Integrand size = 15, antiderivative size = 106 \[ \int \frac {\sqrt [3]{-x+x^3}}{x^2} \, dx=-\frac {3 \sqrt [3]{-x+x^3}}{2 x}-\frac {1}{2} \sqrt {3} \arctan \left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{-x+x^3}}\right )-\frac {1}{2} \log \left (-x+\sqrt [3]{-x+x^3}\right )+\frac {1}{4} \log \left (x^2+x \sqrt [3]{-x+x^3}+\left (-x+x^3\right )^{2/3}\right ) \]
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Time = 0.05 (sec) , antiderivative size = 125, normalized size of antiderivative = 1.18, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {2045, 2057, 335, 281, 337} \[ \int \frac {\sqrt [3]{-x+x^3}}{x^2} \, dx=-\frac {\sqrt {3} x^{2/3} \left (x^2-1\right )^{2/3} \arctan \left (\frac {\frac {2 x^{2/3}}{\sqrt [3]{x^2-1}}+1}{\sqrt {3}}\right )}{2 \left (x^3-x\right )^{2/3}}-\frac {3 \sqrt [3]{x^3-x}}{2 x}-\frac {3 x^{2/3} \left (x^2-1\right )^{2/3} \log \left (x^{2/3}-\sqrt [3]{x^2-1}\right )}{4 \left (x^3-x\right )^{2/3}} \]
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Rule 281
Rule 335
Rule 337
Rule 2045
Rule 2057
Rubi steps \begin{align*} \text {integral}& = -\frac {3 \sqrt [3]{-x+x^3}}{2 x}+\int \frac {x}{\left (-x+x^3\right )^{2/3}} \, dx \\ & = -\frac {3 \sqrt [3]{-x+x^3}}{2 x}+\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}{\left (-x+x^3\right )^{2/3}} \\ & = -\frac {3 \sqrt [3]{-x+x^3}}{2 x}+\frac {\left (3 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 {3 \sqrt [3]{-x+x^3}}{2 x}+\frac {\left (3 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 {3 \sqrt [3]{-x+x^3}}{2 x}-\frac {\sqrt {3} 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 \left (-x+x^3\right )^{2/3}}-\frac {3 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 = 0.43 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.34 \[ \int \frac {\sqrt [3]{-x+x^3}}{x^2} \, dx=-\frac {\left (-1+x^2\right )^{2/3} \left (6 \sqrt [3]{-1+x^2}+2 \sqrt {3} x^{2/3} \arctan \left (\frac {\sqrt {3} x^{2/3}}{x^{2/3}+2 \sqrt [3]{-1+x^2}}\right )+2 x^{2/3} \log \left (-x^{2/3}+\sqrt [3]{-1+x^2}\right )-x^{2/3} \log \left (x^{4/3}+x^{2/3} \sqrt [3]{-1+x^2}+\left (-1+x^2\right )^{2/3}\right )\right )}{4 \left (x \left (-1+x^2\right )\right )^{2/3}} \]
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Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 1.97 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.31
method | result | size |
meijerg | \(-\frac {3 \operatorname {signum}\left (x^{2}-1\right )^{\frac {1}{3}} \operatorname {hypergeom}\left (\left [-\frac {1}{3}, -\frac {1}{3}\right ], \left [\frac {2}{3}\right ], x^{2}\right )}{2 {\left (-\operatorname {signum}\left (x^{2}-1\right )\right )}^{\frac {1}{3}} x^{\frac {2}{3}}}\) | \(33\) |
pseudoelliptic | \(\frac {-6 \left (x^{3}-x \right )^{\frac {1}{3}}-x \left (-2 \arctan \left (\frac {\sqrt {3}\, \left (x +2 \left (x^{3}-x \right )^{\frac {1}{3}}\right )}{3 x}\right ) \sqrt {3}+2 \ln \left (\frac {-x +\left (x^{3}-x \right )^{\frac {1}{3}}}{x}\right )-\ln \left (\frac {x^{2}+x \left (x^{3}-x \right )^{\frac {1}{3}}+\left (x^{3}-x \right )^{\frac {2}{3}}}{x^{2}}\right )\right )}{4 x}\) | \(100\) |
trager | \(-\frac {3 \left (x^{3}-x \right )^{\frac {1}{3}}}{2 x}-\frac {\ln \left (79344 \operatorname {RootOf}\left (144 \textit {\_Z}^{2}-12 \textit {\_Z} +1\right )^{2} x^{2}-27072 \operatorname {RootOf}\left (144 \textit {\_Z}^{2}-12 \textit {\_Z} +1\right ) \left (x^{3}-x \right )^{\frac {2}{3}}+91332 \left (x^{3}-x \right )^{\frac {1}{3}} \operatorname {RootOf}\left (144 \textit {\_Z}^{2}-12 \textit {\_Z} +1\right ) x -70872 \operatorname {RootOf}\left (144 \textit {\_Z}^{2}-12 \textit {\_Z} +1\right ) x^{2}+7611 \left (x^{3}-x \right )^{\frac {2}{3}}-5355 x \left (x^{3}-x \right )^{\frac {1}{3}}-317376 \operatorname {RootOf}\left (144 \textit {\_Z}^{2}-12 \textit {\_Z} +1\right )^{2}-1705 x^{2}+38844 \operatorname {RootOf}\left (144 \textit {\_Z}^{2}-12 \textit {\_Z} +1\right )+1085\right )}{2}+6 \operatorname {RootOf}\left (144 \textit {\_Z}^{2}-12 \textit {\_Z} +1\right ) \ln \left (-101664 \operatorname {RootOf}\left (144 \textit {\_Z}^{2}-12 \textit {\_Z} +1\right )^{2} x^{2}+27072 \operatorname {RootOf}\left (144 \textit {\_Z}^{2}-12 \textit {\_Z} +1\right ) \left (x^{3}-x \right )^{\frac {2}{3}}+64260 \left (x^{3}-x \right )^{\frac {1}{3}} \operatorname {RootOf}\left (144 \textit {\_Z}^{2}-12 \textit {\_Z} +1\right ) x -82860 \operatorname {RootOf}\left (144 \textit {\_Z}^{2}-12 \textit {\_Z} +1\right ) x^{2}+5355 \left (x^{3}-x \right )^{\frac {2}{3}}-7611 x \left (x^{3}-x \right )^{\frac {1}{3}}+406656 \operatorname {RootOf}\left (144 \textit {\_Z}^{2}-12 \textit {\_Z} +1\right )^{2}+1550 x^{2}+17976 \operatorname {RootOf}\left (144 \textit {\_Z}^{2}-12 \textit {\_Z} +1\right )-465\right )\) | \(307\) |
risch | \(\text {Expression too large to display}\) | \(787\) |
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Time = 0.42 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.98 \[ \int \frac {\sqrt [3]{-x+x^3}}{x^2} \, dx=-\frac {2 \, \sqrt {3} x \arctan \left (-\frac {44032959556 \, \sqrt {3} {\left (x^{3} - x\right )}^{\frac {1}{3}} x + \sqrt {3} {\left (16754327161 \, x^{2} - 2707204793\right )} - 10524305234 \, \sqrt {3} {\left (x^{3} - x\right )}^{\frac {2}{3}}}{81835897185 \, x^{2} - 1102302937}\right ) + x \log \left (-3 \, {\left (x^{3} - x\right )}^{\frac {1}{3}} x + 3 \, {\left (x^{3} - x\right )}^{\frac {2}{3}} + 1\right ) + 6 \, {\left (x^{3} - x\right )}^{\frac {1}{3}}}{4 \, x} \]
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\[ \int \frac {\sqrt [3]{-x+x^3}}{x^2} \, dx=\int \frac {\sqrt [3]{x \left (x - 1\right ) \left (x + 1\right )}}{x^{2}}\, dx \]
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\[ \int \frac {\sqrt [3]{-x+x^3}}{x^2} \, dx=\int { \frac {{\left (x^{3} - x\right )}^{\frac {1}{3}}}{x^{2}} \,d x } \]
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Time = 0.30 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.70 \[ \int \frac {\sqrt [3]{-x+x^3}}{x^2} \, dx=\frac {1}{2} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {1}{3}} + 1\right )}\right ) - \frac {3}{2} \, {\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {1}{3}} + \frac {1}{4} \, \log \left ({\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {2}{3}} + {\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {1}{3}} + 1\right ) - \frac {1}{2} \, \log \left ({\left | {\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {1}{3}} - 1 \right |}\right ) \]
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Timed out. \[ \int \frac {\sqrt [3]{-x+x^3}}{x^2} \, dx=\int \frac {{\left (x^3-x\right )}^{1/3}}{x^2} \,d x \]
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