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

Optimal. Leaf size=53 \[ -\frac{3 \sqrt{1-x^4}}{5 x}-\frac{\sqrt{1-x^4}}{5 x^5}+\frac{3}{5} F\left (\left .\sin ^{-1}(x)\right |-1\right )-\frac{3}{5} E\left (\left .\sin ^{-1}(x)\right |-1\right ) \]

[Out]

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

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Rubi [A]  time = 0.0758939, antiderivative size = 53, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ -\frac{3 \sqrt{1-x^4}}{5 x}-\frac{\sqrt{1-x^4}}{5 x^5}+\frac{3}{5} F\left (\left .\sin ^{-1}(x)\right |-1\right )-\frac{3}{5} E\left (\left .\sin ^{-1}(x)\right |-1\right ) \]

Antiderivative was successfully verified.

[In]  Int[1/(x^6*Sqrt[1 - x^4]),x]

[Out]

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

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Rubi in Sympy [A]  time = 11.6885, size = 46, normalized size = 0.87 \[ - \frac{3 E\left (\operatorname{asin}{\left (x \right )}\middle | -1\right )}{5} + \frac{3 F\left (\operatorname{asin}{\left (x \right )}\middle | -1\right )}{5} - \frac{3 \sqrt{- x^{4} + 1}}{5 x} - \frac{\sqrt{- x^{4} + 1}}{5 x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0676831, size = 70, normalized size = 1.32 \[ -\frac{-3 x^8+2 x^4-3 \sqrt{1-x^4} x^5 F\left (\left .\sin ^{-1}(x)\right |-1\right )+3 \sqrt{1-x^4} x^5 E\left (\left .\sin ^{-1}(x)\right |-1\right )+1}{5 x^5 \sqrt{1-x^4}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.017, size = 68, normalized size = 1.3 \[ -{\frac{1}{5\,{x}^{5}}\sqrt{-{x}^{4}+1}}-{\frac{3}{5\,x}\sqrt{-{x}^{4}+1}}+{\frac{3\,{\it EllipticF} \left ( x,i \right ) -3\,{\it EllipticE} \left ( x,i \right ) }{5}\sqrt{-{x}^{2}+1}\sqrt{{x}^{2}+1}{\frac{1}{\sqrt{-{x}^{4}+1}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^6/(-x^4+1)^(1/2),x)

[Out]

-1/5*(-x^4+1)^(1/2)/x^5-3/5*(-x^4+1)^(1/2)/x+3/5*(-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. \[ \int \frac{1}{\sqrt{-x^{4} + 1} x^{6}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(1/(sqrt(-x^4 + 1)*x^6), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(1/(sqrt(-x^4 + 1)*x^6), x)

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Sympy [A]  time = 2.82634, size = 37, normalized size = 0.7 \[ \frac{\Gamma \left (- \frac{5}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{5}{4}, \frac{1}{2} \\ - \frac{1}{4} \end{matrix}\middle |{x^{4} e^{2 i \pi }} \right )}}{4 x^{5} \Gamma \left (- \frac{1}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

gamma(-5/4)*hyper((-5/4, 1/2), (-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. \[ \int \frac{1}{\sqrt{-x^{4} + 1} x^{6}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(1/(sqrt(-x^4 + 1)*x^6), x)