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

Optimal. Leaf size=16 \[ -\frac{1}{2} \tanh ^{-1}\left (\sqrt{1-x^4}\right ) \]

[Out]

-ArcTanh[Sqrt[1 - x^4]]/2

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Rubi [A]  time = 0.0277166, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ -\frac{1}{2} \tanh ^{-1}\left (\sqrt{1-x^4}\right ) \]

Antiderivative was successfully verified.

[In]  Int[1/(x*Sqrt[1 - x^4]),x]

[Out]

-ArcTanh[Sqrt[1 - x^4]]/2

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Rubi in Sympy [A]  time = 3.80427, size = 12, normalized size = 0.75 \[ - \frac{\operatorname{atanh}{\left (\sqrt{- x^{4} + 1} \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-atanh(sqrt(-x**4 + 1))/2

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Mathematica [A]  time = 0.035617, size = 16, normalized size = 1. \[ -\frac{1}{2} \tanh ^{-1}\left (\sqrt{1-x^4}\right ) \]

Antiderivative was successfully verified.

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

[Out]

-ArcTanh[Sqrt[1 - x^4]]/2

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Maple [A]  time = 0.014, size = 13, normalized size = 0.8 \[ -{\frac{1}{2}{\it Artanh} \left ({\frac{1}{\sqrt{-{x}^{4}+1}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x/(-x^4+1)^(1/2),x)

[Out]

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

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Maxima [A]  time = 1.43414, size = 39, normalized size = 2.44 \[ -\frac{1}{4} \, \log \left (\sqrt{-x^{4} + 1} + 1\right ) + \frac{1}{4} \, \log \left (\sqrt{-x^{4} + 1} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.288008, size = 39, normalized size = 2.44 \[ -\frac{1}{4} \, \log \left (\sqrt{-x^{4} + 1} + 1\right ) + \frac{1}{4} \, \log \left (\sqrt{-x^{4} + 1} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 3.36679, size = 24, normalized size = 1.5 \[ \begin{cases} - \frac{\operatorname{acosh}{\left (\frac{1}{x^{2}} \right )}}{2} & \text{for}\: \left |{\frac{1}{x^{4}}}\right | > 1 \\\frac{i \operatorname{asin}{\left (\frac{1}{x^{2}} \right )}}{2} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((-acosh(x**(-2))/2, Abs(x**(-4)) > 1), (I*asin(x**(-2))/2, True))

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GIAC/XCAS [A]  time = 0.216125, size = 42, normalized size = 2.62 \[ -\frac{1}{4} \,{\rm ln}\left (\sqrt{-x^{4} + 1} + 1\right ) + \frac{1}{4} \,{\rm ln}\left (-\sqrt{-x^{4} + 1} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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