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

Optimal. Leaf size=34 \[ \frac{x^2}{\sqrt{1-x^4}}-\frac{1}{2 x^2 \sqrt{1-x^4}} \]

[Out]

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

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Rubi [A]  time = 0.0276021, antiderivative size = 34, 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 \[ \frac{x^2}{\sqrt{1-x^4}}-\frac{1}{2 x^2 \sqrt{1-x^4}} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^3*(1 - x^4)^(3/2)),x]

[Out]

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

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Rubi in Sympy [A]  time = 3.23965, size = 26, normalized size = 0.76 \[ \frac{x^{2}}{\sqrt{- x^{4} + 1}} - \frac{1}{2 x^{2} \sqrt{- x^{4} + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0168733, size = 25, normalized size = 0.74 \[ \frac{2 x^4-1}{2 x^2 \sqrt{1-x^4}} \]

Antiderivative was successfully verified.

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

[Out]

(-1 + 2*x^4)/(2*x^2*Sqrt[1 - x^4])

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Maple [A]  time = 0.007, size = 33, normalized size = 1. \[ -{\frac{ \left ( -1+x \right ) \left ( 1+x \right ) \left ({x}^{2}+1 \right ) \left ( 2\,{x}^{4}-1 \right ) }{2\,{x}^{2}} \left ( -{x}^{4}+1 \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^3/(-x^4+1)^(3/2),x)

[Out]

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

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Maxima [A]  time = 1.43341, size = 39, normalized size = 1.15 \[ \frac{x^{2}}{2 \, \sqrt{-x^{4} + 1}} - \frac{\sqrt{-x^{4} + 1}}{2 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.29657, size = 88, normalized size = 2.59 \[ -\frac{2 \, x^{8} - 5 \, x^{4} + 2 \,{\left (2 \, x^{4} - 1\right )} \sqrt{-x^{4} + 1} + 2}{2 \,{\left (2 \, x^{6} - 2 \, x^{2} -{\left (x^{6} - 2 \, x^{2}\right )} \sqrt{-x^{4} + 1}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*(2*x^8 - 5*x^4 + 2*(2*x^4 - 1)*sqrt(-x^4 + 1) + 2)/(2*x^6 - 2*x^2 - (x^6 -
2*x^2)*sqrt(-x^4 + 1))

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Sympy [A]  time = 2.43996, size = 90, normalized size = 2.65 \[ \begin{cases} - \frac{2 i x^{4} \sqrt{x^{4} - 1}}{2 x^{6} - 2 x^{2}} + \frac{i \sqrt{x^{4} - 1}}{2 x^{6} - 2 x^{2}} & \text{for}\: \left |{x^{4}}\right | > 1 \\- \frac{2 x^{4} \sqrt{- x^{4} + 1}}{2 x^{6} - 2 x^{2}} + \frac{\sqrt{- x^{4} + 1}}{2 x^{6} - 2 x^{2}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.222744, size = 42, normalized size = 1.24 \[ -\frac{\sqrt{-x^{4} + 1} x^{2}}{2 \,{\left (x^{4} - 1\right )}} - \frac{1}{2} \, \sqrt{\frac{1}{x^{4}} - 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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