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

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

[Out]

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

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Rubi [A]  time = 0.0461495, 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 \[ -\frac{\sqrt{1-x^3}}{3 x^3}-\frac{1}{3} \tanh ^{-1}\left (\sqrt{1-x^3}\right ) \]

Antiderivative was successfully verified.

[In]  Int[1/(x^4*Sqrt[1 - x^3]),x]

[Out]

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

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Rubi in Sympy [A]  time = 4.80611, size = 26, normalized size = 0.74 \[ - \frac{\operatorname{atanh}{\left (\sqrt{- x^{3} + 1} \right )}}{3} - \frac{\sqrt{- x^{3} + 1}}{3 x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0292647, size = 35, normalized size = 1. \[ -\frac{\sqrt{1-x^3}}{3 x^3}-\frac{1}{3} \tanh ^{-1}\left (\sqrt{1-x^3}\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^4/(-x^3+1)^(1/2),x)

[Out]

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

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Maxima [A]  time = 1.43616, size = 58, normalized size = 1.66 \[ -\frac{\sqrt{-x^{3} + 1}}{3 \, x^{3}} - \frac{1}{6} \, \log \left (\sqrt{-x^{3} + 1} + 1\right ) + \frac{1}{6} \, \log \left (\sqrt{-x^{3} + 1} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/3*sqrt(-x^3 + 1)/x^3 - 1/6*log(sqrt(-x^3 + 1) + 1) + 1/6*log(sqrt(-x^3 + 1) -
1)

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Fricas [A]  time = 0.234011, size = 68, normalized size = 1.94 \[ -\frac{x^{3} \log \left (\sqrt{-x^{3} + 1} + 1\right ) - x^{3} \log \left (\sqrt{-x^{3} + 1} - 1\right ) + 2 \, \sqrt{-x^{3} + 1}}{6 \, x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/6*(x^3*log(sqrt(-x^3 + 1) + 1) - x^3*log(sqrt(-x^3 + 1) - 1) + 2*sqrt(-x^3 +
1))/x^3

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Sympy [A]  time = 6.46167, size = 82, normalized size = 2.34 \[ \begin{cases} - \frac{\operatorname{acosh}{\left (\frac{1}{x^{\frac{3}{2}}} \right )}}{3} - \frac{\sqrt{-1 + \frac{1}{x^{3}}}}{3 x^{\frac{3}{2}}} & \text{for}\: \left |{\frac{1}{x^{3}}}\right | > 1 \\\frac{i \operatorname{asin}{\left (\frac{1}{x^{\frac{3}{2}}} \right )}}{3} - \frac{i}{3 x^{\frac{3}{2}} \sqrt{1 - \frac{1}{x^{3}}}} + \frac{i}{3 x^{\frac{9}{2}} \sqrt{1 - \frac{1}{x^{3}}}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((-acosh(x**(-3/2))/3 - sqrt(-1 + x**(-3))/(3*x**(3/2)), Abs(x**(-3)) >
1), (I*asin(x**(-3/2))/3 - I/(3*x**(3/2)*sqrt(1 - 1/x**3)) + I/(3*x**(9/2)*sqrt
(1 - 1/x**3)), True))

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GIAC/XCAS [A]  time = 0.221617, size = 59, normalized size = 1.69 \[ -\frac{\sqrt{-x^{3} + 1}}{3 \, x^{3}} - \frac{1}{6} \,{\rm ln}\left (\sqrt{-x^{3} + 1} + 1\right ) + \frac{1}{6} \,{\rm ln}\left ({\left | \sqrt{-x^{3} + 1} - 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/3*sqrt(-x^3 + 1)/x^3 - 1/6*ln(sqrt(-x^3 + 1) + 1) + 1/6*ln(abs(sqrt(-x^3 + 1)
- 1))