3.907 \(\int \frac{1}{x^8 \left (1-x^4\right )^{3/2}} \, dx\)

Optimal. Leaf size=63 \[ -\frac{9 \sqrt{1-x^4}}{14 x^7}+\frac{1}{2 x^7 \sqrt{1-x^4}}-\frac{15 \sqrt{1-x^4}}{14 x^3}+\frac{15}{14} F\left (\left .\sin ^{-1}(x)\right |-1\right ) \]

[Out]

1/(2*x^7*Sqrt[1 - x^4]) - (9*Sqrt[1 - x^4])/(14*x^7) - (15*Sqrt[1 - x^4])/(14*x^
3) + (15*EllipticF[ArcSin[x], -1])/14

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Rubi [A]  time = 0.0517454, antiderivative size = 63, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ -\frac{9 \sqrt{1-x^4}}{14 x^7}+\frac{1}{2 x^7 \sqrt{1-x^4}}-\frac{15 \sqrt{1-x^4}}{14 x^3}+\frac{15}{14} F\left (\left .\sin ^{-1}(x)\right |-1\right ) \]

Antiderivative was successfully verified.

[In]  Int[1/(x^8*(1 - x^4)^(3/2)),x]

[Out]

1/(2*x^7*Sqrt[1 - x^4]) - (9*Sqrt[1 - x^4])/(14*x^7) - (15*Sqrt[1 - x^4])/(14*x^
3) + (15*EllipticF[ArcSin[x], -1])/14

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Rubi in Sympy [A]  time = 5.65612, size = 54, normalized size = 0.86 \[ \frac{15 F\left (\operatorname{asin}{\left (x \right )}\middle | -1\right )}{14} - \frac{15 \sqrt{- x^{4} + 1}}{14 x^{3}} - \frac{9 \sqrt{- x^{4} + 1}}{14 x^{7}} + \frac{1}{2 x^{7} \sqrt{- x^{4} + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**8/(-x**4+1)**(3/2),x)

[Out]

15*elliptic_f(asin(x), -1)/14 - 15*sqrt(-x**4 + 1)/(14*x**3) - 9*sqrt(-x**4 + 1)
/(14*x**7) + 1/(2*x**7*sqrt(-x**4 + 1))

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Mathematica [A]  time = 0.045149, size = 50, normalized size = 0.79 \[ \frac{15 x^8-6 x^4+15 \sqrt{1-x^4} x^7 F\left (\left .\sin ^{-1}(x)\right |-1\right )-2}{14 x^7 \sqrt{1-x^4}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^8*(1 - x^4)^(3/2)),x]

[Out]

(-2 - 6*x^4 + 15*x^8 + 15*x^7*Sqrt[1 - x^4]*EllipticF[ArcSin[x], -1])/(14*x^7*Sq
rt[1 - x^4])

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Maple [A]  time = 0.023, size = 73, normalized size = 1.2 \[{\frac{x}{2}{\frac{1}{\sqrt{-{x}^{4}+1}}}}-{\frac{1}{7\,{x}^{7}}\sqrt{-{x}^{4}+1}}-{\frac{4}{7\,{x}^{3}}\sqrt{-{x}^{4}+1}}+{\frac{15\,{\it EllipticF} \left ( x,i \right ) }{14}\sqrt{-{x}^{2}+1}\sqrt{{x}^{2}+1}{\frac{1}{\sqrt{-{x}^{4}+1}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^8/(-x^4+1)^(3/2),x)

[Out]

1/2*x/(-x^4+1)^(1/2)-1/7*(-x^4+1)^(1/2)/x^7-4/7*(-x^4+1)^(1/2)/x^3+15/14*(-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. \[ \int \frac{1}{{\left (-x^{4} + 1\right )}^{\frac{3}{2}} x^{8}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((-x^4 + 1)^(3/2)*x^8),x, algorithm="maxima")

[Out]

integrate(1/((-x^4 + 1)^(3/2)*x^8), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (-\frac{1}{{\left (x^{12} - x^{8}\right )} \sqrt{-x^{4} + 1}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((-x^4 + 1)^(3/2)*x^8),x, algorithm="fricas")

[Out]

integral(-1/((x^12 - x^8)*sqrt(-x^4 + 1)), x)

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Sympy [A]  time = 5.94527, size = 37, normalized size = 0.59 \[ \frac{\Gamma \left (- \frac{7}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{7}{4}, \frac{3}{2} \\ - \frac{3}{4} \end{matrix}\middle |{x^{4} e^{2 i \pi }} \right )}}{4 x^{7} \Gamma \left (- \frac{3}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**8/(-x**4+1)**(3/2),x)

[Out]

gamma(-7/4)*hyper((-7/4, 3/2), (-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. \[ \int \frac{1}{{\left (-x^{4} + 1\right )}^{\frac{3}{2}} x^{8}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((-x^4 + 1)^(3/2)*x^8),x, algorithm="giac")

[Out]

integrate(1/((-x^4 + 1)^(3/2)*x^8), x)