Integrand size = 18, antiderivative size = 95 \[ \int \frac {-1+x^2}{x \left (-1+x^3\right )^{2/3}} \, dx=-\frac {\arctan \left (\frac {1+\frac {2 x}{\sqrt [3]{-1+x^3}}}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {\arctan \left (\frac {1-2 \sqrt [3]{-1+x^3}}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {\log (x)}{2}-\frac {1}{2} \log \left (x-\sqrt [3]{-1+x^3}\right )-\frac {1}{2} \log \left (1+\sqrt [3]{-1+x^3}\right ) \]
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Time = 0.05 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.389, Rules used = {1858, 272, 60, 632, 210, 31, 337} \[ \int \frac {-1+x^2}{x \left (-1+x^3\right )^{2/3}} \, dx=-\frac {\arctan \left (\frac {\frac {2 x}{\sqrt [3]{x^3-1}}+1}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {\arctan \left (\frac {1-2 \sqrt [3]{x^3-1}}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {1}{2} \log \left (x-\sqrt [3]{x^3-1}\right )-\frac {1}{2} \log \left (\sqrt [3]{x^3-1}+1\right )+\frac {\log (x)}{2} \]
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Rule 31
Rule 60
Rule 210
Rule 272
Rule 337
Rule 632
Rule 1858
Rubi steps \begin{align*} \text {integral}= \int \left (-\frac {1}{x \left (-1+x^3\right )^{2/3}}+\frac {x}{\left (-1+x^3\right )^{2/3}}\right ) \, dx \\ = -\int \frac {1}{x \left (-1+x^3\right )^{2/3}} \, dx+\int \frac {x}{\left (-1+x^3\right )^{2/3}} \, dx \\ = -\frac {\arctan \left (\frac {1+\frac {2 x}{\sqrt [3]{-1+x^3}}}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {1}{2} \log \left (x-\sqrt [3]{-1+x^3}\right )-\frac {1}{3} \text {Subst}\left (\int \frac {1}{(-1+x)^{2/3} x} \, dx,x,x^3\right ) \\ = -\frac {\arctan \left (\frac {1+\frac {2 x}{\sqrt [3]{-1+x^3}}}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {\log (x)}{2}-\frac {1}{2} \log \left (x-\sqrt [3]{-1+x^3}\right )-\frac {1}{2} \text {Subst}\left (\int \frac {1}{1+x} \, dx,x,\sqrt [3]{-1+x^3}\right )-\frac {1}{2} \text {Subst}\left (\int \frac {1}{1-x+x^2} \, dx,x,\sqrt [3]{-1+x^3}\right ) \\ = -\frac {\arctan \left (\frac {1+\frac {2 x}{\sqrt [3]{-1+x^3}}}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {\log (x)}{2}-\frac {1}{2} \log \left (x-\sqrt [3]{-1+x^3}\right )-\frac {1}{2} \log \left (1+\sqrt [3]{-1+x^3}\right )+\text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,-1+2 \sqrt [3]{-1+x^3}\right ) \\ = -\frac {\arctan \left (\frac {1+\frac {2 x}{\sqrt [3]{-1+x^3}}}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {\arctan \left (\frac {-1+2 \sqrt [3]{-1+x^3}}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {\log (x)}{2}-\frac {1}{2} \log \left (x-\sqrt [3]{-1+x^3}\right )-\frac {1}{2} \log \left (1+\sqrt [3]{-1+x^3}\right ) \\ \end{align*}
Time = 1.15 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.91 \[ \int \frac {-1+x^2}{x \left (-1+x^3\right )^{2/3}} \, dx=\frac {1}{2} \left (-2 \sqrt {3} \arctan \left (\frac {\sqrt {3} (-1+x)}{-1+x+2 \sqrt [3]{-1+x^3}}\right )-2 \log \left (1-x+\sqrt [3]{-1+x^3}\right )+\log \left (1-2 x+x^2+(-1+x) \sqrt [3]{-1+x^3}+\left (-1+x^3\right )^{2/3}\right )\right ) \]
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Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.67 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.04
method | result | size |
meijerg | \(\frac {{\left (-\operatorname {signum}\left (x^{3}-1\right )\right )}^{\frac {2}{3}} x^{2} \operatorname {hypergeom}\left (\left [\frac {2}{3}, \frac {2}{3}\right ], \left [\frac {5}{3}\right ], x^{3}\right )}{2 \operatorname {signum}\left (x^{3}-1\right )^{\frac {2}{3}}}-\frac {{\left (-\operatorname {signum}\left (x^{3}-1\right )\right )}^{\frac {2}{3}} \left (\left (\frac {\pi \sqrt {3}}{6}-\frac {3 \ln \left (3\right )}{2}+3 \ln \left (x \right )+i \pi \right ) \Gamma \left (\frac {2}{3}\right )+\frac {2 \Gamma \left (\frac {2}{3}\right ) x^{3} \operatorname {hypergeom}\left (\left [1, 1, \frac {5}{3}\right ], \left [2, 2\right ], x^{3}\right )}{3}\right )}{3 \Gamma \left (\frac {2}{3}\right ) \operatorname {signum}\left (x^{3}-1\right )^{\frac {2}{3}}}\) | \(99\) |
trager | \(-\ln \left (-\frac {58 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x^{2}-145 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x -99 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{3}-1\right )^{\frac {2}{3}}-99 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{3}-1\right )^{\frac {1}{3}} x -157 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{2}+58 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2}+99 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{3}-1\right )^{\frac {1}{3}}+117 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x -28 \left (x^{3}-1\right )^{\frac {2}{3}}-28 \left (x^{3}-1\right )^{\frac {1}{3}} x +30 x^{2}-157 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )+28 \left (x^{3}-1\right )^{\frac {1}{3}}-18 x +30}{x}\right ) \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )+\operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \ln \left (-\frac {58 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x^{2}-145 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x +99 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{3}-1\right )^{\frac {2}{3}}+99 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{3}-1\right )^{\frac {1}{3}} x +41 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{2}+58 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2}-99 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{3}-1\right )^{\frac {1}{3}}+173 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x -127 \left (x^{3}-1\right )^{\frac {2}{3}}-127 \left (x^{3}-1\right )^{\frac {1}{3}} x -69 x^{2}+41 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )+127 \left (x^{3}-1\right )^{\frac {1}{3}}-46 x -69}{x}\right )+\ln \left (-\frac {58 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x^{2}-145 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x -99 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{3}-1\right )^{\frac {2}{3}}-99 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{3}-1\right )^{\frac {1}{3}} x -157 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{2}+58 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2}+99 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{3}-1\right )^{\frac {1}{3}}+117 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x -28 \left (x^{3}-1\right )^{\frac {2}{3}}-28 \left (x^{3}-1\right )^{\frac {1}{3}} x +30 x^{2}-157 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )+28 \left (x^{3}-1\right )^{\frac {1}{3}}-18 x +30}{x}\right )\) | \(560\) |
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Time = 0.40 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.93 \[ \int \frac {-1+x^2}{x \left (-1+x^3\right )^{2/3}} \, dx=-\sqrt {3} \arctan \left (-\frac {4 \, \sqrt {3} {\left (x^{3} - 1\right )}^{\frac {1}{3}} {\left (x - 1\right )} + \sqrt {3} {\left (x^{2} + x + 1\right )} - 2 \, \sqrt {3} {\left (x^{3} - 1\right )}^{\frac {2}{3}}}{3 \, {\left (3 \, x^{2} - 5 \, x + 3\right )}}\right ) - \frac {1}{2} \, \log \left (\frac {{\left (x^{3} - 1\right )}^{\frac {1}{3}} {\left (x - 1\right )} + x - {\left (x^{3} - 1\right )}^{\frac {2}{3}}}{x}\right ) \]
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Result contains complex when optimal does not.
Time = 1.10 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.68 \[ \int \frac {-1+x^2}{x \left (-1+x^3\right )^{2/3}} \, dx=\frac {x^{2} e^{- \frac {2 i \pi }{3}} \Gamma \left (\frac {2}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {2}{3}, \frac {2}{3} \\ \frac {5}{3} \end {matrix}\middle | {x^{3}} \right )}}{3 \Gamma \left (\frac {5}{3}\right )} + \frac {\Gamma \left (\frac {2}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {2}{3}, \frac {2}{3} \\ \frac {5}{3} \end {matrix}\middle | {\frac {e^{2 i \pi }}{x^{3}}} \right )}}{3 x^{2} \Gamma \left (\frac {5}{3}\right )} \]
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Time = 0.24 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.31 \[ \int \frac {-1+x^2}{x \left (-1+x^3\right )^{2/3}} \, dx=-\frac {1}{3} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (x^{3} - 1\right )}^{\frac {1}{3}} - 1\right )}\right ) + \frac {1}{3} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (\frac {2 \, {\left (x^{3} - 1\right )}^{\frac {1}{3}}}{x} + 1\right )}\right ) + \frac {1}{6} \, \log \left ({\left (x^{3} - 1\right )}^{\frac {2}{3}} - {\left (x^{3} - 1\right )}^{\frac {1}{3}} + 1\right ) - \frac {1}{3} \, \log \left ({\left (x^{3} - 1\right )}^{\frac {1}{3}} + 1\right ) + \frac {1}{6} \, \log \left (\frac {{\left (x^{3} - 1\right )}^{\frac {1}{3}}}{x} + \frac {{\left (x^{3} - 1\right )}^{\frac {2}{3}}}{x^{2}} + 1\right ) - \frac {1}{3} \, \log \left (\frac {{\left (x^{3} - 1\right )}^{\frac {1}{3}}}{x} - 1\right ) \]
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\[ \int \frac {-1+x^2}{x \left (-1+x^3\right )^{2/3}} \, dx=\int { \frac {x^{2} - 1}{{\left (x^{3} - 1\right )}^{\frac {2}{3}} x} \,d x } \]
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Timed out. \[ \int \frac {-1+x^2}{x \left (-1+x^3\right )^{2/3}} \, dx=\int \frac {x^2-1}{x\,{\left (x^3-1\right )}^{2/3}} \,d x \]
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