3.880 \(\int \frac{1}{x^7 \sqrt{1-x^4}} \, dx\)

Optimal. Leaf size=37 \[ -\frac{\sqrt{1-x^4}}{6 x^6}-\frac{\sqrt{1-x^4}}{3 x^2} \]

[Out]

-Sqrt[1 - x^4]/(6*x^6) - Sqrt[1 - x^4]/(3*x^2)

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Rubi [A]  time = 0.0297258, antiderivative size = 37, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ -\frac{\sqrt{1-x^4}}{6 x^6}-\frac{\sqrt{1-x^4}}{3 x^2} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^7*Sqrt[1 - x^4]),x]

[Out]

-Sqrt[1 - x^4]/(6*x^6) - Sqrt[1 - x^4]/(3*x^2)

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Rubi in Sympy [A]  time = 3.53772, size = 27, normalized size = 0.73 \[ - \frac{\sqrt{- x^{4} + 1}}{3 x^{2}} - \frac{\sqrt{- x^{4} + 1}}{6 x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**7/(-x**4+1)**(1/2),x)

[Out]

-sqrt(-x**4 + 1)/(3*x**2) - sqrt(-x**4 + 1)/(6*x**6)

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Mathematica [A]  time = 0.0141308, size = 27, normalized size = 0.73 \[ \left (-\frac{1}{6 x^6}-\frac{1}{3 x^2}\right ) \sqrt{1-x^4} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^7*Sqrt[1 - x^4]),x]

[Out]

(-1/(6*x^6) - 1/(3*x^2))*Sqrt[1 - x^4]

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Maple [A]  time = 0.007, size = 33, normalized size = 0.9 \[{\frac{ \left ( -1+x \right ) \left ( 1+x \right ) \left ({x}^{2}+1 \right ) \left ( 2\,{x}^{4}+1 \right ) }{6\,{x}^{6}}{\frac{1}{\sqrt{-{x}^{4}+1}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^7/(-x^4+1)^(1/2),x)

[Out]

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

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Maxima [A]  time = 1.43055, size = 39, normalized size = 1.05 \[ -\frac{\sqrt{-x^{4} + 1}}{2 \, x^{2}} - \frac{{\left (-x^{4} + 1\right )}^{\frac{3}{2}}}{6 \, x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(-x^4 + 1)*x^7),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.246029, size = 100, normalized size = 2.7 \[ -\frac{2 \, x^{12} - 9 \, x^{8} + 3 \, x^{4} +{\left (6 \, x^{8} - 5 \, x^{4} - 4\right )} \sqrt{-x^{4} + 1} + 4}{6 \,{\left (3 \, x^{10} - 4 \, x^{6} -{\left (x^{10} - 4 \, x^{6}\right )} \sqrt{-x^{4} + 1}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(-x^4 + 1)*x^7),x, algorithm="fricas")

[Out]

-1/6*(2*x^12 - 9*x^8 + 3*x^4 + (6*x^8 - 5*x^4 - 4)*sqrt(-x^4 + 1) + 4)/(3*x^10 -
4*x^6 - (x^10 - 4*x^6)*sqrt(-x^4 + 1))

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Sympy [A]  time = 3.04937, size = 63, normalized size = 1.7 \[ \begin{cases} - \frac{i \sqrt{x^{4} - 1}}{3 x^{2}} - \frac{i \sqrt{x^{4} - 1}}{6 x^{6}} & \text{for}\: \left |{x^{4}}\right | > 1 \\- \frac{\sqrt{- x^{4} + 1}}{3 x^{2}} - \frac{\sqrt{- x^{4} + 1}}{6 x^{6}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**7/(-x**4+1)**(1/2),x)

[Out]

Piecewise((-I*sqrt(x**4 - 1)/(3*x**2) - I*sqrt(x**4 - 1)/(6*x**6), Abs(x**4) > 1
), (-sqrt(-x**4 + 1)/(3*x**2) - sqrt(-x**4 + 1)/(6*x**6), True))

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GIAC/XCAS [A]  time = 0.211441, size = 26, normalized size = 0.7 \[ -\frac{1}{6} \,{\left (\frac{1}{x^{4}} - 1\right )}^{\frac{3}{2}} - \frac{1}{2} \, \sqrt{\frac{1}{x^{4}} - 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(-x^4 + 1)*x^7),x, algorithm="giac")

[Out]

-1/6*(1/x^4 - 1)^(3/2) - 1/2*sqrt(1/x^4 - 1)