Optimal. Leaf size=49 \[ \frac {1}{2} \log \left (\sqrt [3]{1-x^3}+x\right )-\frac {\tan ^{-1}\left (\frac {1-\frac {2 x}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{\sqrt {3}} \]
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Rubi [A] time = 0.01, antiderivative size = 49, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {239} \[ \frac {1}{2} \log \left (\sqrt [3]{1-x^3}+x\right )-\frac {\tan ^{-1}\left (\frac {1-\frac {2 x}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{\sqrt {3}} \]
Antiderivative was successfully verified.
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Rule 239
Rubi steps
\begin {align*} \int \frac {1}{\sqrt [3]{1-x^3}} \, dx &=-\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 )\\ \end {align*}
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Mathematica [A] time = 0.04, size = 86, normalized size = 1.76 \[ \frac {1}{3} \log \left (\frac {x}{\sqrt [3]{1-x^3}}+1\right )+\frac {\tan ^{-1}\left (\frac {\frac {2 x}{\sqrt [3]{1-x^3}}-1}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {1}{6} \log \left (-\frac {x}{\sqrt [3]{1-x^3}}+\frac {x^2}{\left (1-x^3\right )^{2/3}}+1\right ) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.90, size = 82, normalized size = 1.67 \[ -\frac {1}{3} \, \sqrt {3} \arctan \left (-\frac {\sqrt {3} x - 2 \, \sqrt {3} {\left (-x^{3} + 1\right )}^{\frac {1}{3}}}{3 \, x}\right ) + \frac {1}{3} \, \log \left (\frac {x + {\left (-x^{3} + 1\right )}^{\frac {1}{3}}}{x}\right ) - \frac {1}{6} \, \log \left (\frac {x^{2} - {\left (-x^{3} + 1\right )}^{\frac {1}{3}} x + {\left (-x^{3} + 1\right )}^{\frac {2}{3}}}{x^{2}}\right ) \]
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 \[ \int \frac {1}{{\left (-x^{3} + 1\right )}^{\frac {1}{3}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.09, size = 12, normalized size = 0.24 \[ x \hypergeom \left (\left [\frac {1}{3}, \frac {1}{3}\right ], \left [\frac {4}{3}\right ], x^{3}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.36, size = 78, normalized size = 1.59 \[ -\frac {1}{3} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (\frac {2 \, {\left (-x^{3} + 1\right )}^{\frac {1}{3}}}{x} - 1\right )}\right ) + \frac {1}{3} \, \log \left (\frac {{\left (-x^{3} + 1\right )}^{\frac {1}{3}}}{x} + 1\right ) - \frac {1}{6} \, \log \left (-\frac {{\left (-x^{3} + 1\right )}^{\frac {1}{3}}}{x} + \frac {{\left (-x^{3} + 1\right )}^{\frac {2}{3}}}{x^{2}} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.33, size = 10, normalized size = 0.20 \[ x\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{3},\frac {1}{3};\ \frac {4}{3};\ x^3\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 0.89, size = 29, normalized size = 0.59 \[ \frac {x \Gamma \left (\frac {1}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{3}, \frac {1}{3} \\ \frac {4}{3} \end {matrix}\middle | {x^{3} e^{2 i \pi }} \right )}}{3 \Gamma \left (\frac {4}{3}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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