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

Optimal. Leaf size=32 \[ \frac{1}{2 \sqrt{1-x^4}}-\frac{1}{2} \tanh ^{-1}\left (\sqrt{1-x^4}\right ) \]

[Out]

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

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Rubi [A]  time = 0.0423776, antiderivative size = 32, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267 \[ \frac{1}{2 \sqrt{1-x^4}}-\frac{1}{2} \tanh ^{-1}\left (\sqrt{1-x^4}\right ) \]

Antiderivative was successfully verified.

[In]  Int[1/(x*(1 - x^4)^(3/2)),x]

[Out]

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

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Rubi in Sympy [A]  time = 4.7933, size = 22, normalized size = 0.69 \[ - \frac{\operatorname{atanh}{\left (\sqrt{- x^{4} + 1} \right )}}{2} + \frac{1}{2 \sqrt{- x^{4} + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0512818, size = 30, normalized size = 0.94 \[ \frac{1}{2} \left (\frac{1}{\sqrt{1-x^4}}-\tanh ^{-1}\left (\sqrt{1-x^4}\right )\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.024, size = 68, normalized size = 2.1 \[ -{\frac{1}{2}{\it Artanh} \left ({\frac{1}{\sqrt{-{x}^{4}+1}}} \right ) }+{\frac{1}{4\,{x}^{2}+4}\sqrt{- \left ({x}^{2}+1 \right ) ^{2}+2+2\,{x}^{2}}}-{\frac{1}{4\,{x}^{2}-4}\sqrt{- \left ({x}^{2}-1 \right ) ^{2}-2\,{x}^{2}+2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x/(-x^4+1)^(3/2),x)

[Out]

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

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Maxima [A]  time = 1.40954, size = 54, normalized size = 1.69 \[ \frac{1}{2 \, \sqrt{-x^{4} + 1}} - \frac{1}{4} \, \log \left (\sqrt{-x^{4} + 1} + 1\right ) + \frac{1}{4} \, \log \left (\sqrt{-x^{4} + 1} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.3062, size = 78, normalized size = 2.44 \[ -\frac{\sqrt{-x^{4} + 1} \log \left (\sqrt{-x^{4} + 1} + 1\right ) - \sqrt{-x^{4} + 1} \log \left (\sqrt{-x^{4} + 1} - 1\right ) - 2}{4 \, \sqrt{-x^{4} + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 4.8843, size = 228, normalized size = 7.12 \[ \begin{cases} - \frac{2 x^{4} \log{\left (x^{2} \right )}}{4 x^{4} - 4} + \frac{x^{4} \log{\left (x^{4} \right )}}{4 x^{4} - 4} + \frac{2 i x^{4} \operatorname{asin}{\left (\frac{1}{x^{2}} \right )}}{4 x^{4} - 4} - \frac{2 i \sqrt{x^{4} - 1}}{4 x^{4} - 4} + \frac{2 \log{\left (x^{2} \right )}}{4 x^{4} - 4} - \frac{\log{\left (x^{4} \right )}}{4 x^{4} - 4} - \frac{2 i \operatorname{asin}{\left (\frac{1}{x^{2}} \right )}}{4 x^{4} - 4} & \text{for}\: \left |{x^{4}}\right | > 1 \\\frac{x^{4} \log{\left (x^{4} \right )}}{4 x^{4} - 4} - \frac{2 x^{4} \log{\left (\sqrt{- x^{4} + 1} + 1 \right )}}{4 x^{4} - 4} + \frac{i \pi x^{4}}{4 x^{4} - 4} - \frac{2 \sqrt{- x^{4} + 1}}{4 x^{4} - 4} - \frac{\log{\left (x^{4} \right )}}{4 x^{4} - 4} + \frac{2 \log{\left (\sqrt{- x^{4} + 1} + 1 \right )}}{4 x^{4} - 4} - \frac{i \pi }{4 x^{4} - 4} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.213032, size = 57, normalized size = 1.78 \[ \frac{1}{2 \, \sqrt{-x^{4} + 1}} - \frac{1}{4} \,{\rm ln}\left (\sqrt{-x^{4} + 1} + 1\right ) + \frac{1}{4} \,{\rm ln}\left (-\sqrt{-x^{4} + 1} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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