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

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

[Out]

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

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Rubi [A]  time = 0.0207963, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ \frac{x}{2 \sqrt{1-x^4}}-\frac{1}{2} F\left (\left .\sin ^{-1}(x)\right |-1\right ) \]

Antiderivative was successfully verified.

[In]  Int[x^4/(1 - x^4)^(3/2),x]

[Out]

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

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Rubi in Sympy [A]  time = 2.86768, size = 19, normalized size = 0.76 \[ \frac{x}{2 \sqrt{- x^{4} + 1}} - \frac{F\left (\operatorname{asin}{\left (x \right )}\middle | -1\right )}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0393378, size = 24, normalized size = 0.96 \[ \frac{1}{2} \left (\frac{x}{\sqrt{1-x^4}}-F\left (\left .\sin ^{-1}(x)\right |-1\right )\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.013, size = 45, normalized size = 1.8 \[{\frac{x}{2}{\frac{1}{\sqrt{-{x}^{4}+1}}}}-{\frac{{\it EllipticF} \left ( x,i \right ) }{2}\sqrt{-{x}^{2}+1}\sqrt{{x}^{2}+1}{\frac{1}{\sqrt{-{x}^{4}+1}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^4/(-x^4+1)^(3/2),x)

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(x^4/(-x^4 + 1)^(3/2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 2.00247, size = 31, normalized size = 1.24 \[ \frac{x^{5} \Gamma \left (\frac{5}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{5}{4}, \frac{3}{2} \\ \frac{9}{4} \end{matrix}\middle |{x^{4} e^{2 i \pi }} \right )}}{4 \Gamma \left (\frac{9}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x**5*gamma(5/4)*hyper((5/4, 3/2), (9/4,), x**4*exp_polar(2*I*pi))/(4*gamma(9/4))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(x^4/(-x^4 + 1)^(3/2), x)