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

Optimal. Leaf size=35 \[ \frac{x^3}{2 \sqrt{1-x^4}}+\frac{1}{2} F\left (\left .\sin ^{-1}(x)\right |-1\right )-\frac{1}{2} E\left (\left .\sin ^{-1}(x)\right |-1\right ) \]

[Out]

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

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Rubi [A]  time = 0.0600672, antiderivative size = 35, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ \frac{x^3}{2 \sqrt{1-x^4}}+\frac{1}{2} F\left (\left .\sin ^{-1}(x)\right |-1\right )-\frac{1}{2} E\left (\left .\sin ^{-1}(x)\right |-1\right ) \]

Antiderivative was successfully verified.

[In]  Int[x^2/(1 - x^4)^(3/2),x]

[Out]

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

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Rubi in Sympy [A]  time = 10.9784, size = 29, normalized size = 0.83 \[ \frac{x^{3}}{2 \sqrt{- x^{4} + 1}} - \frac{E\left (\operatorname{asin}{\left (x \right )}\middle | -1\right )}{2} + \frac{F\left (\operatorname{asin}{\left (x \right )}\middle | -1\right )}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0506911, size = 30, normalized size = 0.86 \[ \frac{1}{2} \left (\frac{x^3}{\sqrt{1-x^4}}+F\left (\left .\sin ^{-1}(x)\right |-1\right )-E\left (\left .\sin ^{-1}(x)\right |-1\right )\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.014, size = 54, normalized size = 1.5 \[{\frac{{x}^{3}}{2}{\frac{1}{\sqrt{-{x}^{4}+1}}}}+{\frac{{\it EllipticF} \left ( x,i \right ) -{\it EllipticE} \left ( x,i \right ) }{2}\sqrt{-{x}^{2}+1}\sqrt{{x}^{2}+1}{\frac{1}{\sqrt{-{x}^{4}+1}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2/(-x^4+1)^(3/2),x)

[Out]

1/2*x^3/(-x^4+1)^(1/2)+1/2*(-x^2+1)^(1/2)*(x^2+1)^(1/2)/(-x^4+1)^(1/2)*(Elliptic
F(x,I)-EllipticE(x,I))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{2}}{{\left (-x^{4} + 1\right )}^{\frac{3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^2/(-x^4 + 1)^(3/2),x, algorithm="maxima")

[Out]

integrate(x^2/(-x^4 + 1)^(3/2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^2/(-x^4 + 1)^(3/2),x, algorithm="fricas")

[Out]

integral(-x^2/((x^4 - 1)*sqrt(-x^4 + 1)), x)

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Sympy [A]  time = 1.93959, size = 31, normalized size = 0.89 \[ \frac{x^{3} \Gamma \left (\frac{3}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{3}{4}, \frac{3}{2} \\ \frac{7}{4} \end{matrix}\middle |{x^{4} e^{2 i \pi }} \right )}}{4 \Gamma \left (\frac{7}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x**3*gamma(3/4)*hyper((3/4, 3/2), (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. \[ \int \frac{x^{2}}{{\left (-x^{4} + 1\right )}^{\frac{3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^2/(-x^4 + 1)^(3/2),x, algorithm="giac")

[Out]

integrate(x^2/(-x^4 + 1)^(3/2), x)