3.913 \(\int \frac{1}{x^6 \left (1-x^4\right )^{3/2}} \, dx\)

Optimal. Leaf size=71 \[ -\frac{21 \sqrt{1-x^4}}{10 x}-\frac{7 \sqrt{1-x^4}}{10 x^5}+\frac{1}{2 x^5 \sqrt{1-x^4}}+\frac{21}{10} F\left (\left .\sin ^{-1}(x)\right |-1\right )-\frac{21}{10} E\left (\left .\sin ^{-1}(x)\right |-1\right ) \]

[Out]

1/(2*x^5*Sqrt[1 - x^4]) - (7*Sqrt[1 - x^4])/(10*x^5) - (21*Sqrt[1 - x^4])/(10*x)
- (21*EllipticE[ArcSin[x], -1])/10 + (21*EllipticF[ArcSin[x], -1])/10

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Rubi [A]  time = 0.0884827, antiderivative size = 71, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4 \[ -\frac{21 \sqrt{1-x^4}}{10 x}-\frac{7 \sqrt{1-x^4}}{10 x^5}+\frac{1}{2 x^5 \sqrt{1-x^4}}+\frac{21}{10} F\left (\left .\sin ^{-1}(x)\right |-1\right )-\frac{21}{10} E\left (\left .\sin ^{-1}(x)\right |-1\right ) \]

Antiderivative was successfully verified.

[In]  Int[1/(x^6*(1 - x^4)^(3/2)),x]

[Out]

1/(2*x^5*Sqrt[1 - x^4]) - (7*Sqrt[1 - x^4])/(10*x^5) - (21*Sqrt[1 - x^4])/(10*x)
- (21*EllipticE[ArcSin[x], -1])/10 + (21*EllipticF[ArcSin[x], -1])/10

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Rubi in Sympy [A]  time = 13.3491, size = 63, normalized size = 0.89 \[ - \frac{21 E\left (\operatorname{asin}{\left (x \right )}\middle | -1\right )}{10} + \frac{21 F\left (\operatorname{asin}{\left (x \right )}\middle | -1\right )}{10} - \frac{21 \sqrt{- x^{4} + 1}}{10 x} - \frac{7 \sqrt{- x^{4} + 1}}{10 x^{5}} + \frac{1}{2 x^{5} \sqrt{- x^{4} + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**6/(-x**4+1)**(3/2),x)

[Out]

-21*elliptic_e(asin(x), -1)/10 + 21*elliptic_f(asin(x), -1)/10 - 21*sqrt(-x**4 +
1)/(10*x) - 7*sqrt(-x**4 + 1)/(10*x**5) + 1/(2*x**5*sqrt(-x**4 + 1))

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Mathematica [A]  time = 0.0677026, size = 70, normalized size = 0.99 \[ -\frac{-21 x^8+14 x^4-21 \sqrt{1-x^4} x^5 F\left (\left .\sin ^{-1}(x)\right |-1\right )+21 \sqrt{1-x^4} x^5 E\left (\left .\sin ^{-1}(x)\right |-1\right )+2}{10 x^5 \sqrt{1-x^4}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^6*(1 - x^4)^(3/2)),x]

[Out]

-(2 + 14*x^4 - 21*x^8 + 21*x^5*Sqrt[1 - x^4]*EllipticE[ArcSin[x], -1] - 21*x^5*S
qrt[1 - x^4]*EllipticF[ArcSin[x], -1])/(10*x^5*Sqrt[1 - x^4])

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Maple [A]  time = 0.023, size = 82, normalized size = 1.2 \[{\frac{{x}^{3}}{2}{\frac{1}{\sqrt{-{x}^{4}+1}}}}-{\frac{1}{5\,{x}^{5}}\sqrt{-{x}^{4}+1}}-{\frac{8}{5\,x}\sqrt{-{x}^{4}+1}}+{\frac{21\,{\it EllipticF} \left ( x,i \right ) -21\,{\it EllipticE} \left ( x,i \right ) }{10}\sqrt{-{x}^{2}+1}\sqrt{{x}^{2}+1}{\frac{1}{\sqrt{-{x}^{4}+1}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^6/(-x^4+1)^(3/2),x)

[Out]

1/2*x^3/(-x^4+1)^(1/2)-1/5*(-x^4+1)^(1/2)/x^5-8/5*(-x^4+1)^(1/2)/x+21/10*(-x^2+1
)^(1/2)*(x^2+1)^(1/2)/(-x^4+1)^(1/2)*(EllipticF(x,I)-EllipticE(x,I))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (-x^{4} + 1\right )}^{\frac{3}{2}} x^{6}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((-x^4 + 1)^(3/2)*x^6),x, algorithm="maxima")

[Out]

integrate(1/((-x^4 + 1)^(3/2)*x^6), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (-\frac{1}{{\left (x^{10} - x^{6}\right )} \sqrt{-x^{4} + 1}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((-x^4 + 1)^(3/2)*x^6),x, algorithm="fricas")

[Out]

integral(-1/((x^10 - x^6)*sqrt(-x^4 + 1)), x)

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Sympy [A]  time = 4.12242, size = 37, normalized size = 0.52 \[ \frac{\Gamma \left (- \frac{5}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{5}{4}, \frac{3}{2} \\ - \frac{1}{4} \end{matrix}\middle |{x^{4} e^{2 i \pi }} \right )}}{4 x^{5} \Gamma \left (- \frac{1}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**6/(-x**4+1)**(3/2),x)

[Out]

gamma(-5/4)*hyper((-5/4, 3/2), (-1/4,), x**4*exp_polar(2*I*pi))/(4*x**5*gamma(-1
/4))

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (-x^{4} + 1\right )}^{\frac{3}{2}} x^{6}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((-x^4 + 1)^(3/2)*x^6),x, algorithm="giac")

[Out]

integrate(1/((-x^4 + 1)^(3/2)*x^6), x)