Optimal. Leaf size=58 \[ \frac {1}{2} \sqrt {3} \tan ^{-1}\left (\frac {1+2 \sqrt [3]{1-x^2}}{\sqrt {3}}\right )-\frac {\log (x)}{2}+\frac {3}{4} \log \left (1-\sqrt [3]{1-x^2}\right ) \]
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Rubi [A]
time = 0.02, 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 = {272, 57, 632,
210, 31} \begin {gather*} \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} \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 57
Rule 210
Rule 272
Rule 632
Rubi steps
\begin {align*} \int \frac {1}{x \sqrt [3]{1-x^2}} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {1}{\sqrt [3]{1-x} x} \, dx,x,x^2\right )\\ &=-\frac {\log (x)}{2}-\frac {3}{4} \text {Subst}\left (\int \frac {1}{1-x} \, dx,x,\sqrt [3]{1-x^2}\right )+\frac {3}{4} \text {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} \text {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 = 0.04, size = 77, normalized size = 1.33 \begin {gather*} \frac {1}{4} \left (2 \sqrt {3} \tan ^{-1}\left (\frac {1+2 \sqrt [3]{1-x^2}}{\sqrt {3}}\right )+2 \log \left (-1+\sqrt [3]{1-x^2}\right )-\log \left (1+\sqrt [3]{1-x^2}+\left (1-x^2\right )^{2/3}\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Mathics [C] Result contains higher order function than in optimal. Order 9 vs. order 3 in
optimal.
time = 2.03, size = 17, normalized size = 0.29 \begin {gather*} 3-\frac {\text {hyper}\left [\left \{\frac {1}{3},\frac {1}{3}\right \},\left \{\frac {4}{3}\right \},\frac {1}{x^2}\right ]}{2 x^{\frac {2}{3}}} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [C] Result contains higher order function than in optimal. Order 5 vs. order
3.
time = 1.09, size = 65, normalized size = 1.12
method | result | size |
meijerg | \(\frac {\sqrt {3}\, \Gamma \left (\frac {2}{3}\right ) \left (\frac {2 \left (-\frac {\pi \sqrt {3}}{6}-\frac {3 \ln \left (3\right )}{2}+2 \ln \left (x \right )+i \pi \right ) \pi \sqrt {3}}{3 \Gamma \left (\frac {2}{3}\right )}+\frac {2 \pi \sqrt {3}\, x^{2} \hypergeom \left (\left [1, 1, \frac {4}{3}\right ], \left [2, 2\right ], x^{2}\right )}{9 \Gamma \left (\frac {2}{3}\right )}\right )}{4 \pi }\) | \(65\) |
trager | \(\frac {\ln \left (-\frac {4 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{2}+15 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (-x^{2}+1\right )^{\frac {2}{3}}+17 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{2}+24 \left (-x^{2}+1\right )^{\frac {2}{3}}+9 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (-x^{2}+1\right )^{\frac {1}{3}}-16 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2}+15 x^{2}-15 \left (-x^{2}+1\right )^{\frac {1}{3}}-40 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )-25}{x^{2}}\right )}{2}+\frac {\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \ln \left (\frac {5 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{2}+15 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (-x^{2}+1\right )^{\frac {2}{3}}-6 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{2}-9 \left (-x^{2}+1\right )^{\frac {2}{3}}-24 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (-x^{2}+1\right )^{\frac {1}{3}}-20 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2}-8 x^{2}-15 \left (-x^{2}+1\right )^{\frac {1}{3}}-11 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )+4}{x^{2}}\right )}{2}\) | \(246\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.32, size = 62, normalized size = 1.07 \begin {gather*} \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 ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.31, size = 64, normalized size = 1.10 \begin {gather*} \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 ) \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 = 0.42, size = 36, normalized size = 0.62 \begin {gather*} - \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 )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.00, size = 81, normalized size = 1.40 \begin {gather*} \frac {3}{2} \left (-\frac {\ln \left (\left (\left (-x^{2}+1\right )^{\frac {1}{3}}\right )^{2}+\left (-x^{2}+1\right )^{\frac {1}{3}}+1\right )}{6}+\frac {1}{3} \sqrt {3} \arctan \left (\frac {2 \left (\left (-x^{2}+1\right )^{\frac {1}{3}}+\frac 1{2}\right )}{\sqrt {3}}\right )+\frac {\ln \left |\left (-x^{2}+1\right )^{\frac {1}{3}}-1\right |}{3}\right ) \end {gather*}
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 \begin {gather*} \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 ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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