Integrand size = 40, antiderivative size = 102 \[ \int \frac {\left (-1+x^6\right )^{2/3} \left (1+x^6\right ) \left (-2+x^3+2 x^6\right )}{x^6 \left (-1-x^3+x^6\right )} \, dx=\frac {\left (-1+x^6\right )^{2/3} \left (-4+15 x^3+4 x^6\right )}{10 x^5}-\sqrt {3} \arctan \left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{-1+x^6}}\right )+\log \left (-x+\sqrt [3]{-1+x^6}\right )-\frac {1}{2} \log \left (x^2+x \sqrt [3]{-1+x^6}+\left (-1+x^6\right )^{2/3}\right ) \]
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Result contains higher order function than in optimal. Order 6 vs. order 3 in optimal.
Time = 0.78 (sec) , antiderivative size = 389, normalized size of antiderivative = 3.81, number of steps used = 25, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {6860, 252, 251, 372, 371, 281, 1452, 441, 440, 476, 525, 524} \[ \int \frac {\left (-1+x^6\right )^{2/3} \left (1+x^6\right ) \left (-2+x^3+2 x^6\right )}{x^6 \left (-1-x^3+x^6\right )} \, dx=-\frac {3 \left (1-\sqrt {5}\right ) \left (x^6-1\right )^{2/3} x \operatorname {AppellF1}\left (\frac {1}{6},-\frac {2}{3},1,\frac {7}{6},x^6,\frac {2 x^6}{3-\sqrt {5}}\right )}{\left (3-\sqrt {5}\right ) \left (1-x^6\right )^{2/3}}-\frac {3 \left (1+\sqrt {5}\right ) \left (x^6-1\right )^{2/3} x \operatorname {AppellF1}\left (\frac {1}{6},1,-\frac {2}{3},\frac {7}{6},\frac {2 x^6}{3+\sqrt {5}},x^6\right )}{\left (3+\sqrt {5}\right ) \left (1-x^6\right )^{2/3}}-\frac {3 \left (x^6-1\right )^{2/3} x^4 \operatorname {AppellF1}\left (\frac {2}{3},-\frac {2}{3},1,\frac {5}{3},x^6,\frac {2 x^6}{3-\sqrt {5}}\right )}{2 \left (3-\sqrt {5}\right ) \left (1-x^6\right )^{2/3}}-\frac {3 \left (x^6-1\right )^{2/3} x^4 \operatorname {AppellF1}\left (\frac {2}{3},-\frac {2}{3},1,\frac {5}{3},x^6,\frac {2 x^6}{3+\sqrt {5}}\right )}{2 \left (3+\sqrt {5}\right ) \left (1-x^6\right )^{2/3}}+\frac {2 \left (x^6-1\right )^{2/3} x \operatorname {Hypergeometric2F1}\left (-\frac {2}{3},\frac {1}{6},\frac {7}{6},x^6\right )}{\left (1-x^6\right )^{2/3}}-\frac {2 \left (x^6-1\right )^{2/3} \operatorname {Hypergeometric2F1}\left (-\frac {5}{6},-\frac {2}{3},\frac {1}{6},x^6\right )}{5 \left (1-x^6\right )^{2/3} x^5}+\frac {3 \left (x^6-1\right )^{2/3} \operatorname {Hypergeometric2F1}\left (-\frac {2}{3},-\frac {1}{3},\frac {2}{3},x^6\right )}{2 \left (1-x^6\right )^{2/3} x^2} \]
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Rule 251
Rule 252
Rule 281
Rule 371
Rule 372
Rule 440
Rule 441
Rule 476
Rule 524
Rule 525
Rule 1452
Rule 6860
Rubi steps \begin{align*} \text {integral}& = \int \left (2 \left (-1+x^6\right )^{2/3}+\frac {2 \left (-1+x^6\right )^{2/3}}{x^6}-\frac {3 \left (-1+x^6\right )^{2/3}}{x^3}+\frac {3 \left (-1+2 x^3\right ) \left (-1+x^6\right )^{2/3}}{-1-x^3+x^6}\right ) \, dx \\ & = 2 \int \left (-1+x^6\right )^{2/3} \, dx+2 \int \frac {\left (-1+x^6\right )^{2/3}}{x^6} \, dx-3 \int \frac {\left (-1+x^6\right )^{2/3}}{x^3} \, dx+3 \int \frac {\left (-1+2 x^3\right ) \left (-1+x^6\right )^{2/3}}{-1-x^3+x^6} \, dx \\ & = -\left (\frac {3}{2} \text {Subst}\left (\int \frac {\left (-1+x^3\right )^{2/3}}{x^2} \, dx,x,x^2\right )\right )+3 \int \left (\frac {2 \left (-1+x^6\right )^{2/3}}{-1-\sqrt {5}+2 x^3}+\frac {2 \left (-1+x^6\right )^{2/3}}{-1+\sqrt {5}+2 x^3}\right ) \, dx+\frac {\left (2 \left (-1+x^6\right )^{2/3}\right ) \int \left (1-x^6\right )^{2/3} \, dx}{\left (1-x^6\right )^{2/3}}+\frac {\left (2 \left (-1+x^6\right )^{2/3}\right ) \int \frac {\left (1-x^6\right )^{2/3}}{x^6} \, dx}{\left (1-x^6\right )^{2/3}} \\ & = -\frac {2 \left (-1+x^6\right )^{2/3} \operatorname {Hypergeometric2F1}\left (-\frac {5}{6},-\frac {2}{3},\frac {1}{6},x^6\right )}{5 x^5 \left (1-x^6\right )^{2/3}}+\frac {2 x \left (-1+x^6\right )^{2/3} \operatorname {Hypergeometric2F1}\left (-\frac {2}{3},\frac {1}{6},\frac {7}{6},x^6\right )}{\left (1-x^6\right )^{2/3}}+6 \int \frac {\left (-1+x^6\right )^{2/3}}{-1-\sqrt {5}+2 x^3} \, dx+6 \int \frac {\left (-1+x^6\right )^{2/3}}{-1+\sqrt {5}+2 x^3} \, dx-\frac {\left (3 \left (-1+x^6\right )^{2/3}\right ) \text {Subst}\left (\int \frac {\left (1-x^3\right )^{2/3}}{x^2} \, dx,x,x^2\right )}{2 \left (1-x^6\right )^{2/3}} \\ & = -\frac {2 \left (-1+x^6\right )^{2/3} \operatorname {Hypergeometric2F1}\left (-\frac {5}{6},-\frac {2}{3},\frac {1}{6},x^6\right )}{5 x^5 \left (1-x^6\right )^{2/3}}+\frac {3 \left (-1+x^6\right )^{2/3} \operatorname {Hypergeometric2F1}\left (-\frac {2}{3},-\frac {1}{3},\frac {2}{3},x^6\right )}{2 x^2 \left (1-x^6\right )^{2/3}}+\frac {2 x \left (-1+x^6\right )^{2/3} \operatorname {Hypergeometric2F1}\left (-\frac {2}{3},\frac {1}{6},\frac {7}{6},x^6\right )}{\left (1-x^6\right )^{2/3}}+6 \int \left (\frac {\left (-1-\sqrt {5}\right ) \left (-1+x^6\right )^{2/3}}{2 \left (3+\sqrt {5}-2 x^6\right )}+\frac {x^3 \left (-1+x^6\right )^{2/3}}{-3-\sqrt {5}+2 x^6}\right ) \, dx+6 \int \left (\frac {\left (1-\sqrt {5}\right ) \left (-1+x^6\right )^{2/3}}{2 \left (-3+\sqrt {5}+2 x^6\right )}+\frac {x^3 \left (-1+x^6\right )^{2/3}}{-3+\sqrt {5}+2 x^6}\right ) \, dx \\ & = -\frac {2 \left (-1+x^6\right )^{2/3} \operatorname {Hypergeometric2F1}\left (-\frac {5}{6},-\frac {2}{3},\frac {1}{6},x^6\right )}{5 x^5 \left (1-x^6\right )^{2/3}}+\frac {3 \left (-1+x^6\right )^{2/3} \operatorname {Hypergeometric2F1}\left (-\frac {2}{3},-\frac {1}{3},\frac {2}{3},x^6\right )}{2 x^2 \left (1-x^6\right )^{2/3}}+\frac {2 x \left (-1+x^6\right )^{2/3} \operatorname {Hypergeometric2F1}\left (-\frac {2}{3},\frac {1}{6},\frac {7}{6},x^6\right )}{\left (1-x^6\right )^{2/3}}+6 \int \frac {x^3 \left (-1+x^6\right )^{2/3}}{-3-\sqrt {5}+2 x^6} \, dx+6 \int \frac {x^3 \left (-1+x^6\right )^{2/3}}{-3+\sqrt {5}+2 x^6} \, dx+\left (3 \left (1-\sqrt {5}\right )\right ) \int \frac {\left (-1+x^6\right )^{2/3}}{-3+\sqrt {5}+2 x^6} \, dx-\left (3 \left (1+\sqrt {5}\right )\right ) \int \frac {\left (-1+x^6\right )^{2/3}}{3+\sqrt {5}-2 x^6} \, dx \\ & = -\frac {2 \left (-1+x^6\right )^{2/3} \operatorname {Hypergeometric2F1}\left (-\frac {5}{6},-\frac {2}{3},\frac {1}{6},x^6\right )}{5 x^5 \left (1-x^6\right )^{2/3}}+\frac {3 \left (-1+x^6\right )^{2/3} \operatorname {Hypergeometric2F1}\left (-\frac {2}{3},-\frac {1}{3},\frac {2}{3},x^6\right )}{2 x^2 \left (1-x^6\right )^{2/3}}+\frac {2 x \left (-1+x^6\right )^{2/3} \operatorname {Hypergeometric2F1}\left (-\frac {2}{3},\frac {1}{6},\frac {7}{6},x^6\right )}{\left (1-x^6\right )^{2/3}}+3 \text {Subst}\left (\int \frac {x \left (-1+x^3\right )^{2/3}}{-3-\sqrt {5}+2 x^3} \, dx,x,x^2\right )+3 \text {Subst}\left (\int \frac {x \left (-1+x^3\right )^{2/3}}{-3+\sqrt {5}+2 x^3} \, dx,x,x^2\right )+\frac {\left (3 \left (1-\sqrt {5}\right ) \left (-1+x^6\right )^{2/3}\right ) \int \frac {\left (1-x^6\right )^{2/3}}{-3+\sqrt {5}+2 x^6} \, dx}{\left (1-x^6\right )^{2/3}}-\frac {\left (3 \left (1+\sqrt {5}\right ) \left (-1+x^6\right )^{2/3}\right ) \int \frac {\left (1-x^6\right )^{2/3}}{3+\sqrt {5}-2 x^6} \, dx}{\left (1-x^6\right )^{2/3}} \\ & = -\frac {3 \left (1-\sqrt {5}\right ) x \left (-1+x^6\right )^{2/3} \operatorname {AppellF1}\left (\frac {1}{6},-\frac {2}{3},1,\frac {7}{6},x^6,\frac {2 x^6}{3-\sqrt {5}}\right )}{\left (3-\sqrt {5}\right ) \left (1-x^6\right )^{2/3}}-\frac {3 \left (1+\sqrt {5}\right ) x \left (-1+x^6\right )^{2/3} \operatorname {AppellF1}\left (\frac {1}{6},1,-\frac {2}{3},\frac {7}{6},\frac {2 x^6}{3+\sqrt {5}},x^6\right )}{\left (3+\sqrt {5}\right ) \left (1-x^6\right )^{2/3}}-\frac {2 \left (-1+x^6\right )^{2/3} \operatorname {Hypergeometric2F1}\left (-\frac {5}{6},-\frac {2}{3},\frac {1}{6},x^6\right )}{5 x^5 \left (1-x^6\right )^{2/3}}+\frac {3 \left (-1+x^6\right )^{2/3} \operatorname {Hypergeometric2F1}\left (-\frac {2}{3},-\frac {1}{3},\frac {2}{3},x^6\right )}{2 x^2 \left (1-x^6\right )^{2/3}}+\frac {2 x \left (-1+x^6\right )^{2/3} \operatorname {Hypergeometric2F1}\left (-\frac {2}{3},\frac {1}{6},\frac {7}{6},x^6\right )}{\left (1-x^6\right )^{2/3}}+\frac {\left (3 \left (-1+x^6\right )^{2/3}\right ) \text {Subst}\left (\int \frac {x \left (1-x^3\right )^{2/3}}{-3-\sqrt {5}+2 x^3} \, dx,x,x^2\right )}{\left (1-x^6\right )^{2/3}}+\frac {\left (3 \left (-1+x^6\right )^{2/3}\right ) \text {Subst}\left (\int \frac {x \left (1-x^3\right )^{2/3}}{-3+\sqrt {5}+2 x^3} \, dx,x,x^2\right )}{\left (1-x^6\right )^{2/3}} \\ & = -\frac {3 \left (1-\sqrt {5}\right ) x \left (-1+x^6\right )^{2/3} \operatorname {AppellF1}\left (\frac {1}{6},-\frac {2}{3},1,\frac {7}{6},x^6,\frac {2 x^6}{3-\sqrt {5}}\right )}{\left (3-\sqrt {5}\right ) \left (1-x^6\right )^{2/3}}-\frac {3 \left (1+\sqrt {5}\right ) x \left (-1+x^6\right )^{2/3} \operatorname {AppellF1}\left (\frac {1}{6},1,-\frac {2}{3},\frac {7}{6},\frac {2 x^6}{3+\sqrt {5}},x^6\right )}{\left (3+\sqrt {5}\right ) \left (1-x^6\right )^{2/3}}-\frac {3 x^4 \left (-1+x^6\right )^{2/3} \operatorname {AppellF1}\left (\frac {2}{3},-\frac {2}{3},1,\frac {5}{3},x^6,\frac {2 x^6}{3-\sqrt {5}}\right )}{2 \left (3-\sqrt {5}\right ) \left (1-x^6\right )^{2/3}}-\frac {3 x^4 \left (-1+x^6\right )^{2/3} \operatorname {AppellF1}\left (\frac {2}{3},-\frac {2}{3},1,\frac {5}{3},x^6,\frac {2 x^6}{3+\sqrt {5}}\right )}{2 \left (3+\sqrt {5}\right ) \left (1-x^6\right )^{2/3}}-\frac {2 \left (-1+x^6\right )^{2/3} \operatorname {Hypergeometric2F1}\left (-\frac {5}{6},-\frac {2}{3},\frac {1}{6},x^6\right )}{5 x^5 \left (1-x^6\right )^{2/3}}+\frac {3 \left (-1+x^6\right )^{2/3} \operatorname {Hypergeometric2F1}\left (-\frac {2}{3},-\frac {1}{3},\frac {2}{3},x^6\right )}{2 x^2 \left (1-x^6\right )^{2/3}}+\frac {2 x \left (-1+x^6\right )^{2/3} \operatorname {Hypergeometric2F1}\left (-\frac {2}{3},\frac {1}{6},\frac {7}{6},x^6\right )}{\left (1-x^6\right )^{2/3}} \\ \end{align*}
Time = 1.86 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.00 \[ \int \frac {\left (-1+x^6\right )^{2/3} \left (1+x^6\right ) \left (-2+x^3+2 x^6\right )}{x^6 \left (-1-x^3+x^6\right )} \, dx=\frac {\left (-1+x^6\right )^{2/3} \left (-4+15 x^3+4 x^6\right )}{10 x^5}-\sqrt {3} \arctan \left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{-1+x^6}}\right )+\log \left (-x+\sqrt [3]{-1+x^6}\right )-\frac {1}{2} \log \left (x^2+x \sqrt [3]{-1+x^6}+\left (-1+x^6\right )^{2/3}\right ) \]
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Time = 14.64 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.04
method | result | size |
pseudoelliptic | \(\frac {-5 \ln \left (\frac {x^{2}+x \left (x^{6}-1\right )^{\frac {1}{3}}+\left (x^{6}-1\right )^{\frac {2}{3}}}{x^{2}}\right ) x^{5}+10 \sqrt {3}\, \arctan \left (\frac {\left (x +2 \left (x^{6}-1\right )^{\frac {1}{3}}\right ) \sqrt {3}}{3 x}\right ) x^{5}+10 \ln \left (\frac {-x +\left (x^{6}-1\right )^{\frac {1}{3}}}{x}\right ) x^{5}+\left (4 x^{6}+15 x^{3}-4\right ) \left (x^{6}-1\right )^{\frac {2}{3}}}{10 x^{5}}\) | \(106\) |
risch | \(\frac {4 x^{12}+15 x^{9}-8 x^{6}-15 x^{3}+4}{10 x^{5} \left (x^{6}-1\right )^{\frac {1}{3}}}-3 \ln \left (\frac {3 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{6}+2 x^{6}+9 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{3}-9 \left (x^{6}-1\right )^{\frac {2}{3}} \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x +9 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{3}-3 x^{2} \left (x^{6}-1\right )^{\frac {1}{3}}+2 x^{3}-3 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )-2}{x^{6}-x^{3}-1}\right ) \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )+\ln \left (\frac {-3 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{6}-2 x^{6}+9 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{3}+9 \left (x^{6}-1\right )^{\frac {2}{3}} \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x -9 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{6}-1\right )^{\frac {1}{3}} x^{2}+6 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{3}+3 x \left (x^{6}-1\right )^{\frac {2}{3}}+3 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )+2}{x^{6}-x^{3}-1}\right )-\ln \left (\frac {3 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{6}+2 x^{6}+9 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{3}-9 \left (x^{6}-1\right )^{\frac {2}{3}} \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x +9 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{3}-3 x^{2} \left (x^{6}-1\right )^{\frac {1}{3}}+2 x^{3}-3 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )-2}{x^{6}-x^{3}-1}\right )\) | \(433\) |
trager | \(\text {Expression too large to display}\) | \(618\) |
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Time = 8.38 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.45 \[ \int \frac {\left (-1+x^6\right )^{2/3} \left (1+x^6\right ) \left (-2+x^3+2 x^6\right )}{x^6 \left (-1-x^3+x^6\right )} \, dx=-\frac {10 \, \sqrt {3} x^{5} \arctan \left (\frac {473996388635948633452428917614298985996886224511260115036680453514888144148250 \, \sqrt {3} {\left (x^{6} - 1\right )}^{\frac {1}{3}} x^{2} + 19325031480489228255674265966448835967818926087643600184123099965366515892788 \, \sqrt {3} {\left (x^{6} - 1\right )}^{\frac {2}{3}} x + \sqrt {3} {\left (771225779807741020855977802972631216428368740202755221603971931588718036144 \, x^{6} + 245889484278411189833195613987401279765924206559249102388797804808538611984375 \, x^{3} - 771225779807741020855977802972631216428368740202755221603971931588718036144\right )}}{3 \, {\left (15407513785538665202033017569552164636906896740149986002803824712402669144 \, x^{6} - 227351086091515241263579358841494627179170556108548407412281480599473216796875 \, x^{3} - 15407513785538665202033017569552164636906896740149986002803824712402669144\right )}}\right ) - 5 \, x^{5} \log \left (\frac {x^{6} - x^{3} + 3 \, {\left (x^{6} - 1\right )}^{\frac {1}{3}} x^{2} - 3 \, {\left (x^{6} - 1\right )}^{\frac {2}{3}} x - 1}{x^{6} - x^{3} - 1}\right ) - {\left (4 \, x^{6} + 15 \, x^{3} - 4\right )} {\left (x^{6} - 1\right )}^{\frac {2}{3}}}{10 \, x^{5}} \]
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Timed out. \[ \int \frac {\left (-1+x^6\right )^{2/3} \left (1+x^6\right ) \left (-2+x^3+2 x^6\right )}{x^6 \left (-1-x^3+x^6\right )} \, dx=\text {Timed out} \]
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\[ \int \frac {\left (-1+x^6\right )^{2/3} \left (1+x^6\right ) \left (-2+x^3+2 x^6\right )}{x^6 \left (-1-x^3+x^6\right )} \, dx=\int { \frac {{\left (2 \, x^{6} + x^{3} - 2\right )} {\left (x^{6} + 1\right )} {\left (x^{6} - 1\right )}^{\frac {2}{3}}}{{\left (x^{6} - x^{3} - 1\right )} x^{6}} \,d x } \]
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\[ \int \frac {\left (-1+x^6\right )^{2/3} \left (1+x^6\right ) \left (-2+x^3+2 x^6\right )}{x^6 \left (-1-x^3+x^6\right )} \, dx=\int { \frac {{\left (2 \, x^{6} + x^{3} - 2\right )} {\left (x^{6} + 1\right )} {\left (x^{6} - 1\right )}^{\frac {2}{3}}}{{\left (x^{6} - x^{3} - 1\right )} x^{6}} \,d x } \]
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Timed out. \[ \int \frac {\left (-1+x^6\right )^{2/3} \left (1+x^6\right ) \left (-2+x^3+2 x^6\right )}{x^6 \left (-1-x^3+x^6\right )} \, dx=\int -\frac {{\left (x^6-1\right )}^{2/3}\,\left (x^6+1\right )\,\left (2\,x^6+x^3-2\right )}{x^6\,\left (-x^6+x^3+1\right )} \,d x \]
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