Integrand size = 13, antiderivative size = 93 \[ \int \sqrt [3]{x \left (1-x^2\right )} \, dx=\frac {1}{2} x \sqrt [3]{x \left (1-x^2\right )}+\frac {\arctan \left (\frac {2 x-\sqrt [3]{x \left (1-x^2\right )}}{\sqrt {3} \sqrt [3]{x \left (1-x^2\right )}}\right )}{2 \sqrt {3}}+\frac {\log (x)}{12}-\frac {1}{4} \log \left (x+\sqrt [3]{x \left (1-x^2\right )}\right ) \]
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Time = 0.06 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.39, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.462, Rules used = {2004, 2029, 2057, 335, 281, 337} \[ \int \sqrt [3]{x \left (1-x^2\right )} \, dx=-\frac {\left (1-x^2\right )^{2/3} x^{2/3} \arctan \left (\frac {1-\frac {2 x^{2/3}}{\sqrt [3]{1-x^2}}}{\sqrt {3}}\right )}{2 \sqrt {3} \left (x-x^3\right )^{2/3}}+\frac {1}{2} \sqrt [3]{x-x^3} x-\frac {\left (1-x^2\right )^{2/3} x^{2/3} \log \left (x^{2/3}+\sqrt [3]{1-x^2}\right )}{4 \left (x-x^3\right )^{2/3}} \]
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Rule 281
Rule 335
Rule 337
Rule 2004
Rule 2029
Rule 2057
Rubi steps \begin{align*} \text {integral}& = \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 ) \text {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 ) \text {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 {x^{2/3} \left (1-x^2\right )^{2/3} \arctan \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 (x^{2/3}+\sqrt [3]{1-x^2}\right )}{4 \left (x-x^3\right )^{2/3}} \\ \end{align*}
Time = 0.70 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.47 \[ \int \sqrt [3]{x \left (1-x^2\right )} \, dx=\frac {\sqrt [3]{x-x^3} \left (6 x^{4/3} \sqrt [3]{-1+x^2}+2 \sqrt {3} \arctan \left (\frac {\sqrt {3} x^{2/3}}{x^{2/3}+2 \sqrt [3]{-1+x^2}}\right )+2 \log \left (-x^{2/3}+\sqrt [3]{-1+x^2}\right )-\log \left (x^{4/3}+x^{2/3} \sqrt [3]{-1+x^2}+\left (-1+x^2\right )^{2/3}\right )\right )}{12 \sqrt [3]{x} \sqrt [3]{-1+x^2}} \]
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Result contains higher order function than in optimal. Order 5 vs. order 3.
Time = 2.62 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.16
method | result | size |
meijerg | \(\frac {3 x^{\frac {4}{3}} {}_{2}^{}{\moversetsp {}{\mundersetsp {}{F_{1}^{}}}}\left (-\frac {1}{3},\frac {2}{3};\frac {5}{3};x^{2}\right )}{4}\) | \(15\) |
pseudoelliptic | \(\frac {x \left (2 \sqrt {3}\, \arctan \left (\frac {\left (-2 \left (-x^{3}+x \right )^{\frac {1}{3}}+x \right ) \sqrt {3}}{3 x}\right )+6 \left (-x^{3}+x \right )^{\frac {1}{3}} x -2 \ln \left (\frac {\left (-x^{3}+x \right )^{\frac {1}{3}}+x}{x}\right )+\ln \left (\frac {\left (-x^{3}+x \right )^{\frac {2}{3}}-\left (-x^{3}+x \right )^{\frac {1}{3}} x +x^{2}}{x^{2}}\right )\right )}{12 \left (\left (-x^{3}+x \right )^{\frac {1}{3}}+x \right ) \left (\left (-x^{3}+x \right )^{\frac {2}{3}}-\left (-x^{3}+x \right )^{\frac {1}{3}} x +x^{2}\right )}\) | \(132\) |
trager | \(\frac {\left (-x^{3}+x \right )^{\frac {1}{3}} x}{2}-\frac {\ln \left (4959 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )^{2} x^{2}+6768 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) \left (-x^{3}+x \right )^{\frac {2}{3}}+22833 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) \left (-x^{3}+x \right )^{\frac {1}{3}} x +14412 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) x^{2}+5355 \left (-x^{3}+x \right )^{\frac {2}{3}}-2256 \left (-x^{3}+x \right )^{\frac {1}{3}} x -19836 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )^{2}-7060 x^{2}+3513 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )+2118\right )}{6}+\frac {\ln \left (-6354 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )^{2} x^{2}-6768 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) \left (-x^{3}+x \right )^{\frac {2}{3}}+16065 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) \left (-x^{3}+x \right )^{\frac {1}{3}} x +24951 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) x^{2}+7611 \left (-x^{3}+x \right )^{\frac {2}{3}}+2256 \left (-x^{3}+x \right )^{\frac {1}{3}} x +25416 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )^{2}-6061 x^{2}-21438 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )+3857\right )}{6}-\frac {\ln \left (-6354 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )^{2} x^{2}-6768 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) \left (-x^{3}+x \right )^{\frac {2}{3}}+16065 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) \left (-x^{3}+x \right )^{\frac {1}{3}} x +24951 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) x^{2}+7611 \left (-x^{3}+x \right )^{\frac {2}{3}}+2256 \left (-x^{3}+x \right )^{\frac {1}{3}} x +25416 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )^{2}-6061 x^{2}-21438 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )+3857\right ) \operatorname {RootOf}\left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )}{2}\) | \(445\) |
risch | \(\text {Expression too large to display}\) | \(788\) |
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Time = 0.35 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.06 \[ \int \sqrt [3]{x \left (1-x^2\right )} \, dx=-\frac {1}{6} \, \sqrt {3} \arctan \left (\frac {44032959556 \, \sqrt {3} {\left (-x^{3} + x\right )}^{\frac {1}{3}} x - \sqrt {3} {\left (16754327161 \, x^{2} - 2707204793\right )} + 10524305234 \, \sqrt {3} {\left (-x^{3} + x\right )}^{\frac {2}{3}}}{81835897185 \, x^{2} - 1102302937}\right ) + \frac {1}{2} \, {\left (-x^{3} + x\right )}^{\frac {1}{3}} x - \frac {1}{12} \, \log \left (3 \, {\left (-x^{3} + x\right )}^{\frac {1}{3}} x + 3 \, {\left (-x^{3} + x\right )}^{\frac {2}{3}} + 1\right ) \]
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\[ \int \sqrt [3]{x \left (1-x^2\right )} \, dx=\int \sqrt [3]{x \left (1 - x^{2}\right )}\, dx \]
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\[ \int \sqrt [3]{x \left (1-x^2\right )} \, dx=\int { \left (-{\left (x^{2} - 1\right )} x\right )^{\frac {1}{3}} \,d x } \]
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Time = 0.30 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.74 \[ \int \sqrt [3]{x \left (1-x^2\right )} \, dx=\frac {1}{2} \, x^{2} {\left (\frac {1}{x^{2}} - 1\right )}^{\frac {1}{3}} - \frac {1}{6} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (\frac {1}{x^{2}} - 1\right )}^{\frac {1}{3}} - 1\right )}\right ) + \frac {1}{12} \, \log \left ({\left (\frac {1}{x^{2}} - 1\right )}^{\frac {2}{3}} - {\left (\frac {1}{x^{2}} - 1\right )}^{\frac {1}{3}} + 1\right ) - \frac {1}{6} \, \log \left ({\left | {\left (\frac {1}{x^{2}} - 1\right )}^{\frac {1}{3}} + 1 \right |}\right ) \]
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Time = 0.40 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.31 \[ \int \sqrt [3]{x \left (1-x^2\right )} \, dx=\frac {3\,x\,{\left (x-x^3\right )}^{1/3}\,{{}}_2{\mathrm {F}}_1\left (-\frac {1}{3},\frac {2}{3};\ \frac {5}{3};\ x^2\right )}{4\,{\left (1-x^2\right )}^{1/3}} \]
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