3.888 \(\int \frac{x^6}{\sqrt{1-x^4}} \, dx\)

Optimal. Leaf size=35 \[ -\frac{1}{5} \sqrt{1-x^4} x^3-\frac{3}{5} F\left (\left .\sin ^{-1}(x)\right |-1\right )+\frac{3}{5} E\left (\left .\sin ^{-1}(x)\right |-1\right ) \]

[Out]

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

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Rubi [A]  time = 0.0634609, 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 \[ -\frac{1}{5} \sqrt{1-x^4} x^3-\frac{3}{5} F\left (\left .\sin ^{-1}(x)\right |-1\right )+\frac{3}{5} E\left (\left .\sin ^{-1}(x)\right |-1\right ) \]

Antiderivative was successfully verified.

[In]  Int[x^6/Sqrt[1 - x^4],x]

[Out]

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

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Rubi in Sympy [A]  time = 10.6163, size = 32, normalized size = 0.91 \[ - \frac{x^{3} \sqrt{- x^{4} + 1}}{5} + \frac{3 E\left (\operatorname{asin}{\left (x \right )}\middle | -1\right )}{5} - \frac{3 F\left (\operatorname{asin}{\left (x \right )}\middle | -1\right )}{5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**6/(-x**4+1)**(1/2),x)

[Out]

-x**3*sqrt(-x**4 + 1)/5 + 3*elliptic_e(asin(x), -1)/5 - 3*elliptic_f(asin(x), -1
)/5

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Mathematica [A]  time = 0.0691163, size = 48, normalized size = 1.37 \[ \frac{1}{5} \left (\frac{x^7}{\sqrt{1-x^4}}-\frac{x^3}{\sqrt{1-x^4}}-3 F\left (\left .\sin ^{-1}(x)\right |-1\right )+3 E\left (\left .\sin ^{-1}(x)\right |-1\right )\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[x^6/Sqrt[1 - x^4],x]

[Out]

(-(x^3/Sqrt[1 - x^4]) + x^7/Sqrt[1 - x^4] + 3*EllipticE[ArcSin[x], -1] - 3*Ellip
ticF[ArcSin[x], -1])/5

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Maple [A]  time = 0.013, size = 54, normalized size = 1.5 \[ -{\frac{{x}^{3}}{5}\sqrt{-{x}^{4}+1}}-{\frac{3\,{\it EllipticF} \left ( x,i \right ) -3\,{\it EllipticE} \left ( x,i \right ) }{5}\sqrt{-{x}^{2}+1}\sqrt{{x}^{2}+1}{\frac{1}{\sqrt{-{x}^{4}+1}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^6/(-x^4+1)^(1/2),x)

[Out]

-1/5*x^3*(-x^4+1)^(1/2)-3/5*(-x^2+1)^(1/2)*(x^2+1)^(1/2)/(-x^4+1)^(1/2)*(Ellipti
cF(x,I)-EllipticE(x,I))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{6}}{\sqrt{-x^{4} + 1}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^6/sqrt(-x^4 + 1),x, algorithm="maxima")

[Out]

integrate(x^6/sqrt(-x^4 + 1), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{x^{6}}{\sqrt{-x^{4} + 1}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^6/sqrt(-x^4 + 1),x, algorithm="fricas")

[Out]

integral(x^6/sqrt(-x^4 + 1), x)

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Sympy [A]  time = 2.23125, size = 31, normalized size = 0.89 \[ \frac{x^{7} \Gamma \left (\frac{7}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, \frac{7}{4} \\ \frac{11}{4} \end{matrix}\middle |{x^{4} e^{2 i \pi }} \right )}}{4 \Gamma \left (\frac{11}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**6/(-x**4+1)**(1/2),x)

[Out]

x**7*gamma(7/4)*hyper((1/2, 7/4), (11/4,), x**4*exp_polar(2*I*pi))/(4*gamma(11/4
))

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{6}}{\sqrt{-x^{4} + 1}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^6/sqrt(-x^4 + 1),x, algorithm="giac")

[Out]

integrate(x^6/sqrt(-x^4 + 1), x)