Integrand size = 13, antiderivative size = 104 \[ \int \frac {1}{x^{19} \sqrt [3]{-1+x^6}} \, dx=\frac {\left (-1+x^6\right )^{2/3} \left (18+21 x^6+28 x^{12}\right )}{324 x^{18}}-\frac {7 \arctan \left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [3]{-1+x^6}}{\sqrt {3}}\right )}{81 \sqrt {3}}-\frac {7}{243} \log \left (1+\sqrt [3]{-1+x^6}\right )+\frac {7}{486} \log \left (1-\sqrt [3]{-1+x^6}+\left (-1+x^6\right )^{2/3}\right ) \]
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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 = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.462, Rules used = {272, 44, 58, 632, 210, 31} \[ \int \frac {1}{x^{19} \sqrt [3]{-1+x^6}} \, dx=-\frac {7 \arctan \left (\frac {1-2 \sqrt [3]{x^6-1}}{\sqrt {3}}\right )}{81 \sqrt {3}}+\frac {7 \left (x^6-1\right )^{2/3}}{81 x^6}-\frac {7}{162} \log \left (\sqrt [3]{x^6-1}+1\right )+\frac {\left (x^6-1\right )^{2/3}}{18 x^{18}}+\frac {7 \left (x^6-1\right )^{2/3}}{108 x^{12}}+\frac {7 \log (x)}{81} \]
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Rule 31
Rule 44
Rule 58
Rule 210
Rule 272
Rule 632
Rubi steps \begin{align*} \text {integral}& = \frac {1}{6} \text {Subst}\left (\int \frac {1}{\sqrt [3]{-1+x} x^4} \, dx,x,x^6\right ) \\ & = \frac {\left (-1+x^6\right )^{2/3}}{18 x^{18}}+\frac {7}{54} \text {Subst}\left (\int \frac {1}{\sqrt [3]{-1+x} x^3} \, dx,x,x^6\right ) \\ & = \frac {\left (-1+x^6\right )^{2/3}}{18 x^{18}}+\frac {7 \left (-1+x^6\right )^{2/3}}{108 x^{12}}+\frac {7}{81} \text {Subst}\left (\int \frac {1}{\sqrt [3]{-1+x} x^2} \, dx,x,x^6\right ) \\ & = \frac {\left (-1+x^6\right )^{2/3}}{18 x^{18}}+\frac {7 \left (-1+x^6\right )^{2/3}}{108 x^{12}}+\frac {7 \left (-1+x^6\right )^{2/3}}{81 x^6}+\frac {7}{243} \text {Subst}\left (\int \frac {1}{\sqrt [3]{-1+x} x} \, dx,x,x^6\right ) \\ & = \frac {\left (-1+x^6\right )^{2/3}}{18 x^{18}}+\frac {7 \left (-1+x^6\right )^{2/3}}{108 x^{12}}+\frac {7 \left (-1+x^6\right )^{2/3}}{81 x^6}+\frac {7 \log (x)}{81}-\frac {7}{162} \text {Subst}\left (\int \frac {1}{1+x} \, dx,x,\sqrt [3]{-1+x^6}\right )+\frac {7}{162} \text {Subst}\left (\int \frac {1}{1-x+x^2} \, dx,x,\sqrt [3]{-1+x^6}\right ) \\ & = \frac {\left (-1+x^6\right )^{2/3}}{18 x^{18}}+\frac {7 \left (-1+x^6\right )^{2/3}}{108 x^{12}}+\frac {7 \left (-1+x^6\right )^{2/3}}{81 x^6}+\frac {7 \log (x)}{81}-\frac {7}{162} \log \left (1+\sqrt [3]{-1+x^6}\right )-\frac {7}{81} \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,-1+2 \sqrt [3]{-1+x^6}\right ) \\ & = \frac {\left (-1+x^6\right )^{2/3}}{18 x^{18}}+\frac {7 \left (-1+x^6\right )^{2/3}}{108 x^{12}}+\frac {7 \left (-1+x^6\right )^{2/3}}{81 x^6}-\frac {7 \arctan \left (\frac {1-2 \sqrt [3]{-1+x^6}}{\sqrt {3}}\right )}{81 \sqrt {3}}+\frac {7 \log (x)}{81}-\frac {7}{162} \log \left (1+\sqrt [3]{-1+x^6}\right ) \\ \end{align*}
Time = 0.17 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.93 \[ \int \frac {1}{x^{19} \sqrt [3]{-1+x^6}} \, dx=\frac {1}{972} \left (\frac {3 \left (-1+x^6\right )^{2/3} \left (18+21 x^6+28 x^{12}\right )}{x^{18}}-28 \sqrt {3} \arctan \left (\frac {1-2 \sqrt [3]{-1+x^6}}{\sqrt {3}}\right )-28 \log \left (1+\sqrt [3]{-1+x^6}\right )+14 \log \left (1-\sqrt [3]{-1+x^6}+\left (-1+x^6\right )^{2/3}\right )\right ) \]
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Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 4.80 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.09
method | result | size |
risch | \(\frac {28 x^{18}-7 x^{12}-3 x^{6}-18}{324 x^{18} \left (x^{6}-1\right )^{\frac {1}{3}}}+\frac {7 \sqrt {3}\, \Gamma \left (\frac {2}{3}\right ) {\left (-\operatorname {signum}\left (x^{6}-1\right )\right )}^{\frac {1}{3}} \left (\frac {2 \pi \sqrt {3}\, x^{6} \operatorname {hypergeom}\left (\left [1, 1, \frac {4}{3}\right ], \left [2, 2\right ], x^{6}\right )}{9 \Gamma \left (\frac {2}{3}\right )}+\frac {2 \left (-\frac {\pi \sqrt {3}}{6}-\frac {3 \ln \left (3\right )}{2}+6 \ln \left (x \right )+i \pi \right ) \pi \sqrt {3}}{3 \Gamma \left (\frac {2}{3}\right )}\right )}{486 \pi \operatorname {signum}\left (x^{6}-1\right )^{\frac {1}{3}}}\) | \(113\) |
pseudoelliptic | \(\frac {-14 \ln \left (1-\left (x^{6}-1\right )^{\frac {1}{3}}+\left (x^{6}-1\right )^{\frac {2}{3}}\right ) x^{18}-28 \arctan \left (\frac {\left (2 \left (x^{6}-1\right )^{\frac {1}{3}}-1\right ) \sqrt {3}}{3}\right ) \sqrt {3}\, x^{18}+28 \ln \left (1+\left (x^{6}-1\right )^{\frac {1}{3}}\right ) x^{18}+\left (-84 x^{12}-63 x^{6}-54\right ) \left (x^{6}-1\right )^{\frac {2}{3}}}{972 {\left (-1+\left (x^{6}-1\right )^{\frac {1}{3}}-\left (x^{6}-1\right )^{\frac {2}{3}}\right )}^{3} {\left (1+\left (x^{6}-1\right )^{\frac {1}{3}}\right )}^{3}}\) | \(119\) |
meijerg | \(-\frac {\sqrt {3}\, \Gamma \left (\frac {2}{3}\right ) {\left (-\operatorname {signum}\left (x^{6}-1\right )\right )}^{\frac {1}{3}} \left (-\frac {70 \pi \sqrt {3}\, x^{6} \operatorname {hypergeom}\left (\left [1, 1, \frac {13}{3}\right ], \left [2, 5\right ], x^{6}\right )}{729 \Gamma \left (\frac {2}{3}\right )}-\frac {28 \left (\frac {197}{84}-\frac {\pi \sqrt {3}}{6}-\frac {3 \ln \left (3\right )}{2}+6 \ln \left (x \right )+i \pi \right ) \pi \sqrt {3}}{243 \Gamma \left (\frac {2}{3}\right )}+\frac {2 \pi \sqrt {3}}{9 \Gamma \left (\frac {2}{3}\right ) x^{18}}+\frac {\pi \sqrt {3}}{9 \Gamma \left (\frac {2}{3}\right ) x^{12}}+\frac {4 \pi \sqrt {3}}{27 \Gamma \left (\frac {2}{3}\right ) x^{6}}\right )}{12 \pi \operatorname {signum}\left (x^{6}-1\right )^{\frac {1}{3}}}\) | \(123\) |
trager | \(\text {Expression too large to display}\) | \(453\) |
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Time = 0.28 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.89 \[ \int \frac {1}{x^{19} \sqrt [3]{-1+x^6}} \, dx=\frac {28 \, \sqrt {3} x^{18} \arctan \left (\frac {2}{3} \, \sqrt {3} {\left (x^{6} - 1\right )}^{\frac {1}{3}} - \frac {1}{3} \, \sqrt {3}\right ) + 14 \, x^{18} \log \left ({\left (x^{6} - 1\right )}^{\frac {2}{3}} - {\left (x^{6} - 1\right )}^{\frac {1}{3}} + 1\right ) - 28 \, x^{18} \log \left ({\left (x^{6} - 1\right )}^{\frac {1}{3}} + 1\right ) + 3 \, {\left (28 \, x^{12} + 21 \, x^{6} + 18\right )} {\left (x^{6} - 1\right )}^{\frac {2}{3}}}{972 \, x^{18}} \]
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Result contains complex when optimal does not.
Time = 7.12 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.31 \[ \int \frac {1}{x^{19} \sqrt [3]{-1+x^6}} \, dx=- \frac {\Gamma \left (\frac {10}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{3}, \frac {10}{3} \\ \frac {13}{3} \end {matrix}\middle | {\frac {e^{2 i \pi }}{x^{6}}} \right )}}{6 x^{20} \Gamma \left (\frac {13}{3}\right )} \]
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Time = 0.27 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.07 \[ \int \frac {1}{x^{19} \sqrt [3]{-1+x^6}} \, dx=\frac {7}{243} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (x^{6} - 1\right )}^{\frac {1}{3}} - 1\right )}\right ) + \frac {28 \, {\left (x^{6} - 1\right )}^{\frac {8}{3}} + 77 \, {\left (x^{6} - 1\right )}^{\frac {5}{3}} + 67 \, {\left (x^{6} - 1\right )}^{\frac {2}{3}}}{324 \, {\left (3 \, x^{6} + {\left (x^{6} - 1\right )}^{3} + 3 \, {\left (x^{6} - 1\right )}^{2} - 2\right )}} + \frac {7}{486} \, \log \left ({\left (x^{6} - 1\right )}^{\frac {2}{3}} - {\left (x^{6} - 1\right )}^{\frac {1}{3}} + 1\right ) - \frac {7}{243} \, \log \left ({\left (x^{6} - 1\right )}^{\frac {1}{3}} + 1\right ) \]
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Time = 0.28 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.87 \[ \int \frac {1}{x^{19} \sqrt [3]{-1+x^6}} \, dx=\frac {7}{243} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (x^{6} - 1\right )}^{\frac {1}{3}} - 1\right )}\right ) + \frac {28 \, {\left (x^{6} - 1\right )}^{\frac {8}{3}} + 77 \, {\left (x^{6} - 1\right )}^{\frac {5}{3}} + 67 \, {\left (x^{6} - 1\right )}^{\frac {2}{3}}}{324 \, x^{18}} + \frac {7}{486} \, \log \left ({\left (x^{6} - 1\right )}^{\frac {2}{3}} - {\left (x^{6} - 1\right )}^{\frac {1}{3}} + 1\right ) - \frac {7}{243} \, \log \left ({\left | {\left (x^{6} - 1\right )}^{\frac {1}{3}} + 1 \right |}\right ) \]
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Time = 6.46 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.29 \[ \int \frac {1}{x^{19} \sqrt [3]{-1+x^6}} \, dx=\frac {\frac {67\,{\left (x^6-1\right )}^{2/3}}{324}+\frac {77\,{\left (x^6-1\right )}^{5/3}}{324}+\frac {7\,{\left (x^6-1\right )}^{8/3}}{81}}{3\,{\left (x^6-1\right )}^2+{\left (x^6-1\right )}^3+3\,x^6-2}-\ln \left (9\,{\left (-\frac {7}{486}+\frac {\sqrt {3}\,7{}\mathrm {i}}{486}\right )}^2+\frac {49\,{\left (x^6-1\right )}^{1/3}}{6561}\right )\,\left (-\frac {7}{486}+\frac {\sqrt {3}\,7{}\mathrm {i}}{486}\right )+\ln \left (9\,{\left (\frac {7}{486}+\frac {\sqrt {3}\,7{}\mathrm {i}}{486}\right )}^2+\frac {49\,{\left (x^6-1\right )}^{1/3}}{6561}\right )\,\left (\frac {7}{486}+\frac {\sqrt {3}\,7{}\mathrm {i}}{486}\right )-\frac {7\,\ln \left (\frac {49\,{\left (x^6-1\right )}^{1/3}}{6561}+\frac {49}{6561}\right )}{243} \]
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