Optimal. Leaf size=47 \[ \frac{\sqrt{x^3-1}}{4 x^3}+\frac{1}{4} \tan ^{-1}\left (\sqrt{x^3-1}\right )+\frac{\sqrt{x^3-1}}{6 x^6} \]
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Rubi [A] time = 0.0495756, antiderivative size = 47, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308 \[ \frac{\sqrt{x^3-1}}{4 x^3}+\frac{1}{4} \tan ^{-1}\left (\sqrt{x^3-1}\right )+\frac{\sqrt{x^3-1}}{6 x^6} \]
Antiderivative was successfully verified.
[In] Int[1/(x^7*Sqrt[-1 + x^3]),x]
[Out]
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Rubi in Sympy [A] time = 4.84446, size = 37, 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 = 0.0597511, size = 48, normalized size = 1.02 \[ \frac{1}{4} \sqrt{x^3-1} \left (\frac{2}{3 x^6}+\frac{1}{x^3}+\frac{\tanh ^{-1}\left (\sqrt{1-x^3}\right )}{\sqrt{1-x^3}}\right ) \]
Antiderivative was successfully verified.
[In] Integrate[1/(x^7*Sqrt[-1 + x^3]),x]
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Maple [A] time = 0.03, size = 36, 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 = 1.58574, size = 65, normalized size = 1.38 \[ \frac{3 \,{\left (x^{3} - 1\right )}^{\frac{3}{2}} + 5 \, \sqrt{x^{3} - 1}}{12 \,{\left (2 \, x^{3} +{\left (x^{3} - 1\right )}^{2} - 1\right )}} + \frac{1}{4} \, \arctan \left (\sqrt{x^{3} - 1}\right ) \]
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 = 0.239076, size = 46, normalized size = 0.98 \[ \frac{3 \, x^{6} \arctan \left (\sqrt{x^{3} - 1}\right ) +{\left (3 \, x^{3} + 2\right )} \sqrt{x^{3} - 1}}{12 \, x^{6}} \]
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 = 10.6037, size = 138, normalized size = 2.94 \[ \begin{cases} \frac{i \operatorname{acosh}{\left (\frac{1}{x^{\frac{3}{2}}} \right )}}{4} - \frac{i}{4 x^{\frac{3}{2}} \sqrt{-1 + \frac{1}{x^{3}}}} + \frac{i}{12 x^{\frac{9}{2}} \sqrt{-1 + \frac{1}{x^{3}}}} + \frac{i}{6 x^{\frac{15}{2}} \sqrt{-1 + \frac{1}{x^{3}}}} & \text{for}\: \left |{\frac{1}{x^{3}}}\right | > 1 \\- \frac{\operatorname{asin}{\left (\frac{1}{x^{\frac{3}{2}}} \right )}}{4} + \frac{1}{4 x^{\frac{3}{2}} \sqrt{1 - \frac{1}{x^{3}}}} - \frac{1}{12 x^{\frac{9}{2}} \sqrt{1 - \frac{1}{x^{3}}}} - \frac{1}{6 x^{\frac{15}{2}} \sqrt{1 - \frac{1}{x^{3}}}} & \text{otherwise} \end{cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/x**7/(x**3-1)**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.221712, size = 47, normalized size = 1. \[ \frac{3 \,{\left (x^{3} - 1\right )}^{\frac{3}{2}} + 5 \, \sqrt{x^{3} - 1}}{12 \, x^{6}} + \frac{1}{4} \, \arctan \left (\sqrt{x^{3} - 1}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt(x^3 - 1)*x^7),x, algorithm="giac")
[Out]