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