3.889 \(\int \frac{x^2}{\sqrt{1-x^4}} \, dx\)

Optimal. Leaf size=11 \[ E\left (\left .\sin ^{-1}(x)\right |-1\right )-F\left (\left .\sin ^{-1}(x)\right |-1\right ) \]

[Out]

EllipticE[ArcSin[x], -1] - EllipticF[ArcSin[x], -1]

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Rubi [A]  time = 0.0480243, antiderivative size = 11, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267 \[ E\left (\left .\sin ^{-1}(x)\right |-1\right )-F\left (\left .\sin ^{-1}(x)\right |-1\right ) \]

Antiderivative was successfully verified.

[In]  Int[x^2/Sqrt[1 - x^4],x]

[Out]

EllipticE[ArcSin[x], -1] - EllipticF[ArcSin[x], -1]

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Rubi in Sympy [A]  time = 9.35612, size = 12, normalized size = 1.09 \[ E\left (\operatorname{asin}{\left (x \right )}\middle | -1\right ) - F\left (\operatorname{asin}{\left (x \right )}\middle | -1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

elliptic_e(asin(x), -1) - elliptic_f(asin(x), -1)

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Mathematica [A]  time = 0.0280366, size = 11, normalized size = 1. \[ E\left (\left .\sin ^{-1}(x)\right |-1\right )-F\left (\left .\sin ^{-1}(x)\right |-1\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[x^2/Sqrt[1 - x^4],x]

[Out]

EllipticE[ArcSin[x], -1] - EllipticF[ArcSin[x], -1]

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Maple [B]  time = 0.01, size = 39, normalized size = 3.6 \[ -{({\it EllipticF} \left ( x,i \right ) -{\it EllipticE} \left ( x,i \right ) )\sqrt{-{x}^{2}+1}\sqrt{{x}^{2}+1}{\frac{1}{\sqrt{-{x}^{4}+1}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2/(-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)-EllipticE(x,I))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{2}}{\sqrt{-x^{4} + 1}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^2/sqrt(-x^4 + 1),x, algorithm="maxima")

[Out]

integrate(x^2/sqrt(-x^4 + 1), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{x^{2}}{\sqrt{-x^{4} + 1}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^2/sqrt(-x^4 + 1),x, algorithm="fricas")

[Out]

integral(x^2/sqrt(-x^4 + 1), x)

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Sympy [A]  time = 1.77979, size = 31, normalized size = 2.82 \[ \frac{x^{3} \Gamma \left (\frac{3}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, \frac{3}{4} \\ \frac{7}{4} \end{matrix}\middle |{x^{4} e^{2 i \pi }} \right )}}{4 \Gamma \left (\frac{7}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x**3*gamma(3/4)*hyper((1/2, 3/4), (7/4,), x**4*exp_polar(2*I*pi))/(4*gamma(7/4))

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{2}}{\sqrt{-x^{4} + 1}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^2/sqrt(-x^4 + 1),x, algorithm="giac")

[Out]

integrate(x^2/sqrt(-x^4 + 1), x)