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