Integrand size = 11, antiderivative size = 67 \[ \int \left (1-x^3\right )^{2/3} \, dx=\frac {1}{3} x \left (1-x^3\right )^{2/3}-\frac {2 \arctan \left (\frac {1-\frac {2 x}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{3 \sqrt {3}}+\frac {1}{3} \log \left (x+\sqrt [3]{1-x^3}\right ) \]
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Time = 0.01 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {201, 245} \[ \int \left (1-x^3\right )^{2/3} \, dx=-\frac {2 \arctan \left (\frac {1-\frac {2 x}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{3 \sqrt {3}}+\frac {1}{3} \left (1-x^3\right )^{2/3} x+\frac {1}{3} \log \left (\sqrt [3]{1-x^3}+x\right ) \]
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Rule 201
Rule 245
Rubi steps \begin{align*} \text {integral}& = \frac {1}{3} x \left (1-x^3\right )^{2/3}+\frac {2}{3} \int \frac {1}{\sqrt [3]{1-x^3}} \, dx \\ & = \frac {1}{3} x \left (1-x^3\right )^{2/3}-\frac {2 \arctan \left (\frac {1-\frac {2 x}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{3 \sqrt {3}}+\frac {1}{3} \log \left (x+\sqrt [3]{1-x^3}\right ) \\ \end{align*}
Result contains higher order function than in optimal. Order 6 vs. order 3 in optimal.
Time = 0.08 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.51 \[ \int \left (1-x^3\right )^{2/3} \, dx=\frac {3 (-1+x) \left (1-x^3\right )^{2/3} \operatorname {AppellF1}\left (\frac {5}{3},-\frac {2}{3},-\frac {2}{3},\frac {8}{3},-\frac {-1+x}{1-(-1)^{2/3}},-\frac {-1+x}{1+\sqrt [3]{-1}}\right )}{5 \left (1+\frac {-1+x}{1+\sqrt [3]{-1}}\right )^{2/3} \left (1+\frac {-1+x}{1-(-1)^{2/3}}\right )^{2/3}} \]
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Result contains higher order function than in optimal. Order 5 vs. order 3.
Time = 1.28 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.18
method | result | size |
meijerg | \(x {}_{2}^{}{\moversetsp {}{\mundersetsp {}{F_{1}^{}}}}\left (-\frac {2}{3},\frac {1}{3};\frac {4}{3};x^{3}\right )\) | \(12\) |
risch | \(-\frac {x \left (x^{3}-1\right )}{3 \left (-x^{3}+1\right )^{\frac {1}{3}}}+\frac {2 x {}_{2}^{}{\moversetsp {}{\mundersetsp {}{F_{1}^{}}}}\left (\frac {1}{3},\frac {1}{3};\frac {4}{3};x^{3}\right )}{3}\) | \(31\) |
pseudoelliptic | \(\frac {3 x \left (-x^{3}+1\right )^{\frac {2}{3}}+2 \sqrt {3}\, \arctan \left (\frac {\left (-2 \left (-x^{3}+1\right )^{\frac {1}{3}}+x \right ) \sqrt {3}}{3 x}\right )+2 \ln \left (\frac {x +\left (-x^{3}+1\right )^{\frac {1}{3}}}{x}\right )-\ln \left (\frac {\left (-x^{3}+1\right )^{\frac {2}{3}}-x \left (-x^{3}+1\right )^{\frac {1}{3}}+x^{2}}{x^{2}}\right )}{9 \left (x +\left (-x^{3}+1\right )^{\frac {1}{3}}\right ) \left (\left (-x^{3}+1\right )^{\frac {2}{3}}-x \left (-x^{3}+1\right )^{\frac {1}{3}}+x^{2}\right )}\) | \(133\) |
trager | \(\frac {x \left (-x^{3}+1\right )^{\frac {2}{3}}}{3}+\frac {2 \ln \left (-2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{3}+3 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (-x^{3}+1\right )^{\frac {2}{3}} x -5 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{3}+3 x \left (-x^{3}+1\right )^{\frac {2}{3}}+3 x^{2} \left (-x^{3}+1\right )^{\frac {1}{3}}-2 x^{3}+2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )+1\right )}{9}+\frac {2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \ln \left (\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{3}-3 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (-x^{3}+1\right )^{\frac {2}{3}} x -3 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (-x^{3}+1\right )^{\frac {1}{3}} x^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{3}-3 x^{2} \left (-x^{3}+1\right )^{\frac {1}{3}}-2 x^{3}+\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )+2\right )}{9}\) | \(206\) |
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none
Time = 0.25 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.40 \[ \int \left (1-x^3\right )^{2/3} \, dx=\frac {1}{3} \, {\left (-x^{3} + 1\right )}^{\frac {2}{3}} x - \frac {2}{9} \, \sqrt {3} \arctan \left (-\frac {\sqrt {3} x - 2 \, \sqrt {3} {\left (-x^{3} + 1\right )}^{\frac {1}{3}}}{3 \, x}\right ) + \frac {2}{9} \, \log \left (\frac {x + {\left (-x^{3} + 1\right )}^{\frac {1}{3}}}{x}\right ) - \frac {1}{9} \, \log \left (\frac {x^{2} - {\left (-x^{3} + 1\right )}^{\frac {1}{3}} x + {\left (-x^{3} + 1\right )}^{\frac {2}{3}}}{x^{2}}\right ) \]
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Result contains complex when optimal does not.
Time = 0.55 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.46 \[ \int \left (1-x^3\right )^{2/3} \, dx=\frac {x \Gamma \left (\frac {1}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {2}{3}, \frac {1}{3} \\ \frac {4}{3} \end {matrix}\middle | {x^{3} e^{2 i \pi }} \right )}}{3 \Gamma \left (\frac {4}{3}\right )} \]
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Leaf count of result is larger than twice the leaf count of optimal. 105 vs. \(2 (52) = 104\).
Time = 0.29 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.57 \[ \int \left (1-x^3\right )^{2/3} \, dx=-\frac {2}{9} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (\frac {2 \, {\left (-x^{3} + 1\right )}^{\frac {1}{3}}}{x} - 1\right )}\right ) - \frac {{\left (-x^{3} + 1\right )}^{\frac {2}{3}}}{3 \, x^{2} {\left (\frac {x^{3} - 1}{x^{3}} - 1\right )}} + \frac {2}{9} \, \log \left (\frac {{\left (-x^{3} + 1\right )}^{\frac {1}{3}}}{x} + 1\right ) - \frac {1}{9} \, \log \left (-\frac {{\left (-x^{3} + 1\right )}^{\frac {1}{3}}}{x} + \frac {{\left (-x^{3} + 1\right )}^{\frac {2}{3}}}{x^{2}} + 1\right ) \]
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\[ \int \left (1-x^3\right )^{2/3} \, dx=\int { {\left (-x^{3} + 1\right )}^{\frac {2}{3}} \,d x } \]
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Time = 0.46 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.15 \[ \int \left (1-x^3\right )^{2/3} \, dx=x\,{{}}_2{\mathrm {F}}_1\left (-\frac {2}{3},\frac {1}{3};\ \frac {4}{3};\ x^3\right ) \]
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