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 = 0.04, antiderivative size = 58, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {266, 55, 618, 204, 31} \[ \frac {3}{4} \log \left (1-\sqrt [3]{1-x^2}\right )+\frac {1}{2} \sqrt {3} \tan ^{-1}\left (\frac {2 \sqrt [3]{1-x^2}+1}{\sqrt {3}}\right )-\frac {\log (x)}{2} \]
Antiderivative was successfully verified.
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Rule 31
Rule 55
Rule 204
Rule 266
Rule 618
Rubi steps
\begin {align*} \int \frac {1}{x \sqrt [3]{1-x^2}} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{\sqrt [3]{1-x} x} \, dx,x,x^2\right )\\ &=-\frac {\log (x)}{2}-\frac {3}{4} \operatorname {Subst}\left (\int \frac {1}{1-x} \, dx,x,\sqrt [3]{1-x^2}\right )+\frac {3}{4} \operatorname {Subst}\left (\int \frac {1}{1+x+x^2} \, dx,x,\sqrt [3]{1-x^2}\right )\\ &=-\frac {\log (x)}{2}+\frac {3}{4} \log \left (1-\sqrt [3]{1-x^2}\right )-\frac {3}{2} \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+2 \sqrt [3]{1-x^2}\right )\\ &=\frac {1}{2} \sqrt {3} \tan ^{-1}\left (\frac {1+2 \sqrt [3]{1-x^2}}{\sqrt {3}}\right )-\frac {\log (x)}{2}+\frac {3}{4} \log \left (1-\sqrt [3]{1-x^2}\right )\\ \end {align*}
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Mathematica [A] time = 0.01, size = 57, normalized size = 0.98 \[ \frac {1}{2} \left (\frac {3}{2} \log \left (1-\sqrt [3]{1-x^2}\right )+\sqrt {3} \tan ^{-1}\left (\frac {2 \sqrt [3]{1-x^2}+1}{\sqrt {3}}\right )-\log (x)\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.95, size = 64, normalized size = 1.10 \[ \frac {1}{2} \, \sqrt {3} \arctan \left (\frac {2}{3} \, \sqrt {3} {\left (-x^{2} + 1\right )}^{\frac {1}{3}} + \frac {1}{3} \, \sqrt {3}\right ) - \frac {1}{4} \, \log \left ({\left (-x^{2} + 1\right )}^{\frac {2}{3}} + {\left (-x^{2} + 1\right )}^{\frac {1}{3}} + 1\right ) + \frac {1}{2} \, \log \left ({\left (-x^{2} + 1\right )}^{\frac {1}{3}} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.92, size = 64, normalized size = 1.10 \[ \frac {1}{2} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (-x^{2} + 1\right )}^{\frac {1}{3}} + 1\right )}\right ) - \frac {1}{4} \, \log \left ({\left (-x^{2} + 1\right )}^{\frac {2}{3}} + {\left (-x^{2} + 1\right )}^{\frac {1}{3}} + 1\right ) + \frac {1}{2} \, \log \left (-{\left (-x^{2} + 1\right )}^{\frac {1}{3}} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.10, size = 65, normalized size = 1.12 \[ \frac {\sqrt {3}\, \Gamma \left (\frac {2}{3}\right ) \left (\frac {2 \pi \sqrt {3}\, x^{2} \hypergeom \left (\left [1, 1, \frac {4}{3}\right ], \left [2, 2\right ], x^{2}\right )}{9 \Gamma \left (\frac {2}{3}\right )}+\frac {2 \left (2 \ln \relax (x )-\frac {\pi \sqrt {3}}{6}-\frac {3 \ln \relax (3)}{2}+i \pi \right ) \pi \sqrt {3}}{3 \Gamma \left (\frac {2}{3}\right )}\right )}{4 \pi } \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.09, size = 62, normalized size = 1.07 \[ \frac {1}{2} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (-x^{2} + 1\right )}^{\frac {1}{3}} + 1\right )}\right ) - \frac {1}{4} \, \log \left ({\left (-x^{2} + 1\right )}^{\frac {2}{3}} + {\left (-x^{2} + 1\right )}^{\frac {1}{3}} + 1\right ) + \frac {1}{2} \, \log \left ({\left (-x^{2} + 1\right )}^{\frac {1}{3}} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.54, size = 86, normalized size = 1.48 \[ \frac {\ln \left (\frac {9\,{\left (1-x^2\right )}^{1/3}}{4}-\frac {9}{4}\right )}{2}+\ln \left (\frac {9\,{\left (1-x^2\right )}^{1/3}}{4}-9\,{\left (-\frac {1}{4}+\frac {\sqrt {3}\,1{}\mathrm {i}}{4}\right )}^2\right )\,\left (-\frac {1}{4}+\frac {\sqrt {3}\,1{}\mathrm {i}}{4}\right )-\ln \left (\frac {9\,{\left (1-x^2\right )}^{1/3}}{4}-9\,{\left (\frac {1}{4}+\frac {\sqrt {3}\,1{}\mathrm {i}}{4}\right )}^2\right )\,\left (\frac {1}{4}+\frac {\sqrt {3}\,1{}\mathrm {i}}{4}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 0.94, size = 36, normalized size = 0.62 \[ - \frac {e^{- \frac {i \pi }{3}} \Gamma \left (\frac {1}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{3}, \frac {1}{3} \\ \frac {4}{3} \end {matrix}\middle | {\frac {1}{x^{2}}} \right )}}{2 x^{\frac {2}{3}} \Gamma \left (\frac {4}{3}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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