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

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

[Out]

(2*ArcTan[Sqrt[-1 - x^3]])/3

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Rubi [A]  time = 0.0278808, 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 \[ \frac{2}{3} \tan ^{-1}\left (\sqrt{-x^3-1}\right ) \]

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 = 4.06261, size = 14, normalized size = 0.88 \[ \frac{2 \operatorname{atan}{\left (\sqrt{- x^{3} - 1} \right )}}{3} \]

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 = 0.0251471, size = 34, normalized size = 2.12 \[ \frac{2 \sqrt{-x^3-1} \tanh ^{-1}\left (\sqrt{x^3+1}\right )}{3 \sqrt{x^3+1}} \]

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 = 0.036, size = 13, 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 = 1.57058, size = 16, 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 = 0.23419, size = 16, 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 = 3.46753, size = 12, normalized size = 0.75 \[ \frac{2 i \operatorname{asinh}{\left (\frac{1}{x^{\frac{3}{2}}} \right )}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*I*asinh(x**(-3/2))/3

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GIAC/XCAS [A]  time = 0.228708, size = 16, 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))