3.886 \(\int \frac{1}{x^8 \sqrt{1-x^4}} \, dx\)

Optimal. Leaf size=45 \[ -\frac{\sqrt{1-x^4}}{7 x^7}-\frac{5 \sqrt{1-x^4}}{21 x^3}+\frac{5}{21} F\left (\left .\sin ^{-1}(x)\right |-1\right ) \]

[Out]

-Sqrt[1 - x^4]/(7*x^7) - (5*Sqrt[1 - x^4])/(21*x^3) + (5*EllipticF[ArcSin[x], -1
])/21

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Rubi [A]  time = 0.0358512, antiderivative size = 45, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ -\frac{\sqrt{1-x^4}}{7 x^7}-\frac{5 \sqrt{1-x^4}}{21 x^3}+\frac{5}{21} F\left (\left .\sin ^{-1}(x)\right |-1\right ) \]

Antiderivative was successfully verified.

[In]  Int[1/(x^8*Sqrt[1 - x^4]),x]

[Out]

-Sqrt[1 - x^4]/(7*x^7) - (5*Sqrt[1 - x^4])/(21*x^3) + (5*EllipticF[ArcSin[x], -1
])/21

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Rubi in Sympy [A]  time = 4.06708, size = 37, normalized size = 0.82 \[ \frac{5 F\left (\operatorname{asin}{\left (x \right )}\middle | -1\right )}{21} - \frac{5 \sqrt{- x^{4} + 1}}{21 x^{3}} - \frac{\sqrt{- x^{4} + 1}}{7 x^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**8/(-x**4+1)**(1/2),x)

[Out]

5*elliptic_f(asin(x), -1)/21 - 5*sqrt(-x**4 + 1)/(21*x**3) - sqrt(-x**4 + 1)/(7*
x**7)

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Mathematica [A]  time = 0.0359872, size = 50, normalized size = 1.11 \[ \frac{5 x^8-2 x^4+5 \sqrt{1-x^4} x^7 F\left (\left .\sin ^{-1}(x)\right |-1\right )-3}{21 x^7 \sqrt{1-x^4}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^8*Sqrt[1 - x^4]),x]

[Out]

(-3 - 2*x^4 + 5*x^8 + 5*x^7*Sqrt[1 - x^4]*EllipticF[ArcSin[x], -1])/(21*x^7*Sqrt
[1 - x^4])

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Maple [A]  time = 0.018, size = 61, normalized size = 1.4 \[ -{\frac{1}{7\,{x}^{7}}\sqrt{-{x}^{4}+1}}-{\frac{5}{21\,{x}^{3}}\sqrt{-{x}^{4}+1}}+{\frac{5\,{\it EllipticF} \left ( x,i \right ) }{21}\sqrt{-{x}^{2}+1}\sqrt{{x}^{2}+1}{\frac{1}{\sqrt{-{x}^{4}+1}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^8/(-x^4+1)^(1/2),x)

[Out]

-1/7*(-x^4+1)^(1/2)/x^7-5/21*(-x^4+1)^(1/2)/x^3+5/21*(-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. \[ \int \frac{1}{\sqrt{-x^{4} + 1} x^{8}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(-x^4 + 1)*x^8),x, algorithm="maxima")

[Out]

integrate(1/(sqrt(-x^4 + 1)*x^8), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{\sqrt{-x^{4} + 1} x^{8}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(-x^4 + 1)*x^8),x, algorithm="fricas")

[Out]

integral(1/(sqrt(-x^4 + 1)*x^8), x)

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Sympy [A]  time = 3.72186, size = 37, normalized size = 0.82 \[ \frac{\Gamma \left (- \frac{7}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{7}{4}, \frac{1}{2} \\ - \frac{3}{4} \end{matrix}\middle |{x^{4} e^{2 i \pi }} \right )}}{4 x^{7} \Gamma \left (- \frac{3}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**8/(-x**4+1)**(1/2),x)

[Out]

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

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{-x^{4} + 1} x^{8}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(-x^4 + 1)*x^8),x, algorithm="giac")

[Out]

integrate(1/(sqrt(-x^4 + 1)*x^8), x)