3.497 \(\int \frac{1}{x^4 \sqrt{-1-x^3}} \, dx\)

Optimal. Leaf size=35 \[ \frac{\sqrt{-x^3-1}}{3 x^3}-\frac{1}{3} \tan ^{-1}\left (\sqrt{-x^3-1}\right ) \]

[Out]

Sqrt[-1 - x^3]/(3*x^3) - ArcTan[Sqrt[-1 - x^3]]/3

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Rubi [A]  time = 0.046122, antiderivative size = 35, 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 \[ \frac{\sqrt{-x^3-1}}{3 x^3}-\frac{1}{3} \tan ^{-1}\left (\sqrt{-x^3-1}\right ) \]

Antiderivative was successfully verified.

[In]  Int[1/(x^4*Sqrt[-1 - x^3]),x]

[Out]

Sqrt[-1 - x^3]/(3*x^3) - ArcTan[Sqrt[-1 - x^3]]/3

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Rubi in Sympy [A]  time = 4.95253, size = 27, normalized size = 0.77 \[ - \frac{\operatorname{atan}{\left (\sqrt{- x^{3} - 1} \right )}}{3} + \frac{\sqrt{- x^{3} - 1}}{3 x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-atan(sqrt(-x**3 - 1))/3 + sqrt(-x**3 - 1)/(3*x**3)

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Mathematica [A]  time = 0.0360163, size = 53, normalized size = 1.51 \[ \frac{\sqrt{-x^3-1}}{3 x^3}-\frac{\sqrt{-x^3-1} \tanh ^{-1}\left (\sqrt{x^3+1}\right )}{3 \sqrt{x^3+1}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.034, size = 28, normalized size = 0.8 \[ -{\frac{1}{3}\arctan \left ( \sqrt{-{x}^{3}-1} \right ) }+{\frac{1}{3\,{x}^{3}}\sqrt{-{x}^{3}-1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^4/(-x^3-1)^(1/2),x)

[Out]

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

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Maxima [A]  time = 1.63732, size = 36, normalized size = 1.03 \[ \frac{\sqrt{-x^{3} - 1}}{3 \, x^{3}} - \frac{1}{3} \, \arctan \left (\sqrt{-x^{3} - 1}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.234282, size = 42, normalized size = 1.2 \[ -\frac{x^{3} \arctan \left (\sqrt{-x^{3} - 1}\right ) - \sqrt{-x^{3} - 1}}{3 \, x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 6.32696, size = 29, normalized size = 0.83 \[ - \frac{i \operatorname{asinh}{\left (\frac{1}{x^{\frac{3}{2}}} \right )}}{3} + \frac{i \sqrt{1 + \frac{1}{x^{3}}}}{3 x^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-I*asinh(x**(-3/2))/3 + I*sqrt(1 + x**(-3))/(3*x**(3/2))

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GIAC/XCAS [A]  time = 0.21876, size = 36, normalized size = 1.03 \[ \frac{\sqrt{-x^{3} - 1}}{3 \, x^{3}} - \frac{1}{3} \, \arctan \left (\sqrt{-x^{3} - 1}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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