Optimal. Leaf size=53 \[ -\frac {\tan ^{-1}\left (\frac {1-\frac {2 x}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {1}{2} \log \left (-x-\sqrt [3]{1-x^3}\right ) \]
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Rubi [A]
time = 0.00, antiderivative size = 53, normalized size of antiderivative = 1.00, number of steps
used = 1, number of rules used = 1, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {337}
\begin {gather*} -\frac {\text {ArcTan}\left (\frac {1-\frac {2 x}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {1}{2} \log \left (-\sqrt [3]{1-x^3}-x\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 337
Rubi steps
\begin {align*} \int \frac {x}{\left (1-x^3\right )^{2/3}} \, dx &=\text {Subst}\left (\int \frac {x}{1+x^3} \, dx,x,\frac {x}{\sqrt [3]{1-x^3}}\right )\\ &=-\left (\frac {1}{3} \text {Subst}\left (\int \frac {1}{1+x} \, dx,x,\frac {x}{\sqrt [3]{1-x^3}}\right )\right )+\frac {1}{3} \text {Subst}\left (\int \frac {1+x}{1-x+x^2} \, dx,x,\frac {x}{\sqrt [3]{1-x^3}}\right )\\ &=-\frac {1}{3} \log \left (1+\frac {x}{\sqrt [3]{1-x^3}}\right )+\frac {1}{6} \text {Subst}\left (\int \frac {-1+2 x}{1-x+x^2} \, dx,x,\frac {x}{\sqrt [3]{1-x^3}}\right )+\frac {1}{2} \text {Subst}\left (\int \frac {1}{1-x+x^2} \, dx,x,\frac {x}{\sqrt [3]{1-x^3}}\right )\\ &=\frac {1}{6} \log \left (1+\frac {x^2}{\left (1-x^3\right )^{2/3}}-\frac {x}{\sqrt [3]{1-x^3}}\right )-\frac {1}{3} \log \left (1+\frac {x}{\sqrt [3]{1-x^3}}\right )-\text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,-1+\frac {2 x}{\sqrt [3]{1-x^3}}\right )\\ &=\frac {\tan ^{-1}\left (\frac {-1+\frac {2 x}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {1}{6} \log \left (1+\frac {x^2}{\left (1-x^3\right )^{2/3}}-\frac {x}{\sqrt [3]{1-x^3}}\right )-\frac {1}{3} \log \left (1+\frac {x}{\sqrt [3]{1-x^3}}\right )\\ \end {align*}
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Mathematica [A]
time = 0.13, size = 83, normalized size = 1.57 \begin {gather*} \frac {1}{6} \left (-2 \sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x}{x-2 \sqrt [3]{1-x^3}}\right )-2 \log \left (x+\sqrt [3]{1-x^3}\right )+\log \left (x^2-x \sqrt [3]{1-x^3}+\left (1-x^3\right )^{2/3}\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 5 vs. order
3.
time = 1.57, size = 15, normalized size = 0.28
method | result | size |
meijerg | \(\frac {x^{2} \hypergeom \left (\left [\frac {2}{3}, \frac {2}{3}\right ], \left [\frac {5}{3}\right ], x^{3}\right )}{2}\) | \(15\) |
trager | \(\frac {\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \ln \left (\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x^{3}+3 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (-x^{3}+1\right )^{\frac {2}{3}} x -3 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (-x^{3}+1\right )^{\frac {1}{3}} x^{2}+2 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{3}+x^{3}-\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )-1\right )}{3}-\frac {\ln \left (\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x^{3}-3 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (-x^{3}+1\right )^{\frac {2}{3}} x +3 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (-x^{3}+1\right )^{\frac {1}{3}} x^{2}-4 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{3}+3 \left (-x^{3}+1\right )^{\frac {2}{3}} x -3 \left (-x^{3}+1\right )^{\frac {1}{3}} x^{2}+4 x^{3}+\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )-2\right ) \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )}{3}+\frac {\ln \left (\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x^{3}-3 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (-x^{3}+1\right )^{\frac {2}{3}} x +3 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (-x^{3}+1\right )^{\frac {1}{3}} x^{2}-4 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{3}+3 \left (-x^{3}+1\right )^{\frac {2}{3}} x -3 \left (-x^{3}+1\right )^{\frac {1}{3}} x^{2}+4 x^{3}+\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )-2\right )}{3}\) | \(348\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.50, size = 78, normalized size = 1.47 \begin {gather*} -\frac {1}{3} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (\frac {2 \, {\left (-x^{3} + 1\right )}^{\frac {1}{3}}}{x} - 1\right )}\right ) - \frac {1}{3} \, \log \left (\frac {{\left (-x^{3} + 1\right )}^{\frac {1}{3}}}{x} + 1\right ) + \frac {1}{6} \, \log \left (-\frac {{\left (-x^{3} + 1\right )}^{\frac {1}{3}}}{x} + \frac {{\left (-x^{3} + 1\right )}^{\frac {2}{3}}}{x^{2}} + 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.38, size = 82, normalized size = 1.55 \begin {gather*} -\frac {1}{3} \, \sqrt {3} \arctan \left (-\frac {\sqrt {3} x - 2 \, \sqrt {3} {\left (-x^{3} + 1\right )}^{\frac {1}{3}}}{3 \, x}\right ) - \frac {1}{3} \, \log \left (\frac {x + {\left (-x^{3} + 1\right )}^{\frac {1}{3}}}{x}\right ) + \frac {1}{6} \, \log \left (\frac {x^{2} - {\left (-x^{3} + 1\right )}^{\frac {1}{3}} x + {\left (-x^{3} + 1\right )}^{\frac {2}{3}}}{x^{2}}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] Result contains complex when optimal does not.
time = 0.38, size = 31, normalized size = 0.58 \begin {gather*} \frac {x^{2} \Gamma \left (\frac {2}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {2}{3}, \frac {2}{3} \\ \frac {5}{3} \end {matrix}\middle | {x^{3} e^{2 i \pi }} \right )}}{3 \Gamma \left (\frac {5}{3}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {x}{{\left (1-x^3\right )}^{2/3}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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