Optimal. Leaf size=58 \[ \frac{3}{4} \log \left (1-\sqrt [3]{1-x^2}\right )-\frac{1}{2} \sqrt{3} \tan ^{-1}\left (\frac{2 \sqrt [3]{1-x^2}+1}{\sqrt{3}}\right )-\frac{\log (x)}{2} \]
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Rubi [A] time = 0.069181, antiderivative size = 58, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ \frac{3}{4} \log \left (1-\sqrt [3]{1-x^2}\right )-\frac{1}{2} \sqrt{3} \tan ^{-1}\left (\frac{2 \sqrt [3]{1-x^2}+1}{\sqrt{3}}\right )-\frac{\log (x)}{2} \]
Antiderivative was successfully verified.
[In] Int[1/(x*(1 - x^2)^(2/3)),x]
[Out]
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Rubi in Sympy [A] time = 2.45289, size = 49, normalized size = 0.84 \[ - \frac{\log{\left (x^{2} \right )}}{4} + \frac{3 \log{\left (- \sqrt [3]{- x^{2} + 1} + 1 \right )}}{4} - \frac{\sqrt{3} \operatorname{atan}{\left (\sqrt{3} \left (\frac{2 \sqrt [3]{- x^{2} + 1}}{3} + \frac{1}{3}\right ) \right )}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/x/(-x**2+1)**(2/3),x)
[Out]
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Mathematica [C] time = 0.0202607, size = 41, normalized size = 0.71 \[ -\frac{3 \left (\frac{x^2-1}{x^2}\right )^{2/3} \, _2F_1\left (\frac{2}{3},\frac{2}{3};\frac{5}{3};\frac{1}{x^2}\right )}{4 \left (1-x^2\right )^{2/3}} \]
Antiderivative was successfully verified.
[In] Integrate[1/(x*(1 - x^2)^(2/3)),x]
[Out]
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Maple [C] time = 0.063, size = 48, normalized size = 0.8 \[{\frac{1}{2\,\Gamma \left ( 2/3 \right ) } \left ( \left ({\frac{\pi \,\sqrt{3}}{6}}-{\frac{3\,\ln \left ( 3 \right ) }{2}}+2\,\ln \left ( x \right ) +i\pi \right ) \Gamma \left ({\frac{2}{3}} \right ) +{\frac{2\,\Gamma \left ( 2/3 \right ){x}^{2}}{3}{\mbox{$_3$F$_2$}(1,1,{\frac{5}{3}};\,2,2;\,{x}^{2})}} \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/x/(-x^2+1)^(2/3),x)
[Out]
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Maxima [A] time = 1.49964, size = 84, normalized size = 1.45 \[ -\frac{1}{2} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \,{\left (-x^{2} + 1\right )}^{\frac{1}{3}} + 1\right )}\right ) - \frac{1}{4} \, \log \left ({\left (-x^{2} + 1\right )}^{\frac{2}{3}} +{\left (-x^{2} + 1\right )}^{\frac{1}{3}} + 1\right ) + \frac{1}{2} \, \log \left ({\left (-x^{2} + 1\right )}^{\frac{1}{3}} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((-x^2 + 1)^(2/3)*x),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.21626, size = 84, normalized size = 1.45 \[ -\frac{1}{2} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \,{\left (-x^{2} + 1\right )}^{\frac{1}{3}} + 1\right )}\right ) - \frac{1}{4} \, \log \left ({\left (-x^{2} + 1\right )}^{\frac{2}{3}} +{\left (-x^{2} + 1\right )}^{\frac{1}{3}} + 1\right ) + \frac{1}{2} \, \log \left ({\left (-x^{2} + 1\right )}^{\frac{1}{3}} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((-x^2 + 1)^(2/3)*x),x, algorithm="fricas")
[Out]
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Sympy [A] time = 1.71959, size = 37, normalized size = 0.64 \[ - \frac{e^{- \frac{2 i \pi }{3}} \Gamma \left (\frac{2}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{2}{3}, \frac{2}{3} \\ \frac{5}{3} \end{matrix}\middle |{\frac{1}{x^{2}}} \right )}}{2 x^{\frac{4}{3}} \Gamma \left (\frac{5}{3}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/x/(-x**2+1)**(2/3),x)
[Out]
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GIAC/XCAS [A] time = 0.204186, size = 86, normalized size = 1.48 \[ -\frac{1}{2} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \,{\left (-x^{2} + 1\right )}^{\frac{1}{3}} + 1\right )}\right ) - \frac{1}{4} \,{\rm ln}\left ({\left (-x^{2} + 1\right )}^{\frac{2}{3}} +{\left (-x^{2} + 1\right )}^{\frac{1}{3}} + 1\right ) + \frac{1}{2} \,{\rm ln}\left (-{\left (-x^{2} + 1\right )}^{\frac{1}{3}} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((-x^2 + 1)^(2/3)*x),x, algorithm="giac")
[Out]