Integrand size = 18, antiderivative size = 111 \[ \int \frac {\sqrt [3]{-1+x^6} \left (1+x^6\right )}{x^9} \, dx=\frac {\left (-1-3 x^6\right ) \sqrt [3]{-1+x^6}}{8 x^8}-\frac {\arctan \left (\frac {\sqrt {3} x^2}{x^2+2 \sqrt [3]{-1+x^6}}\right )}{2 \sqrt {3}}-\frac {1}{6} \log \left (-x^2+\sqrt [3]{-1+x^6}\right )+\frac {1}{12} \log \left (x^4+x^2 \sqrt [3]{-1+x^6}+\left (-1+x^6\right )^{2/3}\right ) \]
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Time = 0.02 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.77, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {462, 281, 283, 337} \[ \int \frac {\sqrt [3]{-1+x^6} \left (1+x^6\right )}{x^9} \, dx=-\frac {\arctan \left (\frac {\frac {2 x^2}{\sqrt [3]{x^6-1}}+1}{\sqrt {3}}\right )}{2 \sqrt {3}}+\frac {\left (x^6-1\right )^{4/3}}{8 x^8}-\frac {\sqrt [3]{x^6-1}}{2 x^2}-\frac {1}{4} \log \left (x^2-\sqrt [3]{x^6-1}\right ) \]
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Rule 281
Rule 283
Rule 337
Rule 462
Rubi steps \begin{align*} \text {integral}& = \frac {\left (-1+x^6\right )^{4/3}}{8 x^8}+\int \frac {\sqrt [3]{-1+x^6}}{x^3} \, dx \\ & = \frac {\left (-1+x^6\right )^{4/3}}{8 x^8}+\frac {1}{2} \text {Subst}\left (\int \frac {\sqrt [3]{-1+x^3}}{x^2} \, dx,x,x^2\right ) \\ & = -\frac {\sqrt [3]{-1+x^6}}{2 x^2}+\frac {\left (-1+x^6\right )^{4/3}}{8 x^8}+\frac {1}{2} \text {Subst}\left (\int \frac {x}{\left (-1+x^3\right )^{2/3}} \, dx,x,x^2\right ) \\ & = -\frac {\sqrt [3]{-1+x^6}}{2 x^2}+\frac {\left (-1+x^6\right )^{4/3}}{8 x^8}-\frac {\arctan \left (\frac {1+\frac {2 x^2}{\sqrt [3]{-1+x^6}}}{\sqrt {3}}\right )}{2 \sqrt {3}}-\frac {1}{4} \log \left (x^2-\sqrt [3]{-1+x^6}\right ) \\ \end{align*}
Time = 0.43 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.96 \[ \int \frac {\sqrt [3]{-1+x^6} \left (1+x^6\right )}{x^9} \, dx=\frac {1}{24} \left (-\frac {3 \sqrt [3]{-1+x^6} \left (1+3 x^6\right )}{x^8}-4 \sqrt {3} \arctan \left (\frac {\sqrt {3} x^2}{x^2+2 \sqrt [3]{-1+x^6}}\right )-4 \log \left (-x^2+\sqrt [3]{-1+x^6}\right )+2 \log \left (x^4+x^2 \sqrt [3]{-1+x^6}+\left (-1+x^6\right )^{2/3}\right )\right ) \]
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Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 3.51 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.52
method | result | size |
risch | \(-\frac {3 x^{12}-2 x^{6}-1}{8 x^{8} \left (x^{6}-1\right )^{\frac {2}{3}}}+\frac {{\left (-\operatorname {signum}\left (x^{6}-1\right )\right )}^{\frac {2}{3}} x^{4} \operatorname {hypergeom}\left (\left [\frac {2}{3}, \frac {2}{3}\right ], \left [\frac {5}{3}\right ], x^{6}\right )}{4 \operatorname {signum}\left (x^{6}-1\right )^{\frac {2}{3}}}\) | \(58\) |
meijerg | \(-\frac {\operatorname {signum}\left (x^{6}-1\right )^{\frac {1}{3}} \operatorname {hypergeom}\left (\left [-\frac {1}{3}, -\frac {1}{3}\right ], \left [\frac {2}{3}\right ], x^{6}\right )}{2 {\left (-\operatorname {signum}\left (x^{6}-1\right )\right )}^{\frac {1}{3}} x^{2}}-\frac {\operatorname {signum}\left (x^{6}-1\right )^{\frac {1}{3}} \left (-x^{6}+1\right )^{\frac {4}{3}}}{8 {\left (-\operatorname {signum}\left (x^{6}-1\right )\right )}^{\frac {1}{3}} x^{8}}\) | \(66\) |
pseudoelliptic | \(\frac {4 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (x^{2}+2 \left (x^{6}-1\right )^{\frac {1}{3}}\right )}{3 x^{2}}\right ) x^{8}+2 \ln \left (\frac {x^{4}+x^{2} \left (x^{6}-1\right )^{\frac {1}{3}}+\left (x^{6}-1\right )^{\frac {2}{3}}}{x^{4}}\right ) x^{8}-4 \ln \left (\frac {-x^{2}+\left (x^{6}-1\right )^{\frac {1}{3}}}{x^{2}}\right ) x^{8}-9 \left (x^{6}-1\right )^{\frac {1}{3}} x^{6}-3 \left (x^{6}-1\right )^{\frac {1}{3}}}{24 x^{8}}\) | \(113\) |
trager | \(-\frac {\left (3 x^{6}+1\right ) \left (x^{6}-1\right )^{\frac {1}{3}}}{8 x^{8}}+\frac {128 \operatorname {RootOf}\left (65536 \textit {\_Z}^{2}-256 \textit {\_Z} +1\right ) \ln \left (-102774784196608 \operatorname {RootOf}\left (65536 \textit {\_Z}^{2}-256 \textit {\_Z} +1\right )^{2} x^{6}+77455459320064 \operatorname {RootOf}\left (65536 \textit {\_Z}^{2}-256 \textit {\_Z} +1\right ) x^{6}+77053995319296 \operatorname {RootOf}\left (65536 \textit {\_Z}^{2}-256 \textit {\_Z} +1\right ) \left (x^{6}-1\right )^{\frac {1}{3}} x^{4}-578845773886 x^{6}+77053995319296 \operatorname {RootOf}\left (65536 \textit {\_Z}^{2}-256 \textit {\_Z} +1\right ) \left (x^{6}-1\right )^{\frac {2}{3}} x^{2}-577277555133 x^{4} \left (x^{6}-1\right )^{\frac {1}{3}}-577277555133 x^{2} \left (x^{6}-1\right )^{\frac {2}{3}}+6577586188582912 \operatorname {RootOf}\left (65536 \textit {\_Z}^{2}-256 \textit {\_Z} +1\right )^{2}-100639379193600 \operatorname {RootOf}\left (65536 \textit {\_Z}^{2}-256 \textit {\_Z} +1\right )+384886980542\right )}{3}+\frac {\ln \left (-102774784196608 \operatorname {RootOf}\left (65536 \textit {\_Z}^{2}-256 \textit {\_Z} +1\right )^{2} x^{6}-76652531318528 \operatorname {RootOf}\left (65536 \textit {\_Z}^{2}-256 \textit {\_Z} +1\right ) x^{6}-77053995319296 \operatorname {RootOf}\left (65536 \textit {\_Z}^{2}-256 \textit {\_Z} +1\right ) \left (x^{6}-1\right )^{\frac {1}{3}} x^{4}-277853604670 x^{6}-77053995319296 \operatorname {RootOf}\left (65536 \textit {\_Z}^{2}-256 \textit {\_Z} +1\right ) \left (x^{6}-1\right )^{\frac {2}{3}} x^{2}-276285385917 x^{4} \left (x^{6}-1\right )^{\frac {1}{3}}-276285385917 x^{2} \left (x^{6}-1\right )^{\frac {2}{3}}+6577586188582912 \operatorname {RootOf}\left (65536 \textit {\_Z}^{2}-256 \textit {\_Z} +1\right )^{2}+49251987095296 \operatorname {RootOf}\left (65536 \textit {\_Z}^{2}-256 \textit {\_Z} +1\right )+92130405759\right )}{6}-\frac {128 \ln \left (-102774784196608 \operatorname {RootOf}\left (65536 \textit {\_Z}^{2}-256 \textit {\_Z} +1\right )^{2} x^{6}-76652531318528 \operatorname {RootOf}\left (65536 \textit {\_Z}^{2}-256 \textit {\_Z} +1\right ) x^{6}-77053995319296 \operatorname {RootOf}\left (65536 \textit {\_Z}^{2}-256 \textit {\_Z} +1\right ) \left (x^{6}-1\right )^{\frac {1}{3}} x^{4}-277853604670 x^{6}-77053995319296 \operatorname {RootOf}\left (65536 \textit {\_Z}^{2}-256 \textit {\_Z} +1\right ) \left (x^{6}-1\right )^{\frac {2}{3}} x^{2}-276285385917 x^{4} \left (x^{6}-1\right )^{\frac {1}{3}}-276285385917 x^{2} \left (x^{6}-1\right )^{\frac {2}{3}}+6577586188582912 \operatorname {RootOf}\left (65536 \textit {\_Z}^{2}-256 \textit {\_Z} +1\right )^{2}+49251987095296 \operatorname {RootOf}\left (65536 \textit {\_Z}^{2}-256 \textit {\_Z} +1\right )+92130405759\right ) \operatorname {RootOf}\left (65536 \textit {\_Z}^{2}-256 \textit {\_Z} +1\right )}{3}\) | \(469\) |
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Time = 0.43 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.05 \[ \int \frac {\sqrt [3]{-1+x^6} \left (1+x^6\right )}{x^9} \, dx=-\frac {4 \, \sqrt {3} x^{8} \arctan \left (-\frac {25382 \, \sqrt {3} {\left (x^{6} - 1\right )}^{\frac {1}{3}} x^{4} - 13720 \, \sqrt {3} {\left (x^{6} - 1\right )}^{\frac {2}{3}} x^{2} + \sqrt {3} {\left (5831 \, x^{6} - 7200\right )}}{58653 \, x^{6} - 8000}\right ) + 2 \, x^{8} \log \left (-3 \, {\left (x^{6} - 1\right )}^{\frac {1}{3}} x^{4} + 3 \, {\left (x^{6} - 1\right )}^{\frac {2}{3}} x^{2} + 1\right ) + 3 \, {\left (3 \, x^{6} + 1\right )} {\left (x^{6} - 1\right )}^{\frac {1}{3}}}{24 \, x^{8}} \]
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Result contains complex when optimal does not.
Time = 1.78 (sec) , antiderivative size = 167, normalized size of antiderivative = 1.50 \[ \int \frac {\sqrt [3]{-1+x^6} \left (1+x^6\right )}{x^9} \, dx=\begin {cases} \frac {\sqrt [3]{-1 + \frac {1}{x^{6}}} e^{- \frac {2 i \pi }{3}} \Gamma \left (- \frac {4}{3}\right )}{6 \Gamma \left (- \frac {1}{3}\right )} - \frac {\sqrt [3]{-1 + \frac {1}{x^{6}}} e^{- \frac {2 i \pi }{3}} \Gamma \left (- \frac {4}{3}\right )}{6 x^{6} \Gamma \left (- \frac {1}{3}\right )} & \text {for}\: \frac {1}{\left |{x^{6}}\right |} > 1 \\- \frac {\sqrt [3]{1 - \frac {1}{x^{6}}} \Gamma \left (- \frac {4}{3}\right )}{6 \Gamma \left (- \frac {1}{3}\right )} + \frac {\sqrt [3]{1 - \frac {1}{x^{6}}} \Gamma \left (- \frac {4}{3}\right )}{6 x^{6} \Gamma \left (- \frac {1}{3}\right )} & \text {otherwise} \end {cases} + \frac {e^{\frac {i \pi }{3}} \Gamma \left (- \frac {1}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{3}, - \frac {1}{3} \\ \frac {2}{3} \end {matrix}\middle | {x^{6}} \right )}}{6 x^{2} \Gamma \left (\frac {2}{3}\right )} \]
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Time = 0.27 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.84 \[ \int \frac {\sqrt [3]{-1+x^6} \left (1+x^6\right )}{x^9} \, dx=\frac {1}{6} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (\frac {2 \, {\left (x^{6} - 1\right )}^{\frac {1}{3}}}{x^{2}} + 1\right )}\right ) - \frac {{\left (x^{6} - 1\right )}^{\frac {1}{3}}}{2 \, x^{2}} + \frac {{\left (x^{6} - 1\right )}^{\frac {4}{3}}}{8 \, x^{8}} + \frac {1}{12} \, \log \left (\frac {{\left (x^{6} - 1\right )}^{\frac {1}{3}}}{x^{2}} + \frac {{\left (x^{6} - 1\right )}^{\frac {2}{3}}}{x^{4}} + 1\right ) - \frac {1}{6} \, \log \left (\frac {{\left (x^{6} - 1\right )}^{\frac {1}{3}}}{x^{2}} - 1\right ) \]
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\[ \int \frac {\sqrt [3]{-1+x^6} \left (1+x^6\right )}{x^9} \, dx=\int { \frac {{\left (x^{6} + 1\right )} {\left (x^{6} - 1\right )}^{\frac {1}{3}}}{x^{9}} \,d x } \]
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Timed out. \[ \int \frac {\sqrt [3]{-1+x^6} \left (1+x^6\right )}{x^9} \, dx=\int \frac {{\left (x^6-1\right )}^{1/3}\,\left (x^6+1\right )}{x^9} \,d x \]
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