3.460 \(\int \frac{1}{x \sqrt{1-x^3}} \, dx\)

Optimal. Leaf size=16 \[ -\frac{2}{3} \tanh ^{-1}\left (\sqrt{1-x^3}\right ) \]

[Out]

(-2*ArcTanh[Sqrt[1 - x^3]])/3

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Rubi [A]  time = 0.029688, 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 \[ -\frac{2}{3} \tanh ^{-1}\left (\sqrt{1-x^3}\right ) \]

Antiderivative was successfully verified.

[In]  Int[1/(x*Sqrt[1 - x^3]),x]

[Out]

(-2*ArcTanh[Sqrt[1 - x^3]])/3

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Rubi in Sympy [A]  time = 3.89293, size = 14, normalized size = 0.88 \[ - \frac{2 \operatorname{atanh}{\left (\sqrt{- x^{3} + 1} \right )}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*atanh(sqrt(-x**3 + 1))/3

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Mathematica [A]  time = 0.0199535, size = 16, normalized size = 1. \[ -\frac{2}{3} \tanh ^{-1}\left (\sqrt{1-x^3}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(-2*ArcTanh[Sqrt[1 - x^3]])/3

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Maple [A]  time = 0.118, size = 13, normalized size = 0.8 \[ -{\frac{2}{3}{\it Artanh} \left ( \sqrt{-{x}^{3}+1} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x/(-x^3+1)^(1/2),x)

[Out]

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

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Maxima [A]  time = 1.43147, size = 39, normalized size = 2.44 \[ -\frac{1}{3} \, \log \left (\sqrt{-x^{3} + 1} + 1\right ) + \frac{1}{3} \, \log \left (\sqrt{-x^{3} + 1} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/3*log(sqrt(-x^3 + 1) + 1) + 1/3*log(sqrt(-x^3 + 1) - 1)

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Fricas [A]  time = 0.23471, size = 39, normalized size = 2.44 \[ -\frac{1}{3} \, \log \left (\sqrt{-x^{3} + 1} + 1\right ) + \frac{1}{3} \, \log \left (\sqrt{-x^{3} + 1} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/3*log(sqrt(-x^3 + 1) + 1) + 1/3*log(sqrt(-x^3 + 1) - 1)

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Sympy [A]  time = 3.45071, size = 31, normalized size = 1.94 \[ \begin{cases} - \frac{2 \operatorname{acosh}{\left (\frac{1}{x^{\frac{3}{2}}} \right )}}{3} & \text{for}\: \left |{\frac{1}{x^{3}}}\right | > 1 \\\frac{2 i \operatorname{asin}{\left (\frac{1}{x^{\frac{3}{2}}} \right )}}{3} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((-2*acosh(x**(-3/2))/3, Abs(x**(-3)) > 1), (2*I*asin(x**(-3/2))/3, Tru
e))

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GIAC/XCAS [A]  time = 0.218494, size = 41, normalized size = 2.56 \[ -\frac{1}{3} \,{\rm ln}\left (\sqrt{-x^{3} + 1} + 1\right ) + \frac{1}{3} \,{\rm ln}\left ({\left | \sqrt{-x^{3} + 1} - 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/3*ln(sqrt(-x^3 + 1) + 1) + 1/3*ln(abs(sqrt(-x^3 + 1) - 1))