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

Optimal. Leaf size=31 \[ \frac{\sqrt{x^3-1}}{3 x^3}+\frac{1}{3} \tan ^{-1}\left (\sqrt{x^3-1}\right ) \]

[Out]

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

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Rubi [A]  time = 0.0368371, antiderivative size = 31, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308 \[ \frac{\sqrt{x^3-1}}{3 x^3}+\frac{1}{3} \tan ^{-1}\left (\sqrt{x^3-1}\right ) \]

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 = 4.08683, size = 24, 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 = 0.0374508, size = 41, normalized size = 1.32 \[ \frac{1}{3} \sqrt{x^3-1} \left (\frac{1}{x^3}+\frac{\tanh ^{-1}\left (\sqrt{1-x^3}\right )}{\sqrt{1-x^3}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.033, size = 24, 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 = 1.59887, size = 31, normalized size = 1. \[ \frac{\sqrt{x^{3} - 1}}{3 \, x^{3}} + \frac{1}{3} \, \arctan \left (\sqrt{x^{3} - 1}\right ) \]

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 = 0.234763, size = 34, normalized size = 1.1 \[ \frac{x^{3} \arctan \left (\sqrt{x^{3} - 1}\right ) + \sqrt{x^{3} - 1}}{3 \, x^{3}} \]

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 = 6.35515, size = 82, normalized size = 2.65 \[ \begin{cases} \frac{i \operatorname{acosh}{\left (\frac{1}{x^{\frac{3}{2}}} \right )}}{3} + \frac{i \sqrt{-1 + \frac{1}{x^{3}}}}{3 x^{\frac{3}{2}}} & \text{for}\: \left |{\frac{1}{x^{3}}}\right | > 1 \\- \frac{\operatorname{asin}{\left (\frac{1}{x^{\frac{3}{2}}} \right )}}{3} + \frac{1}{3 x^{\frac{3}{2}} \sqrt{1 - \frac{1}{x^{3}}}} - \frac{1}{3 x^{\frac{9}{2}} \sqrt{1 - \frac{1}{x^{3}}}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.211367, size = 31, normalized size = 1. \[ \frac{\sqrt{x^{3} - 1}}{3 \, x^{3}} + \frac{1}{3} \, \arctan \left (\sqrt{x^{3} - 1}\right ) \]

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))