Integrand size = 21, antiderivative size = 105 \[ \int \frac {\left (1+x^3\right )^{2/3} \left (-2+x^3+x^6\right )}{x^9} \, dx=\frac {\left (1+x^3\right )^{2/3} \left (5-2 x^3-17 x^6\right )}{20 x^8}+\frac {\arctan \left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{1+x^3}}\right )}{\sqrt {3}}-\frac {1}{3} \log \left (-x+\sqrt [3]{1+x^3}\right )+\frac {1}{6} \log \left (x^2+x \sqrt [3]{1+x^3}+\left (1+x^3\right )^{2/3}\right ) \]
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Time = 0.02 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.90, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {1502, 277, 270, 283, 245} \[ \int \frac {\left (1+x^3\right )^{2/3} \left (-2+x^3+x^6\right )}{x^9} \, dx=\frac {\arctan \left (\frac {\frac {2 x}{\sqrt [3]{x^3+1}}+1}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {1}{2} \log \left (\sqrt [3]{x^3+1}-x\right )+\frac {\left (x^3+1\right )^{5/3}}{4 x^8}-\frac {7 \left (x^3+1\right )^{5/3}}{20 x^5}-\frac {\left (x^3+1\right )^{2/3}}{2 x^2} \]
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Rule 245
Rule 270
Rule 277
Rule 283
Rule 1502
Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {2 \left (1+x^3\right )^{2/3}}{x^9}+\frac {\left (1+x^3\right )^{2/3}}{x^6}+\frac {\left (1+x^3\right )^{2/3}}{x^3}\right ) \, dx \\ & = -\left (2 \int \frac {\left (1+x^3\right )^{2/3}}{x^9} \, dx\right )+\int \frac {\left (1+x^3\right )^{2/3}}{x^6} \, dx+\int \frac {\left (1+x^3\right )^{2/3}}{x^3} \, dx \\ & = -\frac {\left (1+x^3\right )^{2/3}}{2 x^2}+\frac {\left (1+x^3\right )^{5/3}}{4 x^8}-\frac {\left (1+x^3\right )^{5/3}}{5 x^5}+\frac {3}{4} \int \frac {\left (1+x^3\right )^{2/3}}{x^6} \, dx+\int \frac {1}{\sqrt [3]{1+x^3}} \, dx \\ & = -\frac {\left (1+x^3\right )^{2/3}}{2 x^2}+\frac {\left (1+x^3\right )^{5/3}}{4 x^8}-\frac {7 \left (1+x^3\right )^{5/3}}{20 x^5}+\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 ) \\ \end{align*}
Time = 0.20 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.00 \[ \int \frac {\left (1+x^3\right )^{2/3} \left (-2+x^3+x^6\right )}{x^9} \, dx=\frac {\left (1+x^3\right )^{2/3} \left (5-2 x^3-17 x^6\right )}{20 x^8}+\frac {\arctan \left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{1+x^3}}\right )}{\sqrt {3}}-\frac {1}{3} \log \left (-x+\sqrt [3]{1+x^3}\right )+\frac {1}{6} \log \left (x^2+x \sqrt [3]{1+x^3}+\left (1+x^3\right )^{2/3}\right ) \]
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Result contains higher order function than in optimal. Order 5 vs. order 3.
Time = 1.87 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.42
method | result | size |
risch | \(-\frac {17 x^{9}+19 x^{6}-3 x^{3}-5}{20 x^{8} \left (x^{3}+1\right )^{\frac {1}{3}}}+x \operatorname {hypergeom}\left (\left [\frac {1}{3}, \frac {1}{3}\right ], \left [\frac {4}{3}\right ], -x^{3}\right )\) | \(44\) |
meijerg | \(-\frac {\operatorname {hypergeom}\left (\left [-\frac {2}{3}, -\frac {2}{3}\right ], \left [\frac {1}{3}\right ], -x^{3}\right )}{2 x^{2}}-\frac {\left (x^{3}+1\right )^{\frac {5}{3}}}{5 x^{5}}+\frac {\left (-\frac {3}{5} x^{6}+\frac {2}{5} x^{3}+1\right ) \left (x^{3}+1\right )^{\frac {2}{3}}}{4 x^{8}}\) | \(54\) |
pseudoelliptic | \(\frac {10 \ln \left (\frac {x^{2}+x \left (x^{3}+1\right )^{\frac {1}{3}}+\left (x^{3}+1\right )^{\frac {2}{3}}}{x^{2}}\right ) x^{8}-20 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (x +2 \left (x^{3}+1\right )^{\frac {1}{3}}\right )}{3 x}\right ) x^{8}-20 \ln \left (\frac {-x +\left (x^{3}+1\right )^{\frac {1}{3}}}{x}\right ) x^{8}+\left (-51 x^{6}-6 x^{3}+15\right ) \left (x^{3}+1\right )^{\frac {2}{3}}}{60 x^{8}}\) | \(106\) |
trager | \(-\frac {\left (17 x^{6}+2 x^{3}-5\right ) \left (x^{3}+1\right )^{\frac {2}{3}}}{20 x^{8}}-\frac {\ln \left (317 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x^{3}-555 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{3}+1\right )^{\frac {2}{3}} x +2358 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{3}+1\right )^{\frac {1}{3}} x^{2}-2120 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{3}-1803 x \left (x^{3}+1\right )^{\frac {2}{3}}-555 x^{2} \left (x^{3}+1\right )^{\frac {1}{3}}+2675 x^{3}-317 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2}-99 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )+1070\right )}{3}+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \ln \left (-535 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x^{3}+555 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{3}+1\right )^{\frac {2}{3}} x +1803 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{3}+1\right )^{\frac {1}{3}} x^{2}-1823 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{3}-2358 x \left (x^{3}+1\right )^{\frac {2}{3}}+555 x^{2} \left (x^{3}+1\right )^{\frac {1}{3}}+1268 x^{3}+535 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2}-1922 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )+951\right )}{3}\) | \(287\) |
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Time = 0.44 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.11 \[ \int \frac {\left (1+x^3\right )^{2/3} \left (-2+x^3+x^6\right )}{x^9} \, dx=\frac {20 \, \sqrt {3} x^{8} \arctan \left (-\frac {25382 \, \sqrt {3} {\left (x^{3} + 1\right )}^{\frac {1}{3}} x^{2} - 13720 \, \sqrt {3} {\left (x^{3} + 1\right )}^{\frac {2}{3}} x + \sqrt {3} {\left (5831 \, x^{3} + 7200\right )}}{58653 \, x^{3} + 8000}\right ) - 10 \, x^{8} \log \left (3 \, {\left (x^{3} + 1\right )}^{\frac {1}{3}} x^{2} - 3 \, {\left (x^{3} + 1\right )}^{\frac {2}{3}} x + 1\right ) - 3 \, {\left (17 \, x^{6} + 2 \, x^{3} - 5\right )} {\left (x^{3} + 1\right )}^{\frac {2}{3}}}{60 \, x^{8}} \]
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Result contains complex when optimal does not.
Time = 1.85 (sec) , antiderivative size = 175, normalized size of antiderivative = 1.67 \[ \int \frac {\left (1+x^3\right )^{2/3} \left (-2+x^3+x^6\right )}{x^9} \, dx=\frac {\left (1 + \frac {1}{x^{3}}\right )^{\frac {2}{3}} \Gamma \left (- \frac {5}{3}\right )}{3 \Gamma \left (- \frac {2}{3}\right )} - \frac {2 \left (x^{3} + 1\right )^{\frac {2}{3}} \Gamma \left (- \frac {8}{3}\right )}{3 x^{2} \Gamma \left (- \frac {2}{3}\right )} + \frac {\Gamma \left (- \frac {2}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {2}{3}, - \frac {2}{3} \\ \frac {1}{3} \end {matrix}\middle | {x^{3} e^{i \pi }} \right )}}{3 x^{2} \Gamma \left (\frac {1}{3}\right )} + \frac {\left (1 + \frac {1}{x^{3}}\right )^{\frac {2}{3}} \Gamma \left (- \frac {5}{3}\right )}{3 x^{3} \Gamma \left (- \frac {2}{3}\right )} + \frac {4 \left (x^{3} + 1\right )^{\frac {2}{3}} \Gamma \left (- \frac {8}{3}\right )}{9 x^{5} \Gamma \left (- \frac {2}{3}\right )} + \frac {10 \left (x^{3} + 1\right )^{\frac {2}{3}} \Gamma \left (- \frac {8}{3}\right )}{9 x^{8} \Gamma \left (- \frac {2}{3}\right )} \]
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Time = 0.27 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.00 \[ \int \frac {\left (1+x^3\right )^{2/3} \left (-2+x^3+x^6\right )}{x^9} \, dx=-\frac {1}{3} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (\frac {2 \, {\left (x^{3} + 1\right )}^{\frac {1}{3}}}{x} + 1\right )}\right ) - \frac {{\left (x^{3} + 1\right )}^{\frac {2}{3}}}{2 \, x^{2}} - \frac {3 \, {\left (x^{3} + 1\right )}^{\frac {5}{3}}}{5 \, x^{5}} + \frac {{\left (x^{3} + 1\right )}^{\frac {8}{3}}}{4 \, x^{8}} + \frac {1}{6} \, \log \left (\frac {{\left (x^{3} + 1\right )}^{\frac {1}{3}}}{x} + \frac {{\left (x^{3} + 1\right )}^{\frac {2}{3}}}{x^{2}} + 1\right ) - \frac {1}{3} \, \log \left (\frac {{\left (x^{3} + 1\right )}^{\frac {1}{3}}}{x} - 1\right ) \]
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\[ \int \frac {\left (1+x^3\right )^{2/3} \left (-2+x^3+x^6\right )}{x^9} \, dx=\int { \frac {{\left (x^{6} + x^{3} - 2\right )} {\left (x^{3} + 1\right )}^{\frac {2}{3}}}{x^{9}} \,d x } \]
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Timed out. \[ \int \frac {\left (1+x^3\right )^{2/3} \left (-2+x^3+x^6\right )}{x^9} \, dx=\int \frac {{\left (x^3+1\right )}^{2/3}\,\left (x^6+x^3-2\right )}{x^9} \,d x \]
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