Integrand size = 13, antiderivative size = 97 \[ \int \frac {1}{x^{13} \sqrt [3]{1+x^6}} \, dx=\frac {\left (1+x^6\right )^{2/3} \left (-3+4 x^6\right )}{36 x^{12}}+\frac {\arctan \left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{1+x^6}}{\sqrt {3}}\right )}{9 \sqrt {3}}+\frac {1}{27} \log \left (-1+\sqrt [3]{1+x^6}\right )-\frac {1}{54} \log \left (1+\sqrt [3]{1+x^6}+\left (1+x^6\right )^{2/3}\right ) \]
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Time = 0.03 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.89, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.462, Rules used = {272, 44, 57, 632, 210, 31} \[ \int \frac {1}{x^{13} \sqrt [3]{1+x^6}} \, dx=\frac {\arctan \left (\frac {2 \sqrt [3]{x^6+1}+1}{\sqrt {3}}\right )}{9 \sqrt {3}}+\frac {\left (x^6+1\right )^{2/3}}{9 x^6}+\frac {1}{18} \log \left (1-\sqrt [3]{x^6+1}\right )-\frac {\left (x^6+1\right )^{2/3}}{12 x^{12}}-\frac {\log (x)}{9} \]
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Rule 31
Rule 44
Rule 57
Rule 210
Rule 272
Rule 632
Rubi steps \begin{align*} \text {integral}& = \frac {1}{6} \text {Subst}\left (\int \frac {1}{x^3 \sqrt [3]{1+x}} \, dx,x,x^6\right ) \\ & = -\frac {\left (1+x^6\right )^{2/3}}{12 x^{12}}-\frac {1}{9} \text {Subst}\left (\int \frac {1}{x^2 \sqrt [3]{1+x}} \, dx,x,x^6\right ) \\ & = -\frac {\left (1+x^6\right )^{2/3}}{12 x^{12}}+\frac {\left (1+x^6\right )^{2/3}}{9 x^6}+\frac {1}{27} \text {Subst}\left (\int \frac {1}{x \sqrt [3]{1+x}} \, dx,x,x^6\right ) \\ & = -\frac {\left (1+x^6\right )^{2/3}}{12 x^{12}}+\frac {\left (1+x^6\right )^{2/3}}{9 x^6}-\frac {\log (x)}{9}-\frac {1}{18} \text {Subst}\left (\int \frac {1}{1-x} \, dx,x,\sqrt [3]{1+x^6}\right )+\frac {1}{18} \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}}{12 x^{12}}+\frac {\left (1+x^6\right )^{2/3}}{9 x^6}-\frac {\log (x)}{9}+\frac {1}{18} \log \left (1-\sqrt [3]{1+x^6}\right )-\frac {1}{9} \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}}{12 x^{12}}+\frac {\left (1+x^6\right )^{2/3}}{9 x^6}+\frac {\arctan \left (\frac {1+2 \sqrt [3]{1+x^6}}{\sqrt {3}}\right )}{9 \sqrt {3}}-\frac {\log (x)}{9}+\frac {1}{18} \log \left (1-\sqrt [3]{1+x^6}\right ) \\ \end{align*}
Time = 0.12 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.93 \[ \int \frac {1}{x^{13} \sqrt [3]{1+x^6}} \, dx=\frac {1}{108} \left (\frac {3 \left (1+x^6\right )^{2/3} \left (-3+4 x^6\right )}{x^{12}}+4 \sqrt {3} \arctan \left (\frac {1+2 \sqrt [3]{1+x^6}}{\sqrt {3}}\right )+4 \log \left (-1+\sqrt [3]{1+x^6}\right )-2 \log \left (1+\sqrt [3]{1+x^6}+\left (1+x^6\right )^{2/3}\right )\right ) \]
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Result contains higher order function than in optimal. Order 5 vs. order 3.
Time = 6.30 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.89
method | result | size |
risch | \(\frac {4 x^{12}+x^{6}-3}{36 x^{12} \left (x^{6}+1\right )^{\frac {1}{3}}}+\frac {\sqrt {3}\, \Gamma \left (\frac {2}{3}\right ) \left (-\frac {2 \pi \sqrt {3}\, x^{6} \operatorname {hypergeom}\left (\left [1, 1, \frac {4}{3}\right ], \left [2, 2\right ], -x^{6}\right )}{9 \Gamma \left (\frac {2}{3}\right )}+\frac {2 \left (-\frac {\pi \sqrt {3}}{6}-\frac {3 \ln \left (3\right )}{2}+6 \ln \left (x \right )\right ) \pi \sqrt {3}}{3 \Gamma \left (\frac {2}{3}\right )}\right )}{54 \pi }\) | \(86\) |
meijerg | \(\frac {\sqrt {3}\, \Gamma \left (\frac {2}{3}\right ) \left (-\frac {28 \pi \sqrt {3}\, x^{6} \operatorname {hypergeom}\left (\left [1, 1, \frac {10}{3}\right ], \left [2, 4\right ], -x^{6}\right )}{243 \Gamma \left (\frac {2}{3}\right )}+\frac {4 \left (\frac {9}{4}-\frac {\pi \sqrt {3}}{6}-\frac {3 \ln \left (3\right )}{2}+6 \ln \left (x \right )\right ) \pi \sqrt {3}}{27 \Gamma \left (\frac {2}{3}\right )}-\frac {\pi \sqrt {3}}{3 \Gamma \left (\frac {2}{3}\right ) x^{12}}+\frac {2 \pi \sqrt {3}}{9 \Gamma \left (\frac {2}{3}\right ) x^{6}}\right )}{12 \pi }\) | \(90\) |
pseudoelliptic | \(\frac {4 \sqrt {3}\, \arctan \left (\frac {\left (2 \left (x^{6}+1\right )^{\frac {1}{3}}+1\right ) \sqrt {3}}{3}\right ) x^{12}+4 \ln \left (-1+\left (x^{6}+1\right )^{\frac {1}{3}}\right ) x^{12}-2 \ln \left (1+\left (x^{6}+1\right )^{\frac {1}{3}}+\left (x^{6}+1\right )^{\frac {2}{3}}\right ) x^{12}+12 \left (x^{6}+1\right )^{\frac {2}{3}} x^{6}-9 \left (x^{6}+1\right )^{\frac {2}{3}}}{108 {\left (-1+\left (x^{6}+1\right )^{\frac {1}{3}}\right )}^{2} {\left (1+\left (x^{6}+1\right )^{\frac {1}{3}}+\left (x^{6}+1\right )^{\frac {2}{3}}\right )}^{2}}\) | \(116\) |
trager | \(\frac {\left (x^{6}+1\right )^{\frac {2}{3}} \left (4 x^{6}-3\right )}{36 x^{12}}+\frac {2 \operatorname {RootOf}\left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) \ln \left (-\frac {1415351072 \operatorname {RootOf}\left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )^{2} x^{6}-1046043282 \operatorname {RootOf}\left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) x^{6}-369307790 x^{6}-3200009934 \left (x^{6}+1\right )^{\frac {2}{3}} \operatorname {RootOf}\left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )-1415351072 \operatorname {RootOf}\left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )^{2}-3200009934 \left (x^{6}+1\right )^{\frac {1}{3}} \operatorname {RootOf}\left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )-569431707 \left (x^{6}+1\right )^{\frac {2}{3}}-3907685470 \operatorname {RootOf}\left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )-569431707 \left (x^{6}+1\right )^{\frac {1}{3}}-923269475}{x^{6}}\right )}{27}-\frac {\ln \left (-\frac {1415351072 \operatorname {RootOf}\left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )^{2} x^{6}+2461394354 \operatorname {RootOf}\left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) x^{6}+507551619 x^{6}+3200009934 \left (x^{6}+1\right )^{\frac {2}{3}} \operatorname {RootOf}\left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )-1415351072 \operatorname {RootOf}\left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )^{2}+3200009934 \left (x^{6}+1\right )^{\frac {1}{3}} \operatorname {RootOf}\left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )+1030573260 \left (x^{6}+1\right )^{\frac {2}{3}}+2492334398 \operatorname {RootOf}\left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )+1030573260 \left (x^{6}+1\right )^{\frac {1}{3}}+676735492}{x^{6}}\right )}{27}-\frac {2 \ln \left (-\frac {1415351072 \operatorname {RootOf}\left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )^{2} x^{6}+2461394354 \operatorname {RootOf}\left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) x^{6}+507551619 x^{6}+3200009934 \left (x^{6}+1\right )^{\frac {2}{3}} \operatorname {RootOf}\left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )-1415351072 \operatorname {RootOf}\left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )^{2}+3200009934 \left (x^{6}+1\right )^{\frac {1}{3}} \operatorname {RootOf}\left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )+1030573260 \left (x^{6}+1\right )^{\frac {2}{3}}+2492334398 \operatorname {RootOf}\left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )+1030573260 \left (x^{6}+1\right )^{\frac {1}{3}}+676735492}{x^{6}}\right ) \operatorname {RootOf}\left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )}{27}\) | \(448\) |
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Time = 0.27 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.89 \[ \int \frac {1}{x^{13} \sqrt [3]{1+x^6}} \, dx=\frac {4 \, \sqrt {3} x^{12} \arctan \left (\frac {2}{3} \, \sqrt {3} {\left (x^{6} + 1\right )}^{\frac {1}{3}} + \frac {1}{3} \, \sqrt {3}\right ) - 2 \, x^{12} \log \left ({\left (x^{6} + 1\right )}^{\frac {2}{3}} + {\left (x^{6} + 1\right )}^{\frac {1}{3}} + 1\right ) + 4 \, x^{12} \log \left ({\left (x^{6} + 1\right )}^{\frac {1}{3}} - 1\right ) + 3 \, {\left (4 \, x^{6} - 3\right )} {\left (x^{6} + 1\right )}^{\frac {2}{3}}}{108 \, x^{12}} \]
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Result contains complex when optimal does not.
Time = 1.77 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.32 \[ \int \frac {1}{x^{13} \sqrt [3]{1+x^6}} \, dx=- \frac {\Gamma \left (\frac {7}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{3}, \frac {7}{3} \\ \frac {10}{3} \end {matrix}\middle | {\frac {e^{i \pi }}{x^{6}}} \right )}}{6 x^{14} \Gamma \left (\frac {10}{3}\right )} \]
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Time = 0.28 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.96 \[ \int \frac {1}{x^{13} \sqrt [3]{1+x^6}} \, dx=\frac {1}{27} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (x^{6} + 1\right )}^{\frac {1}{3}} + 1\right )}\right ) - \frac {4 \, {\left (x^{6} + 1\right )}^{\frac {5}{3}} - 7 \, {\left (x^{6} + 1\right )}^{\frac {2}{3}}}{36 \, {\left (2 \, x^{6} - {\left (x^{6} + 1\right )}^{2} + 1\right )}} - \frac {1}{54} \, \log \left ({\left (x^{6} + 1\right )}^{\frac {2}{3}} + {\left (x^{6} + 1\right )}^{\frac {1}{3}} + 1\right ) + \frac {1}{27} \, \log \left ({\left (x^{6} + 1\right )}^{\frac {1}{3}} - 1\right ) \]
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Time = 0.27 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.80 \[ \int \frac {1}{x^{13} \sqrt [3]{1+x^6}} \, dx=\frac {1}{27} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (x^{6} + 1\right )}^{\frac {1}{3}} + 1\right )}\right ) + \frac {4 \, {\left (x^{6} + 1\right )}^{\frac {5}{3}} - 7 \, {\left (x^{6} + 1\right )}^{\frac {2}{3}}}{36 \, x^{12}} - \frac {1}{54} \, \log \left ({\left (x^{6} + 1\right )}^{\frac {2}{3}} + {\left (x^{6} + 1\right )}^{\frac {1}{3}} + 1\right ) + \frac {1}{27} \, \log \left ({\left (x^{6} + 1\right )}^{\frac {1}{3}} - 1\right ) \]
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Time = 6.37 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.22 \[ \int \frac {1}{x^{13} \sqrt [3]{1+x^6}} \, dx=\frac {\ln \left (\frac {{\left (x^6+1\right )}^{1/3}}{81}-\frac {1}{81}\right )}{27}+\ln \left (\frac {{\left (x^6+1\right )}^{1/3}}{81}-9\,{\left (-\frac {1}{54}+\frac {\sqrt {3}\,1{}\mathrm {i}}{54}\right )}^2\right )\,\left (-\frac {1}{54}+\frac {\sqrt {3}\,1{}\mathrm {i}}{54}\right )-\ln \left (\frac {{\left (x^6+1\right )}^{1/3}}{81}-9\,{\left (\frac {1}{54}+\frac {\sqrt {3}\,1{}\mathrm {i}}{54}\right )}^2\right )\,\left (\frac {1}{54}+\frac {\sqrt {3}\,1{}\mathrm {i}}{54}\right )+\frac {\frac {7\,{\left (x^6+1\right )}^{2/3}}{36}-\frac {{\left (x^6+1\right )}^{5/3}}{9}}{2\,x^6-{\left (x^6+1\right )}^2+1} \]
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