Integrand size = 13, antiderivative size = 104 \[ \int \frac {\sqrt [3]{-1+x^3}}{x^{10}} \, dx=\frac {\sqrt [3]{-1+x^3} \left (-18+3 x^3+5 x^6\right )}{162 x^9}-\frac {5 \arctan \left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [3]{-1+x^3}}{\sqrt {3}}\right )}{81 \sqrt {3}}+\frac {5}{243} \log \left (1+\sqrt [3]{-1+x^3}\right )-\frac {5}{486} \log \left (1-\sqrt [3]{-1+x^3}+\left (-1+x^3\right )^{2/3}\right ) \]
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Time = 0.04 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.96, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.538, Rules used = {272, 43, 44, 60, 632, 210, 31} \[ \int \frac {\sqrt [3]{-1+x^3}}{x^{10}} \, dx=-\frac {5 \arctan \left (\frac {1-2 \sqrt [3]{x^3-1}}{\sqrt {3}}\right )}{81 \sqrt {3}}+\frac {5 \sqrt [3]{x^3-1}}{162 x^3}+\frac {5}{162} \log \left (\sqrt [3]{x^3-1}+1\right )-\frac {\sqrt [3]{x^3-1}}{9 x^9}+\frac {\sqrt [3]{x^3-1}}{54 x^6}-\frac {5 \log (x)}{162} \]
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Rule 31
Rule 43
Rule 44
Rule 60
Rule 210
Rule 272
Rule 632
Rubi steps \begin{align*} \text {integral}& = \frac {1}{3} \text {Subst}\left (\int \frac {\sqrt [3]{-1+x}}{x^4} \, dx,x,x^3\right ) \\ & = -\frac {\sqrt [3]{-1+x^3}}{9 x^9}+\frac {1}{27} \text {Subst}\left (\int \frac {1}{(-1+x)^{2/3} x^3} \, dx,x,x^3\right ) \\ & = -\frac {\sqrt [3]{-1+x^3}}{9 x^9}+\frac {\sqrt [3]{-1+x^3}}{54 x^6}+\frac {5}{162} \text {Subst}\left (\int \frac {1}{(-1+x)^{2/3} x^2} \, dx,x,x^3\right ) \\ & = -\frac {\sqrt [3]{-1+x^3}}{9 x^9}+\frac {\sqrt [3]{-1+x^3}}{54 x^6}+\frac {5 \sqrt [3]{-1+x^3}}{162 x^3}+\frac {5}{243} \text {Subst}\left (\int \frac {1}{(-1+x)^{2/3} x} \, dx,x,x^3\right ) \\ & = -\frac {\sqrt [3]{-1+x^3}}{9 x^9}+\frac {\sqrt [3]{-1+x^3}}{54 x^6}+\frac {5 \sqrt [3]{-1+x^3}}{162 x^3}-\frac {5 \log (x)}{162}+\frac {5}{162} \text {Subst}\left (\int \frac {1}{1+x} \, dx,x,\sqrt [3]{-1+x^3}\right )+\frac {5}{162} \text {Subst}\left (\int \frac {1}{1-x+x^2} \, dx,x,\sqrt [3]{-1+x^3}\right ) \\ & = -\frac {\sqrt [3]{-1+x^3}}{9 x^9}+\frac {\sqrt [3]{-1+x^3}}{54 x^6}+\frac {5 \sqrt [3]{-1+x^3}}{162 x^3}-\frac {5 \log (x)}{162}+\frac {5}{162} \log \left (1+\sqrt [3]{-1+x^3}\right )-\frac {5}{81} \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,-1+2 \sqrt [3]{-1+x^3}\right ) \\ & = -\frac {\sqrt [3]{-1+x^3}}{9 x^9}+\frac {\sqrt [3]{-1+x^3}}{54 x^6}+\frac {5 \sqrt [3]{-1+x^3}}{162 x^3}-\frac {5 \arctan \left (\frac {1-2 \sqrt [3]{-1+x^3}}{\sqrt {3}}\right )}{81 \sqrt {3}}-\frac {5 \log (x)}{162}+\frac {5}{162} \log \left (1+\sqrt [3]{-1+x^3}\right ) \\ \end{align*}
Time = 0.17 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.93 \[ \int \frac {\sqrt [3]{-1+x^3}}{x^{10}} \, dx=\frac {1}{486} \left (\frac {3 \sqrt [3]{-1+x^3} \left (-18+3 x^3+5 x^6\right )}{x^9}-10 \sqrt {3} \arctan \left (\frac {1-2 \sqrt [3]{-1+x^3}}{\sqrt {3}}\right )+10 \log \left (1+\sqrt [3]{-1+x^3}\right )-5 \log \left (1-\sqrt [3]{-1+x^3}+\left (-1+x^3\right )^{2/3}\right )\right ) \]
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Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 1.80 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.86
method | result | size |
meijerg | \(\frac {\operatorname {signum}\left (x^{3}-1\right )^{\frac {1}{3}} \left (-\frac {10 \Gamma \left (\frac {2}{3}\right ) x^{3} \operatorname {hypergeom}\left (\left [1, 1, \frac {11}{3}\right ], \left [2, 5\right ], x^{3}\right )}{81}-\frac {5 \left (\frac {4}{15}+\frac {\pi \sqrt {3}}{6}-\frac {3 \ln \left (3\right )}{2}+3 \ln \left (x \right )+i \pi \right ) \Gamma \left (\frac {2}{3}\right )}{27}-\frac {\Gamma \left (\frac {2}{3}\right )}{x^{9}}+\frac {\Gamma \left (\frac {2}{3}\right )}{2 x^{6}}+\frac {\Gamma \left (\frac {2}{3}\right )}{3 x^{3}}\right )}{9 \Gamma \left (\frac {2}{3}\right ) {\left (-\operatorname {signum}\left (x^{3}-1\right )\right )}^{\frac {1}{3}}}\) | \(89\) |
risch | \(\frac {5 x^{9}-2 x^{6}-21 x^{3}+18}{162 x^{9} \left (x^{3}-1\right )^{\frac {2}{3}}}+\frac {5 {\left (-\operatorname {signum}\left (x^{3}-1\right )\right )}^{\frac {2}{3}} \left (\frac {2 \Gamma \left (\frac {2}{3}\right ) x^{3} \operatorname {hypergeom}\left (\left [1, 1, \frac {5}{3}\right ], \left [2, 2\right ], x^{3}\right )}{3}+\left (\frac {\pi \sqrt {3}}{6}-\frac {3 \ln \left (3\right )}{2}+3 \ln \left (x \right )+i \pi \right ) \Gamma \left (\frac {2}{3}\right )\right )}{243 \Gamma \left (\frac {2}{3}\right ) \operatorname {signum}\left (x^{3}-1\right )^{\frac {2}{3}}}\) | \(96\) |
pseudoelliptic | \(\frac {\left (15 x^{6}+9 x^{3}-54\right ) \left (x^{3}-1\right )^{\frac {1}{3}}+5 x^{9} \left (2 \sqrt {3}\, \arctan \left (\frac {\left (2 \left (x^{3}-1\right )^{\frac {1}{3}}-1\right ) \sqrt {3}}{3}\right )+2 \ln \left (1+\left (x^{3}-1\right )^{\frac {1}{3}}\right )-\ln \left (1-\left (x^{3}-1\right )^{\frac {1}{3}}+\left (x^{3}-1\right )^{\frac {2}{3}}\right )\right )}{486 {\left (1-\left (x^{3}-1\right )^{\frac {1}{3}}+\left (x^{3}-1\right )^{\frac {2}{3}}\right )}^{3} {\left (1+\left (x^{3}-1\right )^{\frac {1}{3}}\right )}^{3}}\) | \(116\) |
trager | \(\frac {\left (x^{3}-1\right )^{\frac {1}{3}} \left (5 x^{6}+3 x^{3}-18\right )}{162 x^{9}}+\frac {5 \ln \left (-\frac {55312384 \operatorname {RootOf}\left (262144 \textit {\_Z}^{2}+512 \textit {\_Z} +1\right )^{2} x^{3}-1384448 \operatorname {RootOf}\left (262144 \textit {\_Z}^{2}+512 \textit {\_Z} +1\right ) x^{3}+2817024 \operatorname {RootOf}\left (262144 \textit {\_Z}^{2}+512 \textit {\_Z} +1\right ) \left (x^{3}-1\right )^{\frac {2}{3}}+8628 x^{3}-442499072 \operatorname {RootOf}\left (262144 \textit {\_Z}^{2}+512 \textit {\_Z} +1\right )^{2}+10111488 \left (x^{3}-1\right )^{\frac {1}{3}} \operatorname {RootOf}\left (262144 \textit {\_Z}^{2}+512 \textit {\_Z} +1\right )+19749 \left (x^{3}-1\right )^{\frac {2}{3}}+6430208 \operatorname {RootOf}\left (262144 \textit {\_Z}^{2}+512 \textit {\_Z} +1\right )+14247 \left (x^{3}-1\right )^{\frac {1}{3}}-7190}{x^{3}}\right )}{243}+\frac {2560 \operatorname {RootOf}\left (262144 \textit {\_Z}^{2}+512 \textit {\_Z} +1\right ) \ln \left (\frac {432275456 \operatorname {RootOf}\left (262144 \textit {\_Z}^{2}+512 \textit {\_Z} +1\right )^{2} x^{3}+6646272 \operatorname {RootOf}\left (262144 \textit {\_Z}^{2}+512 \textit {\_Z} +1\right ) x^{3}+2817024 \operatorname {RootOf}\left (262144 \textit {\_Z}^{2}+512 \textit {\_Z} +1\right ) \left (x^{3}-1\right )^{\frac {2}{3}}+10066 x^{3}-3458203648 \operatorname {RootOf}\left (262144 \textit {\_Z}^{2}+512 \textit {\_Z} +1\right )^{2}-7294464 \left (x^{3}-1\right )^{\frac {1}{3}} \operatorname {RootOf}\left (262144 \textit {\_Z}^{2}+512 \textit {\_Z} +1\right )-14247 \left (x^{3}-1\right )^{\frac {2}{3}}-16865792 \operatorname {RootOf}\left (262144 \textit {\_Z}^{2}+512 \textit {\_Z} +1\right )-19749 \left (x^{3}-1\right )^{\frac {1}{3}}-18694}{x^{3}}\right )}{243}\) | \(306\) |
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Time = 0.31 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.89 \[ \int \frac {\sqrt [3]{-1+x^3}}{x^{10}} \, dx=\frac {10 \, \sqrt {3} x^{9} \arctan \left (\frac {2}{3} \, \sqrt {3} {\left (x^{3} - 1\right )}^{\frac {1}{3}} - \frac {1}{3} \, \sqrt {3}\right ) - 5 \, x^{9} \log \left ({\left (x^{3} - 1\right )}^{\frac {2}{3}} - {\left (x^{3} - 1\right )}^{\frac {1}{3}} + 1\right ) + 10 \, x^{9} \log \left ({\left (x^{3} - 1\right )}^{\frac {1}{3}} + 1\right ) + 3 \, {\left (5 \, x^{6} + 3 \, x^{3} - 18\right )} {\left (x^{3} - 1\right )}^{\frac {1}{3}}}{486 \, x^{9}} \]
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Result contains complex when optimal does not.
Time = 3.72 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.33 \[ \int \frac {\sqrt [3]{-1+x^3}}{x^{10}} \, dx=- \frac {\Gamma \left (\frac {8}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{3}, \frac {8}{3} \\ \frac {11}{3} \end {matrix}\middle | {\frac {e^{2 i \pi }}{x^{3}}} \right )}}{3 x^{8} \Gamma \left (\frac {11}{3}\right )} \]
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Time = 0.27 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.07 \[ \int \frac {\sqrt [3]{-1+x^3}}{x^{10}} \, dx=\frac {5}{243} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (x^{3} - 1\right )}^{\frac {1}{3}} - 1\right )}\right ) + \frac {5 \, {\left (x^{3} - 1\right )}^{\frac {7}{3}} + 13 \, {\left (x^{3} - 1\right )}^{\frac {4}{3}} - 10 \, {\left (x^{3} - 1\right )}^{\frac {1}{3}}}{162 \, {\left ({\left (x^{3} - 1\right )}^{3} + 3 \, x^{3} + 3 \, {\left (x^{3} - 1\right )}^{2} - 2\right )}} - \frac {5}{486} \, \log \left ({\left (x^{3} - 1\right )}^{\frac {2}{3}} - {\left (x^{3} - 1\right )}^{\frac {1}{3}} + 1\right ) + \frac {5}{243} \, \log \left ({\left (x^{3} - 1\right )}^{\frac {1}{3}} + 1\right ) \]
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Time = 0.28 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.87 \[ \int \frac {\sqrt [3]{-1+x^3}}{x^{10}} \, dx=\frac {5}{243} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (x^{3} - 1\right )}^{\frac {1}{3}} - 1\right )}\right ) + \frac {5 \, {\left (x^{3} - 1\right )}^{\frac {7}{3}} + 13 \, {\left (x^{3} - 1\right )}^{\frac {4}{3}} - 10 \, {\left (x^{3} - 1\right )}^{\frac {1}{3}}}{162 \, x^{9}} - \frac {5}{486} \, \log \left ({\left (x^{3} - 1\right )}^{\frac {2}{3}} - {\left (x^{3} - 1\right )}^{\frac {1}{3}} + 1\right ) + \frac {5}{243} \, \log \left ({\left | {\left (x^{3} - 1\right )}^{\frac {1}{3}} + 1 \right |}\right ) \]
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Time = 6.33 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.19 \[ \int \frac {\sqrt [3]{-1+x^3}}{x^{10}} \, dx=\frac {5\,\ln \left (\frac {25\,{\left (x^3-1\right )}^{1/3}}{6561}+\frac {25}{6561}\right )}{243}+\frac {\frac {13\,{\left (x^3-1\right )}^{4/3}}{162}-\frac {5\,{\left (x^3-1\right )}^{1/3}}{81}+\frac {5\,{\left (x^3-1\right )}^{7/3}}{162}}{3\,{\left (x^3-1\right )}^2+{\left (x^3-1\right )}^3+3\,x^3-2}-\ln \left (\frac {5}{54}-\frac {5\,{\left (x^3-1\right )}^{1/3}}{27}+\frac {\sqrt {3}\,5{}\mathrm {i}}{54}\right )\,\left (\frac {5}{486}+\frac {\sqrt {3}\,5{}\mathrm {i}}{486}\right )+\ln \left (\frac {5\,{\left (x^3-1\right )}^{1/3}}{27}-\frac {5}{54}+\frac {\sqrt {3}\,5{}\mathrm {i}}{54}\right )\,\left (-\frac {5}{486}+\frac {\sqrt {3}\,5{}\mathrm {i}}{486}\right ) \]
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