Optimal. Leaf size=53 \[ -\frac{\sqrt{-x^3-1}}{4 x^3}+\frac{1}{4} \tan ^{-1}\left (\sqrt{-x^3-1}\right )+\frac{\sqrt{-x^3-1}}{6 x^6} \]
[Out]
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Rubi [A] time = 0.0611939, antiderivative size = 53, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267 \[ -\frac{\sqrt{-x^3-1}}{4 x^3}+\frac{1}{4} \tan ^{-1}\left (\sqrt{-x^3-1}\right )+\frac{\sqrt{-x^3-1}}{6 x^6} \]
Antiderivative was successfully verified.
[In] Int[1/(x^7*Sqrt[-1 - x^3]),x]
[Out]
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Rubi in Sympy [A] time = 6.07326, size = 42, normalized size = 0.79 \[ \frac{\operatorname{atan}{\left (\sqrt{- x^{3} - 1} \right )}}{4} - \frac{\sqrt{- x^{3} - 1}}{4 x^{3}} + \frac{\sqrt{- x^{3} - 1}}{6 x^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/x**7/(-x**3-1)**(1/2),x)
[Out]
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Mathematica [A] time = 0.044372, size = 62, normalized size = 1.17 \[ \frac{\sqrt{-x^3-1} \tanh ^{-1}\left (\sqrt{x^3+1}\right )}{4 \sqrt{x^3+1}}+\sqrt{-x^3-1} \left (\frac{1}{6 x^6}-\frac{1}{4 x^3}\right ) \]
Antiderivative was successfully verified.
[In] Integrate[1/(x^7*Sqrt[-1 - x^3]),x]
[Out]
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Maple [A] time = 0.035, size = 42, normalized size = 0.8 \[{\frac{1}{4}\arctan \left ( \sqrt{-{x}^{3}-1} \right ) }+{\frac{1}{6\,{x}^{6}}\sqrt{-{x}^{3}-1}}-{\frac{1}{4\,{x}^{3}}\sqrt{-{x}^{3}-1}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/x^7/(-x^3-1)^(1/2),x)
[Out]
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Maxima [A] time = 1.57431, size = 76, normalized size = 1.43 \[ -\frac{3 \,{\left (-x^{3} - 1\right )}^{\frac{3}{2}} + 5 \, \sqrt{-x^{3} - 1}}{12 \,{\left (2 \, x^{3} -{\left (x^{3} + 1\right )}^{2} + 1\right )}} + \frac{1}{4} \, \arctan \left (\sqrt{-x^{3} - 1}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt(-x^3 - 1)*x^7),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.22738, size = 53, normalized size = 1. \[ \frac{3 \, x^{6} \arctan \left (\sqrt{-x^{3} - 1}\right ) -{\left (3 \, x^{3} - 2\right )} \sqrt{-x^{3} - 1}}{12 \, x^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt(-x^3 - 1)*x^7),x, algorithm="fricas")
[Out]
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Sympy [A] time = 10.7799, size = 66, normalized size = 1.25 \[ \frac{i \operatorname{asinh}{\left (\frac{1}{x^{\frac{3}{2}}} \right )}}{4} - \frac{i}{4 x^{\frac{3}{2}} \sqrt{1 + \frac{1}{x^{3}}}} - \frac{i}{12 x^{\frac{9}{2}} \sqrt{1 + \frac{1}{x^{3}}}} + \frac{i}{6 x^{\frac{15}{2}} \sqrt{1 + \frac{1}{x^{3}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/x**7/(-x**3-1)**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.213917, size = 55, normalized size = 1.04 \[ \frac{3 \,{\left (-x^{3} - 1\right )}^{\frac{3}{2}} + 5 \, \sqrt{-x^{3} - 1}}{12 \, x^{6}} + \frac{1}{4} \, \arctan \left (\sqrt{-x^{3} - 1}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt(-x^3 - 1)*x^7),x, algorithm="giac")
[Out]