Integrand size = 18, antiderivative size = 109 \[ \int \frac {\sqrt [3]{-1+x^3} \left (1+x^3\right )}{x^{13}} \, dx=\frac {\sqrt [3]{-1+x^3} \left (-81-99 x^3+30 x^6+50 x^9\right )}{972 x^{12}}-\frac {25 \arctan \left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [3]{-1+x^3}}{\sqrt {3}}\right )}{243 \sqrt {3}}+\frac {25}{729} \log \left (1+\sqrt [3]{-1+x^3}\right )-\frac {25 \log \left (1-\sqrt [3]{-1+x^3}+\left (-1+x^3\right )^{2/3}\right )}{1458} \]
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Time = 0.04 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.06, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.444, Rules used = {457, 79, 43, 44, 60, 632, 210, 31} \[ \int \frac {\sqrt [3]{-1+x^3} \left (1+x^3\right )}{x^{13}} \, dx=-\frac {25 \arctan \left (\frac {1-2 \sqrt [3]{x^3-1}}{\sqrt {3}}\right )}{243 \sqrt {3}}+\frac {25 \sqrt [3]{x^3-1}}{486 x^3}+\frac {25}{486} \log \left (\sqrt [3]{x^3-1}+1\right )+\frac {\left (x^3-1\right )^{4/3}}{12 x^{12}}-\frac {5 \sqrt [3]{x^3-1}}{27 x^9}+\frac {5 \sqrt [3]{x^3-1}}{162 x^6}-\frac {25 \log (x)}{486} \]
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Rule 31
Rule 43
Rule 44
Rule 60
Rule 79
Rule 210
Rule 457
Rule 632
Rubi steps \begin{align*} \text {integral}& = \frac {1}{3} \text {Subst}\left (\int \frac {\sqrt [3]{-1+x} (1+x)}{x^5} \, dx,x,x^3\right ) \\ & = \frac {\left (-1+x^3\right )^{4/3}}{12 x^{12}}+\frac {5}{9} \text {Subst}\left (\int \frac {\sqrt [3]{-1+x}}{x^4} \, dx,x,x^3\right ) \\ & = -\frac {5 \sqrt [3]{-1+x^3}}{27 x^9}+\frac {\left (-1+x^3\right )^{4/3}}{12 x^{12}}+\frac {5}{81} \text {Subst}\left (\int \frac {1}{(-1+x)^{2/3} x^3} \, dx,x,x^3\right ) \\ & = -\frac {5 \sqrt [3]{-1+x^3}}{27 x^9}+\frac {5 \sqrt [3]{-1+x^3}}{162 x^6}+\frac {\left (-1+x^3\right )^{4/3}}{12 x^{12}}+\frac {25}{486} \text {Subst}\left (\int \frac {1}{(-1+x)^{2/3} x^2} \, dx,x,x^3\right ) \\ & = -\frac {5 \sqrt [3]{-1+x^3}}{27 x^9}+\frac {5 \sqrt [3]{-1+x^3}}{162 x^6}+\frac {25 \sqrt [3]{-1+x^3}}{486 x^3}+\frac {\left (-1+x^3\right )^{4/3}}{12 x^{12}}+\frac {25}{729} \text {Subst}\left (\int \frac {1}{(-1+x)^{2/3} x} \, dx,x,x^3\right ) \\ & = -\frac {5 \sqrt [3]{-1+x^3}}{27 x^9}+\frac {5 \sqrt [3]{-1+x^3}}{162 x^6}+\frac {25 \sqrt [3]{-1+x^3}}{486 x^3}+\frac {\left (-1+x^3\right )^{4/3}}{12 x^{12}}-\frac {25 \log (x)}{486}+\frac {25}{486} \text {Subst}\left (\int \frac {1}{1+x} \, dx,x,\sqrt [3]{-1+x^3}\right )+\frac {25}{486} \text {Subst}\left (\int \frac {1}{1-x+x^2} \, dx,x,\sqrt [3]{-1+x^3}\right ) \\ & = -\frac {5 \sqrt [3]{-1+x^3}}{27 x^9}+\frac {5 \sqrt [3]{-1+x^3}}{162 x^6}+\frac {25 \sqrt [3]{-1+x^3}}{486 x^3}+\frac {\left (-1+x^3\right )^{4/3}}{12 x^{12}}-\frac {25 \log (x)}{486}+\frac {25}{486} \log \left (1+\sqrt [3]{-1+x^3}\right )-\frac {25}{243} \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,-1+2 \sqrt [3]{-1+x^3}\right ) \\ & = -\frac {5 \sqrt [3]{-1+x^3}}{27 x^9}+\frac {5 \sqrt [3]{-1+x^3}}{162 x^6}+\frac {25 \sqrt [3]{-1+x^3}}{486 x^3}+\frac {\left (-1+x^3\right )^{4/3}}{12 x^{12}}-\frac {25 \arctan \left (\frac {1-2 \sqrt [3]{-1+x^3}}{\sqrt {3}}\right )}{243 \sqrt {3}}-\frac {25 \log (x)}{486}+\frac {25}{486} \log \left (1+\sqrt [3]{-1+x^3}\right ) \\ \end{align*}
Time = 0.18 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.94 \[ \int \frac {\sqrt [3]{-1+x^3} \left (1+x^3\right )}{x^{13}} \, dx=\frac {\frac {3 \sqrt [3]{-1+x^3} \left (-81-99 x^3+30 x^6+50 x^9\right )}{x^{12}}-100 \sqrt {3} \arctan \left (\frac {1-2 \sqrt [3]{-1+x^3}}{\sqrt {3}}\right )+100 \log \left (1+\sqrt [3]{-1+x^3}\right )-50 \log \left (1-\sqrt [3]{-1+x^3}+\left (-1+x^3\right )^{2/3}\right )}{2916} \]
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Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 1.65 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.93
method | result | size |
risch | \(\frac {50 x^{12}-20 x^{9}-129 x^{6}+18 x^{3}+81}{972 x^{12} \left (x^{3}-1\right )^{\frac {2}{3}}}+\frac {25 {\left (-\operatorname {signum}\left (x^{3}-1\right )\right )}^{\frac {2}{3}} \left (\frac {2 \Gamma \left (\frac {2}{3}\right ) x^{3} \operatorname {hypergeom}\left (\left [1, 1, \frac {5}{3}\right ], \left [2, 2\right ], x^{3}\right )}{3}+\left (\frac {\pi \sqrt {3}}{6}-\frac {3 \ln \left (3\right )}{2}+3 \ln \left (x \right )+i \pi \right ) \Gamma \left (\frac {2}{3}\right )\right )}{729 \Gamma \left (\frac {2}{3}\right ) \operatorname {signum}\left (x^{3}-1\right )^{\frac {2}{3}}}\) | \(101\) |
pseudoelliptic | \(\frac {\left (150 x^{9}+90 x^{6}-297 x^{3}-243\right ) \left (x^{3}-1\right )^{\frac {1}{3}}-50 x^{12} \left (-2 \sqrt {3}\, \arctan \left (\frac {\left (2 \left (x^{3}-1\right )^{\frac {1}{3}}-1\right ) \sqrt {3}}{3}\right )+\ln \left (1-\left (x^{3}-1\right )^{\frac {1}{3}}+\left (x^{3}-1\right )^{\frac {2}{3}}\right )-2 \ln \left (1+\left (x^{3}-1\right )^{\frac {1}{3}}\right )\right )}{2916 {\left (1-\left (x^{3}-1\right )^{\frac {1}{3}}+\left (x^{3}-1\right )^{\frac {2}{3}}\right )}^{4} {\left (1+\left (x^{3}-1\right )^{\frac {1}{3}}\right )}^{4}}\) | \(119\) |
meijerg | \(\frac {\operatorname {signum}\left (x^{3}-1\right )^{\frac {1}{3}} \left (-\frac {10 \Gamma \left (\frac {2}{3}\right ) x^{3} \operatorname {hypergeom}\left (\left [1, 1, \frac {11}{3}\right ], \left [2, 5\right ], x^{3}\right )}{81}-\frac {5 \left (\frac {4}{15}+\frac {\pi \sqrt {3}}{6}-\frac {3 \ln \left (3\right )}{2}+3 \ln \left (x \right )+i \pi \right ) \Gamma \left (\frac {2}{3}\right )}{27}-\frac {\Gamma \left (\frac {2}{3}\right )}{x^{9}}+\frac {\Gamma \left (\frac {2}{3}\right )}{2 x^{6}}+\frac {\Gamma \left (\frac {2}{3}\right )}{3 x^{3}}\right )}{9 \Gamma \left (\frac {2}{3}\right ) {\left (-\operatorname {signum}\left (x^{3}-1\right )\right )}^{\frac {1}{3}}}-\frac {\operatorname {signum}\left (x^{3}-1\right )^{\frac {1}{3}} \left (\frac {22 \Gamma \left (\frac {2}{3}\right ) x^{3} \operatorname {hypergeom}\left (\left [1, 1, \frac {14}{3}\right ], \left [2, 6\right ], x^{3}\right )}{243}+\frac {10 \left (\frac {47}{120}+\frac {\pi \sqrt {3}}{6}-\frac {3 \ln \left (3\right )}{2}+3 \ln \left (x \right )+i \pi \right ) \Gamma \left (\frac {2}{3}\right )}{81}+\frac {3 \Gamma \left (\frac {2}{3}\right )}{4 x^{12}}-\frac {\Gamma \left (\frac {2}{3}\right )}{3 x^{9}}-\frac {\Gamma \left (\frac {2}{3}\right )}{6 x^{6}}-\frac {5 \Gamma \left (\frac {2}{3}\right )}{27 x^{3}}\right )}{9 \Gamma \left (\frac {2}{3}\right ) {\left (-\operatorname {signum}\left (x^{3}-1\right )\right )}^{\frac {1}{3}}}\) | \(185\) |
trager | \(\frac {\left (x^{3}-1\right )^{\frac {1}{3}} \left (50 x^{9}+30 x^{6}-99 x^{3}-81\right )}{972 x^{12}}-\frac {25 \ln \left (-\frac {24125636608 \operatorname {RootOf}\left (16777216 \textit {\_Z}^{2}+4096 \textit {\_Z} +1\right )^{2} x^{3}+40366080 \operatorname {RootOf}\left (16777216 \textit {\_Z}^{2}+4096 \textit {\_Z} +1\right ) x^{3}-22536192 \operatorname {RootOf}\left (16777216 \textit {\_Z}^{2}+4096 \textit {\_Z} +1\right ) \left (x^{3}-1\right )^{\frac {2}{3}}-1477 x^{3}-193005092864 \operatorname {RootOf}\left (16777216 \textit {\_Z}^{2}+4096 \textit {\_Z} +1\right )^{2}+22536192 \left (x^{3}-1\right )^{\frac {1}{3}} \operatorname {RootOf}\left (16777216 \textit {\_Z}^{2}+4096 \textit {\_Z} +1\right )+14247 \left (x^{3}-1\right )^{\frac {2}{3}}-69656576 \operatorname {RootOf}\left (16777216 \textit {\_Z}^{2}+4096 \textit {\_Z} +1\right )-14247 \left (x^{3}-1\right )^{\frac {1}{3}}+2743}{x^{3}}\right )}{729}-\frac {102400 \ln \left (-\frac {24125636608 \operatorname {RootOf}\left (16777216 \textit {\_Z}^{2}+4096 \textit {\_Z} +1\right )^{2} x^{3}+40366080 \operatorname {RootOf}\left (16777216 \textit {\_Z}^{2}+4096 \textit {\_Z} +1\right ) x^{3}-22536192 \operatorname {RootOf}\left (16777216 \textit {\_Z}^{2}+4096 \textit {\_Z} +1\right ) \left (x^{3}-1\right )^{\frac {2}{3}}-1477 x^{3}-193005092864 \operatorname {RootOf}\left (16777216 \textit {\_Z}^{2}+4096 \textit {\_Z} +1\right )^{2}+22536192 \left (x^{3}-1\right )^{\frac {1}{3}} \operatorname {RootOf}\left (16777216 \textit {\_Z}^{2}+4096 \textit {\_Z} +1\right )+14247 \left (x^{3}-1\right )^{\frac {2}{3}}-69656576 \operatorname {RootOf}\left (16777216 \textit {\_Z}^{2}+4096 \textit {\_Z} +1\right )-14247 \left (x^{3}-1\right )^{\frac {1}{3}}+2743}{x^{3}}\right ) \operatorname {RootOf}\left (16777216 \textit {\_Z}^{2}+4096 \textit {\_Z} +1\right )}{729}+\frac {102400 \operatorname {RootOf}\left (16777216 \textit {\_Z}^{2}+4096 \textit {\_Z} +1\right ) \ln \left (-\frac {24125636608 \operatorname {RootOf}\left (16777216 \textit {\_Z}^{2}+4096 \textit {\_Z} +1\right )^{2} x^{3}-28585984 \operatorname {RootOf}\left (16777216 \textit {\_Z}^{2}+4096 \textit {\_Z} +1\right ) x^{3}+22536192 \operatorname {RootOf}\left (16777216 \textit {\_Z}^{2}+4096 \textit {\_Z} +1\right ) \left (x^{3}-1\right )^{\frac {2}{3}}-9894 x^{3}-193005092864 \operatorname {RootOf}\left (16777216 \textit {\_Z}^{2}+4096 \textit {\_Z} +1\right )^{2}-22536192 \left (x^{3}-1\right )^{\frac {1}{3}} \operatorname {RootOf}\left (16777216 \textit {\_Z}^{2}+4096 \textit {\_Z} +1\right )+19749 \left (x^{3}-1\right )^{\frac {2}{3}}-24584192 \operatorname {RootOf}\left (16777216 \textit {\_Z}^{2}+4096 \textit {\_Z} +1\right )-19749 \left (x^{3}-1\right )^{\frac {1}{3}}+8245}{x^{3}}\right )}{729}\) | \(458\) |
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Time = 0.26 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.90 \[ \int \frac {\sqrt [3]{-1+x^3} \left (1+x^3\right )}{x^{13}} \, dx=\frac {100 \, \sqrt {3} x^{12} \arctan \left (\frac {2}{3} \, \sqrt {3} {\left (x^{3} - 1\right )}^{\frac {1}{3}} - \frac {1}{3} \, \sqrt {3}\right ) - 50 \, x^{12} \log \left ({\left (x^{3} - 1\right )}^{\frac {2}{3}} - {\left (x^{3} - 1\right )}^{\frac {1}{3}} + 1\right ) + 100 \, x^{12} \log \left ({\left (x^{3} - 1\right )}^{\frac {1}{3}} + 1\right ) + 3 \, {\left (50 \, x^{9} + 30 \, x^{6} - 99 \, x^{3} - 81\right )} {\left (x^{3} - 1\right )}^{\frac {1}{3}}}{2916 \, x^{12}} \]
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Timed out. \[ \int \frac {\sqrt [3]{-1+x^3} \left (1+x^3\right )}{x^{13}} \, dx=\text {Timed out} \]
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Leaf count of result is larger than twice the leaf count of optimal. 184 vs. \(2 (87) = 174\).
Time = 0.27 (sec) , antiderivative size = 184, normalized size of antiderivative = 1.69 \[ \int \frac {\sqrt [3]{-1+x^3} \left (1+x^3\right )}{x^{13}} \, dx=\frac {25}{729} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (x^{3} - 1\right )}^{\frac {1}{3}} - 1\right )}\right ) + \frac {20 \, {\left (x^{3} - 1\right )}^{\frac {10}{3}} + 72 \, {\left (x^{3} - 1\right )}^{\frac {7}{3}} + 93 \, {\left (x^{3} - 1\right )}^{\frac {4}{3}} - 40 \, {\left (x^{3} - 1\right )}^{\frac {1}{3}}}{972 \, {\left ({\left (x^{3} - 1\right )}^{4} + 4 \, {\left (x^{3} - 1\right )}^{3} + 4 \, x^{3} + 6 \, {\left (x^{3} - 1\right )}^{2} - 3\right )}} + \frac {5 \, {\left (x^{3} - 1\right )}^{\frac {7}{3}} + 13 \, {\left (x^{3} - 1\right )}^{\frac {4}{3}} - 10 \, {\left (x^{3} - 1\right )}^{\frac {1}{3}}}{162 \, {\left ({\left (x^{3} - 1\right )}^{3} + 3 \, x^{3} + 3 \, {\left (x^{3} - 1\right )}^{2} - 2\right )}} - \frac {25}{1458} \, \log \left ({\left (x^{3} - 1\right )}^{\frac {2}{3}} - {\left (x^{3} - 1\right )}^{\frac {1}{3}} + 1\right ) + \frac {25}{729} \, \log \left ({\left (x^{3} - 1\right )}^{\frac {1}{3}} + 1\right ) \]
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Time = 0.28 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.91 \[ \int \frac {\sqrt [3]{-1+x^3} \left (1+x^3\right )}{x^{13}} \, dx=\frac {25}{729} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (x^{3} - 1\right )}^{\frac {1}{3}} - 1\right )}\right ) + \frac {50 \, {\left (x^{3} - 1\right )}^{\frac {10}{3}} + 180 \, {\left (x^{3} - 1\right )}^{\frac {7}{3}} + 111 \, {\left (x^{3} - 1\right )}^{\frac {4}{3}} - 100 \, {\left (x^{3} - 1\right )}^{\frac {1}{3}}}{972 \, x^{12}} - \frac {25}{1458} \, \log \left ({\left (x^{3} - 1\right )}^{\frac {2}{3}} - {\left (x^{3} - 1\right )}^{\frac {1}{3}} + 1\right ) + \frac {25}{729} \, \log \left ({\left | {\left (x^{3} - 1\right )}^{\frac {1}{3}} + 1 \right |}\right ) \]
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Time = 6.23 (sec) , antiderivative size = 265, normalized size of antiderivative = 2.43 \[ \int \frac {\sqrt [3]{-1+x^3} \left (1+x^3\right )}{x^{13}} \, dx=\frac {5\,\ln \left (\frac {25\,{\left (x^3-1\right )}^{1/3}}{6561}+\frac {25}{6561}\right )}{243}+\frac {10\,\ln \left (\frac {100\,{\left (x^3-1\right )}^{1/3}}{59049}+\frac {100}{59049}\right )}{729}+\frac {\frac {31\,{\left (x^3-1\right )}^{4/3}}{324}-\frac {10\,{\left (x^3-1\right )}^{1/3}}{243}+\frac {2\,{\left (x^3-1\right )}^{7/3}}{27}+\frac {5\,{\left (x^3-1\right )}^{10/3}}{243}}{6\,{\left (x^3-1\right )}^2+4\,{\left (x^3-1\right )}^3+{\left (x^3-1\right )}^4+4\,x^3-3}+\frac {\frac {13\,{\left (x^3-1\right )}^{4/3}}{162}-\frac {5\,{\left (x^3-1\right )}^{1/3}}{81}+\frac {5\,{\left (x^3-1\right )}^{7/3}}{162}}{3\,{\left (x^3-1\right )}^2+{\left (x^3-1\right )}^3+3\,x^3-2}-\ln \left (\frac {5}{54}-\frac {5\,{\left (x^3-1\right )}^{1/3}}{27}+\frac {\sqrt {3}\,5{}\mathrm {i}}{54}\right )\,\left (\frac {5}{486}+\frac {\sqrt {3}\,5{}\mathrm {i}}{486}\right )+\ln \left (\frac {5\,{\left (x^3-1\right )}^{1/3}}{27}-\frac {5}{54}+\frac {\sqrt {3}\,5{}\mathrm {i}}{54}\right )\,\left (-\frac {5}{486}+\frac {\sqrt {3}\,5{}\mathrm {i}}{486}\right )-\ln \left (\frac {5}{81}-\frac {10\,{\left (x^3-1\right )}^{1/3}}{81}+\frac {\sqrt {3}\,5{}\mathrm {i}}{81}\right )\,\left (\frac {5}{729}+\frac {\sqrt {3}\,5{}\mathrm {i}}{729}\right )+\ln \left (\frac {10\,{\left (x^3-1\right )}^{1/3}}{81}-\frac {5}{81}+\frac {\sqrt {3}\,5{}\mathrm {i}}{81}\right )\,\left (-\frac {5}{729}+\frac {\sqrt {3}\,5{}\mathrm {i}}{729}\right ) \]
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