[ F\left (\left .\sin ^{-1}(x)\right |-1\right ) \]

[Out]

EllipticF[ArcSin[x], -1]

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Rubi [A]  time = 0.00514693, antiderivative size = 4, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ F\left (\left .\sin ^{-1}(x)\right |-1\right ) \]

Antiderivative was successfully verified.

[In]  Int[1/Sqrt[1 - x^4],x]

[Out]

EllipticF[ArcSin[x], -1]

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Rubi in Sympy [A]  time = 0.171468, size = 5, normalized size = 1.25 \[ F\left (\operatorname{asin}{\left (x \right )}\middle | -1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

elliptic_f(asin(x), -1)

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Mathematica [A]  time = 0.0170205, size = 4, normalized size = 1. \[ F\left (\left .\sin ^{-1}(x)\right |-1\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[1/Sqrt[1 - x^4],x]

[Out]

EllipticF[ArcSin[x], -1]

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Maple [B]  time = 0.007, size = 31, normalized size = 7.8 \[{{\it EllipticF} \left ( x,i \right ) \sqrt{-{x}^{2}+1}\sqrt{{x}^{2}+1}{\frac{1}{\sqrt{-{x}^{4}+1}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(-x^4+1)^(1/2),x)

[Out]

(-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. \[ \int \frac{1}{\sqrt{-x^{4} + 1}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(1/sqrt(-x^4 + 1), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(1/sqrt(-x^4 + 1), x)

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Sympy [A]  time = 1.71671, size = 29, normalized size = 7.25 \[ \frac{x \Gamma \left (\frac{1}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{4}, \frac{1}{2} \\ \frac{5}{4} \end{matrix}\middle |{x^{4} e^{2 i \pi }} \right )}}{4 \Gamma \left (\frac{5}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(-x**4+1)**(1/2),x)

[Out]

x*gamma(1/4)*hyper((1/4, 1/2), (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. \[ \int \frac{1}{\sqrt{-x^{4} + 1}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(1/sqrt(-x^4 + 1), x)