Integrand size = 23, antiderivative size = 105 \[ \int \frac {\left (-1+x^3\right )^{2/3} \left (-2+2 x^3+x^6\right )}{x^9} \, dx=\frac {\left (-1+x^3\right )^{2/3} \left (1-2 x^3-x^6\right )}{4 x^8}+\frac {\arctan \left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{-1+x^3}}\right )}{\sqrt {3}}-\frac {1}{3} \log \left (-x+\sqrt [3]{-1+x^3}\right )+\frac {1}{6} \log \left (x^2+x \sqrt [3]{-1+x^3}+\left (-1+x^3\right )^{2/3}\right ) \]
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Time = 0.03 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.90, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {1502, 277, 270, 283, 245} \[ \int \frac {\left (-1+x^3\right )^{2/3} \left (-2+2 x^3+x^6\right )}{x^9} \, dx=\frac {\arctan \left (\frac {\frac {2 x}{\sqrt [3]{x^3-1}}+1}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {1}{2} \log \left (\sqrt [3]{x^3-1}-x\right )-\frac {\left (x^3-1\right )^{5/3}}{4 x^8}+\frac {\left (x^3-1\right )^{5/3}}{4 x^5}-\frac {\left (x^3-1\right )^{2/3}}{2 x^2} \]
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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 {2 \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 )+2 \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 {2 \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 {\left (-1+x^3\right )^{5/3}}{4 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 = 0.19 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.97 \[ \int \frac {\left (-1+x^3\right )^{2/3} \left (-2+2 x^3+x^6\right )}{x^9} \, dx=\frac {1}{12} \left (-\frac {3 \left (-1+x^3\right )^{2/3} \left (-1+2 x^3+x^6\right )}{x^8}+4 \sqrt {3} \arctan \left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{-1+x^3}}\right )-4 \log \left (-x+\sqrt [3]{-1+x^3}\right )+2 \log \left (x^2+x \sqrt [3]{-1+x^3}+\left (-1+x^3\right )^{2/3}\right )\right ) \]
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Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 1.67 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.53
method | result | size |
risch | \(-\frac {x^{9}+x^{6}-3 x^{3}+1}{4 x^{8} \left (x^{3}-1\right )^{\frac {1}{3}}}+\frac {{\left (-\operatorname {signum}\left (x^{3}-1\right )\right )}^{\frac {1}{3}} x \operatorname {hypergeom}\left (\left [\frac {1}{3}, \frac {1}{3}\right ], \left [\frac {4}{3}\right ], x^{3}\right )}{\operatorname {signum}\left (x^{3}-1\right )^{\frac {1}{3}}}\) | \(56\) |
pseudoelliptic | \(\frac {2 \ln \left (\frac {x^{2}+x \left (x^{3}-1\right )^{\frac {1}{3}}+\left (x^{3}-1\right )^{\frac {2}{3}}}{x^{2}}\right ) x^{8}-4 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (x +2 \left (x^{3}-1\right )^{\frac {1}{3}}\right )}{3 x}\right ) x^{8}-4 \ln \left (\frac {-x +\left (x^{3}-1\right )^{\frac {1}{3}}}{x}\right ) x^{8}-3 \left (x^{3}-1\right )^{\frac {2}{3}} \left (x^{6}+2 x^{3}-1\right )}{12 x^{8}}\) | \(105\) |
meijerg | \(-\frac {\operatorname {signum}\left (x^{3}-1\right )^{\frac {2}{3}} \operatorname {hypergeom}\left (\left [-\frac {2}{3}, -\frac {2}{3}\right ], \left [\frac {1}{3}\right ], x^{3}\right )}{2 {\left (-\operatorname {signum}\left (x^{3}-1\right )\right )}^{\frac {2}{3}} x^{2}}-\frac {2 \operatorname {signum}\left (x^{3}-1\right )^{\frac {2}{3}} \left (-x^{3}+1\right )^{\frac {5}{3}}}{5 {\left (-\operatorname {signum}\left (x^{3}-1\right )\right )}^{\frac {2}{3}} x^{5}}+\frac {\operatorname {signum}\left (x^{3}-1\right )^{\frac {2}{3}} \left (-\frac {3}{5} x^{6}-\frac {2}{5} x^{3}+1\right ) \left (-x^{3}+1\right )^{\frac {2}{3}}}{4 {\left (-\operatorname {signum}\left (x^{3}-1\right )\right )}^{\frac {2}{3}} x^{8}}\) | \(110\) |
trager | \(-\frac {\left (x^{6}+2 x^{3}-1\right ) \left (x^{3}-1\right )^{\frac {2}{3}}}{4 x^{8}}-\frac {\ln \left (21954560 \operatorname {RootOf}\left (65536 \textit {\_Z}^{2}-256 \textit {\_Z} +1\right )^{2} x^{3}-2482176 \operatorname {RootOf}\left (65536 \textit {\_Z}^{2}-256 \textit {\_Z} +1\right ) \left (x^{3}-1\right )^{\frac {2}{3}} x +4264704 \operatorname {RootOf}\left (65536 \textit {\_Z}^{2}-256 \textit {\_Z} +1\right ) \left (x^{3}-1\right )^{\frac {1}{3}} x^{2}-1868288 \operatorname {RootOf}\left (65536 \textit {\_Z}^{2}-256 \textit {\_Z} +1\right ) x^{3}+16659 x \left (x^{3}-1\right )^{\frac {2}{3}}-6963 x^{2} \left (x^{3}-1\right )^{\frac {1}{3}}-9361 x^{3}-175636480 \operatorname {RootOf}\left (65536 \textit {\_Z}^{2}-256 \textit {\_Z} +1\right )^{2}+452864 \operatorname {RootOf}\left (65536 \textit {\_Z}^{2}-256 \textit {\_Z} +1\right )+6105\right )}{3}+\frac {\ln \left (3713319698432 \operatorname {RootOf}\left (65536 \textit {\_Z}^{2}-256 \textit {\_Z} +1\right )^{2} x^{3}+49654493184 \operatorname {RootOf}\left (65536 \textit {\_Z}^{2}-256 \textit {\_Z} +1\right ) \left (x^{3}-1\right )^{\frac {2}{3}} x -386189407488 \operatorname {RootOf}\left (65536 \textit {\_Z}^{2}-256 \textit {\_Z} +1\right ) \left (x^{3}-1\right )^{\frac {1}{3}} x^{2}+322029759232 \operatorname {RootOf}\left (65536 \textit {\_Z}^{2}-256 \textit {\_Z} +1\right ) x^{3}-1508552373 x \left (x^{3}-1\right )^{\frac {2}{3}}+1314589509 x^{2} \left (x^{3}-1\right )^{\frac {1}{3}}+250623626 x^{3}-29706557587456 \operatorname {RootOf}\left (65536 \textit {\_Z}^{2}-256 \textit {\_Z} +1\right )^{2}-124866866688 \operatorname {RootOf}\left (65536 \textit {\_Z}^{2}-256 \textit {\_Z} +1\right )-79743881\right )}{3}-\frac {256 \ln \left (3713319698432 \operatorname {RootOf}\left (65536 \textit {\_Z}^{2}-256 \textit {\_Z} +1\right )^{2} x^{3}+49654493184 \operatorname {RootOf}\left (65536 \textit {\_Z}^{2}-256 \textit {\_Z} +1\right ) \left (x^{3}-1\right )^{\frac {2}{3}} x -386189407488 \operatorname {RootOf}\left (65536 \textit {\_Z}^{2}-256 \textit {\_Z} +1\right ) \left (x^{3}-1\right )^{\frac {1}{3}} x^{2}+322029759232 \operatorname {RootOf}\left (65536 \textit {\_Z}^{2}-256 \textit {\_Z} +1\right ) x^{3}-1508552373 x \left (x^{3}-1\right )^{\frac {2}{3}}+1314589509 x^{2} \left (x^{3}-1\right )^{\frac {1}{3}}+250623626 x^{3}-29706557587456 \operatorname {RootOf}\left (65536 \textit {\_Z}^{2}-256 \textit {\_Z} +1\right )^{2}-124866866688 \operatorname {RootOf}\left (65536 \textit {\_Z}^{2}-256 \textit {\_Z} +1\right )-79743881\right ) \operatorname {RootOf}\left (65536 \textit {\_Z}^{2}-256 \textit {\_Z} +1\right )}{3}\) | \(449\) |
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Time = 0.46 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.10 \[ \int \frac {\left (-1+x^3\right )^{2/3} \left (-2+2 x^3+x^6\right )}{x^9} \, dx=\frac {4 \, \sqrt {3} x^{8} \arctan \left (-\frac {25382 \, \sqrt {3} {\left (x^{3} - 1\right )}^{\frac {1}{3}} x^{2} - 13720 \, \sqrt {3} {\left (x^{3} - 1\right )}^{\frac {2}{3}} x + \sqrt {3} {\left (5831 \, x^{3} - 7200\right )}}{58653 \, x^{3} - 8000}\right ) - 2 \, x^{8} \log \left (-3 \, {\left (x^{3} - 1\right )}^{\frac {1}{3}} x^{2} + 3 \, {\left (x^{3} - 1\right )}^{\frac {2}{3}} x + 1\right ) - 3 \, {\left (x^{6} + 2 \, x^{3} - 1\right )} {\left (x^{3} - 1\right )}^{\frac {2}{3}}}{12 \, x^{8}} \]
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Result contains complex when optimal does not.
Time = 1.97 (sec) , antiderivative size = 469, normalized size of antiderivative = 4.47 \[ \int \frac {\left (-1+x^3\right )^{2/3} \left (-2+2 x^3+x^6\right )}{x^9} \, dx=2 \left (\begin {cases} \frac {\left (-1 + \frac {1}{x^{3}}\right )^{\frac {2}{3}} e^{- \frac {i \pi }{3}} \Gamma \left (- \frac {5}{3}\right )}{3 \Gamma \left (- \frac {2}{3}\right )} - \frac {\left (-1 + \frac {1}{x^{3}}\right )^{\frac {2}{3}} e^{- \frac {i \pi }{3}} \Gamma \left (- \frac {5}{3}\right )}{3 x^{3} \Gamma \left (- \frac {2}{3}\right )} & \text {for}\: \frac {1}{\left |{x^{3}}\right |} > 1 \\- \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 {\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 )} & \text {otherwise} \end {cases}\right ) - 2 \left (\begin {cases} \frac {3 x^{6} \left (-1 + \frac {1}{x^{3}}\right )^{\frac {2}{3}} e^{\frac {2 i \pi }{3}} \Gamma \left (- \frac {8}{3}\right )}{9 x^{6} \Gamma \left (- \frac {2}{3}\right ) - 9 x^{3} \Gamma \left (- \frac {2}{3}\right )} - \frac {x^{3} \left (-1 + \frac {1}{x^{3}}\right )^{\frac {2}{3}} e^{\frac {2 i \pi }{3}} \Gamma \left (- \frac {8}{3}\right )}{9 x^{6} \Gamma \left (- \frac {2}{3}\right ) - 9 x^{3} \Gamma \left (- \frac {2}{3}\right )} + \frac {5 \left (-1 + \frac {1}{x^{3}}\right )^{\frac {2}{3}} e^{\frac {2 i \pi }{3}} \Gamma \left (- \frac {8}{3}\right )}{9 x^{9} \Gamma \left (- \frac {2}{3}\right ) - 9 x^{6} \Gamma \left (- \frac {2}{3}\right )} - \frac {7 \left (-1 + \frac {1}{x^{3}}\right )^{\frac {2}{3}} e^{\frac {2 i \pi }{3}} \Gamma \left (- \frac {8}{3}\right )}{9 x^{6} \Gamma \left (- \frac {2}{3}\right ) - 9 x^{3} \Gamma \left (- \frac {2}{3}\right )} & \text {for}\: \frac {1}{\left |{x^{3}}\right |} > 1 \\\frac {\left (1 - \frac {1}{x^{3}}\right )^{\frac {2}{3}} \Gamma \left (- \frac {8}{3}\right )}{3 \Gamma \left (- \frac {2}{3}\right )} + \frac {2 \left (1 - \frac {1}{x^{3}}\right )^{\frac {2}{3}} \Gamma \left (- \frac {8}{3}\right )}{9 x^{3} \Gamma \left (- \frac {2}{3}\right )} - \frac {5 \left (1 - \frac {1}{x^{3}}\right )^{\frac {2}{3}} \Gamma \left (- \frac {8}{3}\right )}{9 x^{6} \Gamma \left (- \frac {2}{3}\right )} & \text {otherwise} \end {cases}\right ) + \frac {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 {1}{3} \end {matrix}\middle | {x^{3}} \right )}}{3 x^{2} \Gamma \left (\frac {1}{3}\right )} \]
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Time = 0.28 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.89 \[ \int \frac {\left (-1+x^3\right )^{2/3} \left (-2+2 x^3+x^6\right )}{x^9} \, dx=-\frac {1}{3} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (\frac {2 \, {\left (x^{3} - 1\right )}^{\frac {1}{3}}}{x} + 1\right )}\right ) - \frac {{\left (x^{3} - 1\right )}^{\frac {2}{3}}}{2 \, x^{2}} + \frac {{\left (x^{3} - 1\right )}^{\frac {8}{3}}}{4 \, x^{8}} + \frac {1}{6} \, \log \left (\frac {{\left (x^{3} - 1\right )}^{\frac {1}{3}}}{x} + \frac {{\left (x^{3} - 1\right )}^{\frac {2}{3}}}{x^{2}} + 1\right ) - \frac {1}{3} \, \log \left (\frac {{\left (x^{3} - 1\right )}^{\frac {1}{3}}}{x} - 1\right ) \]
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\[ \int \frac {\left (-1+x^3\right )^{2/3} \left (-2+2 x^3+x^6\right )}{x^9} \, dx=\int { \frac {{\left (x^{6} + 2 \, x^{3} - 2\right )} {\left (x^{3} - 1\right )}^{\frac {2}{3}}}{x^{9}} \,d x } \]
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Timed out. \[ \int \frac {\left (-1+x^3\right )^{2/3} \left (-2+2 x^3+x^6\right )}{x^9} \, dx=\int \frac {{\left (x^3-1\right )}^{2/3}\,\left (x^6+2\,x^3-2\right )}{x^9} \,d x \]
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