Integrand size = 13, antiderivative size = 91 \[ \int \frac {1}{\sqrt [3]{-x^2+x^3}} \, dx=\sqrt {3} \arctan \left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{-x^2+x^3}}\right )-\log \left (-x+\sqrt [3]{-x^2+x^3}\right )+\frac {1}{2} \log \left (x^2+x \sqrt [3]{-x^2+x^3}+\left (-x^2+x^3\right )^{2/3}\right ) \]
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Time = 0.01 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.48, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {2036, 61} \[ \int \frac {1}{\sqrt [3]{-x^2+x^3}} \, dx=-\frac {\sqrt {3} \sqrt [3]{x-1} x^{2/3} \arctan \left (\frac {2 \sqrt [3]{x-1}}{\sqrt {3} \sqrt [3]{x}}+\frac {1}{\sqrt {3}}\right )}{\sqrt [3]{x^3-x^2}}-\frac {3 \sqrt [3]{x-1} x^{2/3} \log \left (\frac {\sqrt [3]{x-1}}{\sqrt [3]{x}}-1\right )}{2 \sqrt [3]{x^3-x^2}}-\frac {\sqrt [3]{x-1} x^{2/3} \log (x)}{2 \sqrt [3]{x^3-x^2}} \]
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Rule 61
Rule 2036
Rubi steps \begin{align*} \text {integral}& = \frac {\left (\sqrt [3]{-1+x} x^{2/3}\right ) \int \frac {1}{\sqrt [3]{-1+x} x^{2/3}} \, dx}{\sqrt [3]{-x^2+x^3}} \\ & = -\frac {\sqrt {3} \sqrt [3]{-1+x} x^{2/3} \arctan \left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{-1+x}}{\sqrt {3} \sqrt [3]{x}}\right )}{\sqrt [3]{-x^2+x^3}}-\frac {3 \sqrt [3]{-1+x} x^{2/3} \log \left (-1+\frac {\sqrt [3]{-1+x}}{\sqrt [3]{x}}\right )}{2 \sqrt [3]{-x^2+x^3}}-\frac {\sqrt [3]{-1+x} x^{2/3} \log (x)}{2 \sqrt [3]{-x^2+x^3}} \\ \end{align*}
Time = 0.15 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.20 \[ \int \frac {1}{\sqrt [3]{-x^2+x^3}} \, dx=\frac {\sqrt [3]{-1+x} x^{2/3} \left (2 \sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{x}}{2 \sqrt [3]{-1+x}+\sqrt [3]{x}}\right )-2 \log \left (\sqrt [3]{-1+x}-\sqrt [3]{x}\right )+\log \left ((-1+x)^{2/3}+\sqrt [3]{-1+x} \sqrt [3]{x}+x^{2/3}\right )\right )}{2 \sqrt [3]{(-1+x) x^2}} \]
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Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 3.54 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.30
method | result | size |
meijerg | \(\frac {3 \left (-\operatorname {signum}\left (x -1\right )\right )^{\frac {1}{3}} x^{\frac {1}{3}} \operatorname {hypergeom}\left (\left [\frac {1}{3}, \frac {1}{3}\right ], \left [\frac {4}{3}\right ], x\right )}{\operatorname {signum}\left (x -1\right )^{\frac {1}{3}}}\) | \(27\) |
pseudoelliptic | \(-\ln \left (\frac {\left (\left (x -1\right ) x^{2}\right )^{\frac {1}{3}}-x}{x}\right )+\frac {\ln \left (\frac {\left (\left (x -1\right ) x^{2}\right )^{\frac {2}{3}}+\left (\left (x -1\right ) x^{2}\right )^{\frac {1}{3}} x +x^{2}}{x^{2}}\right )}{2}-\sqrt {3}\, \arctan \left (\frac {\left (2 \left (\left (x -1\right ) x^{2}\right )^{\frac {1}{3}}+x \right ) \sqrt {3}}{3 x}\right )\) | \(80\) |
trager | \(-\ln \left (-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right )^{2} x^{2}-2 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right )^{2} x -48 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) \left (x^{3}-x^{2}\right )^{\frac {2}{3}}+30 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) \left (x^{3}-x^{2}\right )^{\frac {1}{3}} x +16 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) x^{2}-18 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) x +36 \left (x^{3}-x^{2}\right )^{\frac {2}{3}}-96 x \left (x^{3}-x^{2}\right )^{\frac {1}{3}}+64 x^{2}-16 x}{x}\right )+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) \ln \left (-\frac {2 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right )^{2} x^{2}-4 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right )^{2} x +24 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) \left (x^{3}-x^{2}\right )^{\frac {2}{3}}-9 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) \left (x^{3}-x^{2}\right )^{\frac {1}{3}} x -19 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) x^{2}+10 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) x -30 \left (x^{3}-x^{2}\right )^{\frac {2}{3}}+48 x \left (x^{3}-x^{2}\right )^{\frac {1}{3}}-10 x^{2}+6 x}{x}\right )}{2}\) | \(300\) |
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Time = 0.28 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.01 \[ \int \frac {1}{\sqrt [3]{-x^2+x^3}} \, dx=-\sqrt {3} \arctan \left (\frac {\sqrt {3} x + 2 \, \sqrt {3} {\left (x^{3} - x^{2}\right )}^{\frac {1}{3}}}{3 \, x}\right ) - \log \left (-\frac {x - {\left (x^{3} - x^{2}\right )}^{\frac {1}{3}}}{x}\right ) + \frac {1}{2} \, \log \left (\frac {x^{2} + {\left (x^{3} - x^{2}\right )}^{\frac {1}{3}} x + {\left (x^{3} - x^{2}\right )}^{\frac {2}{3}}}{x^{2}}\right ) \]
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\[ \int \frac {1}{\sqrt [3]{-x^2+x^3}} \, dx=\int \frac {1}{\sqrt [3]{x^{3} - x^{2}}}\, dx \]
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\[ \int \frac {1}{\sqrt [3]{-x^2+x^3}} \, dx=\int { \frac {1}{{\left (x^{3} - x^{2}\right )}^{\frac {1}{3}}} \,d x } \]
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Time = 0.28 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.69 \[ \int \frac {1}{\sqrt [3]{-x^2+x^3}} \, dx=-\sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{3}} + 1\right )}\right ) + \frac {1}{2} \, \log \left ({\left (-\frac {1}{x} + 1\right )}^{\frac {2}{3}} + {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{3}} + 1\right ) - \log \left ({\left | {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{3}} - 1 \right |}\right ) \]
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Time = 6.01 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.30 \[ \int \frac {1}{\sqrt [3]{-x^2+x^3}} \, dx=\frac {3\,x\,{\left (1-x\right )}^{1/3}\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{3},\frac {1}{3};\ \frac {4}{3};\ x\right )}{{\left (x^3-x^2\right )}^{1/3}} \]
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