Optimal. Leaf size=93 \[ \frac {1}{2} \sqrt [3]{x \left (1-x^2\right )} x-\frac {1}{4} \log \left (\sqrt [3]{x \left (1-x^2\right )}+x\right )+\frac {\tan ^{-1}\left (\frac {2 x-\sqrt [3]{x \left (1-x^2\right )}}{\sqrt {3} \sqrt [3]{x \left (1-x^2\right )}}\right )}{2 \sqrt {3}}+\frac {\log (x)}{12} \]
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Rubi [B] time = 0.14, antiderivative size = 200, normalized size of antiderivative = 2.15, number of steps used = 12, number of rules used = 12, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.923, Rules used = {1979, 2004, 2032, 329, 275, 331, 292, 31, 634, 618, 204, 628} \[ \frac {1}{2} \sqrt [3]{x-x^3} x+\frac {\left (1-x^2\right )^{2/3} x^{2/3} \log \left (\frac {x^{4/3}}{\left (1-x^2\right )^{2/3}}-\frac {x^{2/3}}{\sqrt [3]{1-x^2}}+1\right )}{12 \left (x-x^3\right )^{2/3}}-\frac {\left (1-x^2\right )^{2/3} x^{2/3} \log \left (\frac {x^{2/3}}{\sqrt [3]{1-x^2}}+1\right )}{6 \left (x-x^3\right )^{2/3}}-\frac {\left (1-x^2\right )^{2/3} x^{2/3} \tan ^{-1}\left (\frac {1-\frac {2 x^{2/3}}{\sqrt [3]{1-x^2}}}{\sqrt {3}}\right )}{2 \sqrt {3} \left (x-x^3\right )^{2/3}} \]
Antiderivative was successfully verified.
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Rule 31
Rule 204
Rule 275
Rule 292
Rule 329
Rule 331
Rule 618
Rule 628
Rule 634
Rule 1979
Rule 2004
Rule 2032
Rubi steps
\begin {align*} \int \sqrt [3]{x \left (1-x^2\right )} \, dx &=\int \sqrt [3]{x-x^3} \, dx\\ &=\frac {1}{2} x \sqrt [3]{x-x^3}+\frac {1}{3} \int \frac {x}{\left (x-x^3\right )^{2/3}} \, dx\\ &=\frac {1}{2} x \sqrt [3]{x-x^3}+\frac {\left (x^{2/3} \left (1-x^2\right )^{2/3}\right ) \int \frac {\sqrt [3]{x}}{\left (1-x^2\right )^{2/3}} \, dx}{3 \left (x-x^3\right )^{2/3}}\\ &=\frac {1}{2} x \sqrt [3]{x-x^3}+\frac {\left (x^{2/3} \left (1-x^2\right )^{2/3}\right ) \operatorname {Subst}\left (\int \frac {x^3}{\left (1-x^6\right )^{2/3}} \, dx,x,\sqrt [3]{x}\right )}{\left (x-x^3\right )^{2/3}}\\ &=\frac {1}{2} x \sqrt [3]{x-x^3}+\frac {\left (x^{2/3} \left (1-x^2\right )^{2/3}\right ) \operatorname {Subst}\left (\int \frac {x}{\left (1-x^3\right )^{2/3}} \, dx,x,x^{2/3}\right )}{2 \left (x-x^3\right )^{2/3}}\\ &=\frac {1}{2} x \sqrt [3]{x-x^3}+\frac {\left (x^{2/3} \left (1-x^2\right )^{2/3}\right ) \operatorname {Subst}\left (\int \frac {x}{1+x^3} \, dx,x,\frac {x^{2/3}}{\sqrt [3]{1-x^2}}\right )}{2 \left (x-x^3\right )^{2/3}}\\ &=\frac {1}{2} x \sqrt [3]{x-x^3}-\frac {\left (x^{2/3} \left (1-x^2\right )^{2/3}\right ) \operatorname {Subst}\left (\int \frac {1}{1+x} \, dx,x,\frac {x^{2/3}}{\sqrt [3]{1-x^2}}\right )}{6 \left (x-x^3\right )^{2/3}}+\frac {\left (x^{2/3} \left (1-x^2\right )^{2/3}\right ) \operatorname {Subst}\left (\int \frac {1+x}{1-x+x^2} \, dx,x,\frac {x^{2/3}}{\sqrt [3]{1-x^2}}\right )}{6 \left (x-x^3\right )^{2/3}}\\ &=\frac {1}{2} x \sqrt [3]{x-x^3}-\frac {x^{2/3} \left (1-x^2\right )^{2/3} \log \left (1+\frac {x^{2/3}}{\sqrt [3]{1-x^2}}\right )}{6 \left (x-x^3\right )^{2/3}}+\frac {\left (x^{2/3} \left (1-x^2\right )^{2/3}\right ) \operatorname {Subst}\left (\int \frac {-1+2 x}{1-x+x^2} \, dx,x,\frac {x^{2/3}}{\sqrt [3]{1-x^2}}\right )}{12 \left (x-x^3\right )^{2/3}}+\frac {\left (x^{2/3} \left (1-x^2\right )^{2/3}\right ) \operatorname {Subst}\left (\int \frac {1}{1-x+x^2} \, dx,x,\frac {x^{2/3}}{\sqrt [3]{1-x^2}}\right )}{4 \left (x-x^3\right )^{2/3}}\\ &=\frac {1}{2} x \sqrt [3]{x-x^3}+\frac {x^{2/3} \left (1-x^2\right )^{2/3} \log \left (1+\frac {x^{4/3}}{\left (1-x^2\right )^{2/3}}-\frac {x^{2/3}}{\sqrt [3]{1-x^2}}\right )}{12 \left (x-x^3\right )^{2/3}}-\frac {x^{2/3} \left (1-x^2\right )^{2/3} \log \left (1+\frac {x^{2/3}}{\sqrt [3]{1-x^2}}\right )}{6 \left (x-x^3\right )^{2/3}}-\frac {\left (x^{2/3} \left (1-x^2\right )^{2/3}\right ) \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,-1+\frac {2 x^{2/3}}{\sqrt [3]{1-x^2}}\right )}{2 \left (x-x^3\right )^{2/3}}\\ &=\frac {1}{2} x \sqrt [3]{x-x^3}-\frac {x^{2/3} \left (1-x^2\right )^{2/3} \tan ^{-1}\left (\frac {1-\frac {2 x^{2/3}}{\sqrt [3]{1-x^2}}}{\sqrt {3}}\right )}{2 \sqrt {3} \left (x-x^3\right )^{2/3}}+\frac {x^{2/3} \left (1-x^2\right )^{2/3} \log \left (1+\frac {x^{4/3}}{\left (1-x^2\right )^{2/3}}-\frac {x^{2/3}}{\sqrt [3]{1-x^2}}\right )}{12 \left (x-x^3\right )^{2/3}}-\frac {x^{2/3} \left (1-x^2\right )^{2/3} \log \left (1+\frac {x^{2/3}}{\sqrt [3]{1-x^2}}\right )}{6 \left (x-x^3\right )^{2/3}}\\ \end {align*}
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Mathematica [C] time = 0.01, size = 40, normalized size = 0.43 \[ \frac {3 x \sqrt [3]{x-x^3} \, _2F_1\left (-\frac {1}{3},\frac {2}{3};\frac {5}{3};x^2\right )}{4 \sqrt [3]{1-x^2}} \]
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.15, size = 105, normalized size = 1.13 \[ \frac {1}{2} \sqrt [3]{x-x^3} x-\frac {1}{6} \log \left (\sqrt [3]{x-x^3}+x\right )+\frac {\tan ^{-1}\left (\frac {\sqrt {3} x}{2 \sqrt [3]{x-x^3}-x}\right )}{2 \sqrt {3}}+\frac {1}{12} \log \left (-\sqrt [3]{x-x^3} x+\left (x-x^3\right )^{2/3}+x^2\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.74, size = 99, normalized size = 1.06 \[ -\frac {1}{6} \, \sqrt {3} \arctan \left (\frac {44032959556 \, \sqrt {3} {\left (-x^{3} + x\right )}^{\frac {1}{3}} x - \sqrt {3} {\left (16754327161 \, x^{2} - 2707204793\right )} + 10524305234 \, \sqrt {3} {\left (-x^{3} + x\right )}^{\frac {2}{3}}}{81835897185 \, x^{2} - 1102302937}\right ) + \frac {1}{2} \, {\left (-x^{3} + x\right )}^{\frac {1}{3}} x - \frac {1}{12} \, \log \left (3 \, {\left (-x^{3} + x\right )}^{\frac {1}{3}} x + 3 \, {\left (-x^{3} + x\right )}^{\frac {2}{3}} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.68, size = 69, normalized size = 0.74 \[ \frac {1}{2} \, x^{2} {\left (\frac {1}{x^{2}} - 1\right )}^{\frac {1}{3}} - \frac {1}{6} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (\frac {1}{x^{2}} - 1\right )}^{\frac {1}{3}} - 1\right )}\right ) + \frac {1}{12} \, \log \left ({\left (\frac {1}{x^{2}} - 1\right )}^{\frac {2}{3}} - {\left (\frac {1}{x^{2}} - 1\right )}^{\frac {1}{3}} + 1\right ) - \frac {1}{6} \, \log \left ({\left | {\left (\frac {1}{x^{2}} - 1\right )}^{\frac {1}{3}} + 1 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 3.23, size = 15, normalized size = 0.16
method | result | size |
meijerg | \(\frac {3 x^{\frac {4}{3}} \hypergeom \left (\left [-\frac {1}{3}, \frac {2}{3}\right ], \left [\frac {5}{3}\right ], x^{2}\right )}{4}\) | \(15\) |
trager | \(\frac {x \left (-x^{3}+x \right )^{\frac {1}{3}}}{2}-\frac {\ln \left (4959 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )^{2} x^{2}-6768 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) \left (-x^{3}+x \right )^{\frac {2}{3}}-22833 \left (-x^{3}+x \right )^{\frac {1}{3}} \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) x -17718 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) x^{2}+7611 \left (-x^{3}+x \right )^{\frac {2}{3}}+5355 x \left (-x^{3}+x \right )^{\frac {1}{3}}-19836 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )^{2}-1705 x^{2}+9711 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )+1085\right )}{6}+\frac {\RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) \ln \left (-6354 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )^{2} x^{2}+6768 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) \left (-x^{3}+x \right )^{\frac {2}{3}}-16065 \left (-x^{3}+x \right )^{\frac {1}{3}} \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) x -20715 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) x^{2}+5355 \left (-x^{3}+x \right )^{\frac {2}{3}}+7611 x \left (-x^{3}+x \right )^{\frac {1}{3}}+25416 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )^{2}+1550 x^{2}+4494 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )-465\right )}{2}\) | \(305\) |
risch | \(\frac {x \left (-x \left (x^{2}-1\right )\right )^{\frac {1}{3}}}{2}+\frac {\left (\frac {\RootOf \left (\textit {\_Z}^{2}+6 \textit {\_Z} +36\right ) \ln \left (-\frac {47 \RootOf \left (\textit {\_Z}^{2}+6 \textit {\_Z} +36\right )^{2} x^{4}+3207 \RootOf \left (\textit {\_Z}^{2}+6 \textit {\_Z} +36\right ) x^{4}+2925 \left (x^{6}-2 x^{4}+x^{2}\right )^{\frac {1}{3}} \RootOf \left (\textit {\_Z}^{2}+6 \textit {\_Z} +36\right ) x^{2}-235 \RootOf \left (\textit {\_Z}^{2}+6 \textit {\_Z} +36\right )^{2} x^{2}+6930 x^{4}+2925 \RootOf \left (\textit {\_Z}^{2}+6 \textit {\_Z} +36\right ) \left (x^{6}-2 x^{4}+x^{2}\right )^{\frac {2}{3}}+5238 \left (x^{6}-2 x^{4}+x^{2}\right )^{\frac {1}{3}} x^{2}-5601 \RootOf \left (\textit {\_Z}^{2}+6 \textit {\_Z} +36\right ) x^{2}+5238 \left (x^{6}-2 x^{4}+x^{2}\right )^{\frac {2}{3}}-2925 \RootOf \left (\textit {\_Z}^{2}+6 \textit {\_Z} +36\right ) \left (x^{6}-2 x^{4}+x^{2}\right )^{\frac {1}{3}}+188 \RootOf \left (\textit {\_Z}^{2}+6 \textit {\_Z} +36\right )^{2}-11340 x^{2}-5238 \left (x^{6}-2 x^{4}+x^{2}\right )^{\frac {1}{3}}+2394 \RootOf \left (\textit {\_Z}^{2}+6 \textit {\_Z} +36\right )+4410}{\left (1+x \right ) \left (-1+x \right )}\right )}{36}-\frac {\ln \left (\frac {-47 \RootOf \left (\textit {\_Z}^{2}+6 \textit {\_Z} +36\right )^{2} x^{4}+2643 \RootOf \left (\textit {\_Z}^{2}+6 \textit {\_Z} +36\right ) x^{4}+2925 \left (x^{6}-2 x^{4}+x^{2}\right )^{\frac {1}{3}} \RootOf \left (\textit {\_Z}^{2}+6 \textit {\_Z} +36\right ) x^{2}+235 \RootOf \left (\textit {\_Z}^{2}+6 \textit {\_Z} +36\right )^{2} x^{2}+10620 x^{4}+2925 \RootOf \left (\textit {\_Z}^{2}+6 \textit {\_Z} +36\right ) \left (x^{6}-2 x^{4}+x^{2}\right )^{\frac {2}{3}}+12312 \left (x^{6}-2 x^{4}+x^{2}\right )^{\frac {1}{3}} x^{2}-2781 \RootOf \left (\textit {\_Z}^{2}+6 \textit {\_Z} +36\right ) x^{2}+12312 \left (x^{6}-2 x^{4}+x^{2}\right )^{\frac {2}{3}}-2925 \RootOf \left (\textit {\_Z}^{2}+6 \textit {\_Z} +36\right ) \left (x^{6}-2 x^{4}+x^{2}\right )^{\frac {1}{3}}-188 \RootOf \left (\textit {\_Z}^{2}+6 \textit {\_Z} +36\right )^{2}-13806 x^{2}-12312 \left (x^{6}-2 x^{4}+x^{2}\right )^{\frac {1}{3}}+138 \RootOf \left (\textit {\_Z}^{2}+6 \textit {\_Z} +36\right )+3186}{\left (1+x \right ) \left (-1+x \right )}\right ) \RootOf \left (\textit {\_Z}^{2}+6 \textit {\_Z} +36\right )}{36}-\frac {\ln \left (\frac {-47 \RootOf \left (\textit {\_Z}^{2}+6 \textit {\_Z} +36\right )^{2} x^{4}+2643 \RootOf \left (\textit {\_Z}^{2}+6 \textit {\_Z} +36\right ) x^{4}+2925 \left (x^{6}-2 x^{4}+x^{2}\right )^{\frac {1}{3}} \RootOf \left (\textit {\_Z}^{2}+6 \textit {\_Z} +36\right ) x^{2}+235 \RootOf \left (\textit {\_Z}^{2}+6 \textit {\_Z} +36\right )^{2} x^{2}+10620 x^{4}+2925 \RootOf \left (\textit {\_Z}^{2}+6 \textit {\_Z} +36\right ) \left (x^{6}-2 x^{4}+x^{2}\right )^{\frac {2}{3}}+12312 \left (x^{6}-2 x^{4}+x^{2}\right )^{\frac {1}{3}} x^{2}-2781 \RootOf \left (\textit {\_Z}^{2}+6 \textit {\_Z} +36\right ) x^{2}+12312 \left (x^{6}-2 x^{4}+x^{2}\right )^{\frac {2}{3}}-2925 \RootOf \left (\textit {\_Z}^{2}+6 \textit {\_Z} +36\right ) \left (x^{6}-2 x^{4}+x^{2}\right )^{\frac {1}{3}}-188 \RootOf \left (\textit {\_Z}^{2}+6 \textit {\_Z} +36\right )^{2}-13806 x^{2}-12312 \left (x^{6}-2 x^{4}+x^{2}\right )^{\frac {1}{3}}+138 \RootOf \left (\textit {\_Z}^{2}+6 \textit {\_Z} +36\right )+3186}{\left (1+x \right ) \left (-1+x \right )}\right )}{6}\right ) \left (-x \left (x^{2}-1\right )\right )^{\frac {1}{3}} \left (x^{2} \left (x^{2}-1\right )^{2}\right )^{\frac {1}{3}}}{x \left (x^{2}-1\right )}\) | \(786\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (-{\left (x^{2} - 1\right )} x\right )^{\frac {1}{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.37, size = 29, normalized size = 0.31 \[ \frac {3\,x\,{\left (x-x^3\right )}^{1/3}\,{{}}_2{\mathrm {F}}_1\left (-\frac {1}{3},\frac {2}{3};\ \frac {5}{3};\ x^2\right )}{4\,{\left (1-x^2\right )}^{1/3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt [3]{x \left (1 - x^{2}\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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