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

Optimal. Leaf size=14 \[ \frac{2}{3} \tan ^{-1}\left (\sqrt{x^3-1}\right ) \]

[Out]

(2*ArcTan[Sqrt[-1 + x^3]])/3

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Rubi [A]  time = 0.0239718, antiderivative size = 14, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ \frac{2}{3} \tan ^{-1}\left (\sqrt{x^3-1}\right ) \]

Antiderivative was successfully verified.

[In]  Int[1/(x*Sqrt[-1 + x^3]),x]

[Out]

(2*ArcTan[Sqrt[-1 + x^3]])/3

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Rubi in Sympy [A]  time = 3.26201, size = 12, normalized size = 0.86 \[ \frac{2 \operatorname{atan}{\left (\sqrt{x^{3} - 1} \right )}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*atan(sqrt(x**3 - 1))/3

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Mathematica [B]  time = 0.0230429, size = 36, normalized size = 2.57 \[ \frac{2 \sqrt{x^3-1} \tanh ^{-1}\left (\sqrt{1-x^3}\right )}{3 \sqrt{1-x^3}} \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[-1 + x^3]*ArcTanh[Sqrt[1 - x^3]])/(3*Sqrt[1 - x^3])

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Maple [A]  time = 0.029, size = 11, normalized size = 0.8 \[{\frac{2}{3}\arctan \left ( \sqrt{{x}^{3}-1} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x/(x^3-1)^(1/2),x)

[Out]

2/3*arctan((x^3-1)^(1/2))

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Maxima [A]  time = 1.59016, size = 14, normalized size = 1. \[ \frac{2}{3} \, \arctan \left (\sqrt{x^{3} - 1}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/3*arctan(sqrt(x^3 - 1))

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Fricas [A]  time = 0.230918, size = 14, normalized size = 1. \[ \frac{2}{3} \, \arctan \left (\sqrt{x^{3} - 1}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/3*arctan(sqrt(x^3 - 1))

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Sympy [A]  time = 3.45547, size = 31, normalized size = 2.21 \[ \begin{cases} \frac{2 i \operatorname{acosh}{\left (\frac{1}{x^{\frac{3}{2}}} \right )}}{3} & \text{for}\: \left |{\frac{1}{x^{3}}}\right | > 1 \\- \frac{2 \operatorname{asin}{\left (\frac{1}{x^{\frac{3}{2}}} \right )}}{3} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.214851, size = 14, normalized size = 1. \[ \frac{2}{3} \, \arctan \left (\sqrt{x^{3} - 1}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/3*arctan(sqrt(x^3 - 1))