Optimal. Leaf size=67 \[ \sqrt [3]{1-x^3}+\frac{1}{2} \log \left (1-\sqrt [3]{1-x^3}\right )-\frac{\tan ^{-1}\left (\frac{2 \sqrt [3]{1-x^3}+1}{\sqrt{3}}\right )}{\sqrt{3}}-\frac{\log (x)}{2} \]
[Out]
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Rubi [A] time = 0.0840064, antiderivative size = 67, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4 \[ \sqrt [3]{1-x^3}+\frac{1}{2} \log \left (1-\sqrt [3]{1-x^3}\right )-\frac{\tan ^{-1}\left (\frac{2 \sqrt [3]{1-x^3}+1}{\sqrt{3}}\right )}{\sqrt{3}}-\frac{\log (x)}{2} \]
Antiderivative was successfully verified.
[In] Int[(1 - x^3)^(1/3)/x,x]
[Out]
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Rubi in Sympy [A] time = 2.76762, size = 56, normalized size = 0.84 \[ \sqrt [3]{- x^{3} + 1} - \frac{\log{\left (x^{3} \right )}}{6} + \frac{\log{\left (- \sqrt [3]{- x^{3} + 1} + 1 \right )}}{2} - \frac{\sqrt{3} \operatorname{atan}{\left (\sqrt{3} \left (\frac{2 \sqrt [3]{- x^{3} + 1}}{3} + \frac{1}{3}\right ) \right )}}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((-x**3+1)**(1/3)/x,x)
[Out]
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Mathematica [C] time = 0.0246976, size = 48, normalized size = 0.72 \[ \frac{-\left (1-\frac{1}{x^3}\right )^{2/3} \, _2F_1\left (\frac{2}{3},\frac{2}{3};\frac{5}{3};\frac{1}{x^3}\right )-2 x^3+2}{2 \left (1-x^3\right )^{2/3}} \]
Antiderivative was successfully verified.
[In] Integrate[(1 - x^3)^(1/3)/x,x]
[Out]
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Maple [C] time = 0.073, size = 49, normalized size = 0.7 \[ -{\frac{1}{9\,\Gamma \left ( 2/3 \right ) } \left ( -3\, \left ( 3+1/6\,\pi \,\sqrt{3}-3/2\,\ln \left ( 3 \right ) +3\,\ln \left ( x \right ) +i\pi \right ) \Gamma \left ( 2/3 \right ) +\Gamma \left ({\frac{2}{3}} \right ){x}^{3}{\mbox{$_3$F$_2$}({\frac{2}{3}},1,1;\,2,2;\,{x}^{3})} \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((-x^3+1)^(1/3)/x,x)
[Out]
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Maxima [A] time = 1.53367, size = 96, normalized size = 1.43 \[ -\frac{1}{3} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \,{\left (-x^{3} + 1\right )}^{\frac{1}{3}} + 1\right )}\right ) +{\left (-x^{3} + 1\right )}^{\frac{1}{3}} - \frac{1}{6} \, \log \left ({\left (-x^{3} + 1\right )}^{\frac{2}{3}} +{\left (-x^{3} + 1\right )}^{\frac{1}{3}} + 1\right ) + \frac{1}{3} \, \log \left ({\left (-x^{3} + 1\right )}^{\frac{1}{3}} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((-x^3 + 1)^(1/3)/x,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.208472, size = 115, normalized size = 1.72 \[ -\frac{1}{18} \, \sqrt{3}{\left (\sqrt{3} \log \left ({\left (-x^{3} + 1\right )}^{\frac{2}{3}} +{\left (-x^{3} + 1\right )}^{\frac{1}{3}} + 1\right ) - 2 \, \sqrt{3} \log \left ({\left (-x^{3} + 1\right )}^{\frac{1}{3}} - 1\right ) - 6 \, \sqrt{3}{\left (-x^{3} + 1\right )}^{\frac{1}{3}} + 6 \, \arctan \left (\frac{2}{3} \, \sqrt{3}{\left (-x^{3} + 1\right )}^{\frac{1}{3}} + \frac{1}{3} \, \sqrt{3}\right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((-x^3 + 1)^(1/3)/x,x, algorithm="fricas")
[Out]
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Sympy [A] time = 1.67805, size = 37, normalized size = 0.55 \[ - \frac{x e^{\frac{i \pi }{3}} \Gamma \left (- \frac{1}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{3}, - \frac{1}{3} \\ \frac{2}{3} \end{matrix}\middle |{\frac{1}{x^{3}}} \right )}}{3 \Gamma \left (\frac{2}{3}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((-x**3+1)**(1/3)/x,x)
[Out]
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GIAC/XCAS [A] time = 0.24, size = 97, normalized size = 1.45 \[ -\frac{1}{3} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \,{\left (-x^{3} + 1\right )}^{\frac{1}{3}} + 1\right )}\right ) +{\left (-x^{3} + 1\right )}^{\frac{1}{3}} - \frac{1}{6} \,{\rm ln}\left ({\left (-x^{3} + 1\right )}^{\frac{2}{3}} +{\left (-x^{3} + 1\right )}^{\frac{1}{3}} + 1\right ) + \frac{1}{3} \,{\rm ln}\left ({\left |{\left (-x^{3} + 1\right )}^{\frac{1}{3}} - 1 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((-x^3 + 1)^(1/3)/x,x, algorithm="giac")
[Out]