Integrand size = 30, antiderivative size = 61 \[ \int \frac {\left (1+x^6\right ) \left (-1+x^3+x^6\right )^{2/3}}{1-x^6+x^{12}} \, dx=\frac {1}{6} \text {RootSum}\left [2-2 \text {$\#$1}^3+\text {$\#$1}^6\&,\frac {-\log (x) \text {$\#$1}^2+\log \left (\sqrt [3]{-1+x^3+x^6}-x \text {$\#$1}\right ) \text {$\#$1}^2}{-1+\text {$\#$1}^3}\&\right ] \]
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\[ \int \frac {\left (1+x^6\right ) \left (-1+x^3+x^6\right )^{2/3}}{1-x^6+x^{12}} \, dx=\int \frac {\left (1+x^6\right ) \left (-1+x^3+x^6\right )^{2/3}}{1-x^6+x^{12}} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {\left (1-i \sqrt {3}\right ) \left (-1+x^3+x^6\right )^{2/3}}{-1-i \sqrt {3}+2 x^6}+\frac {\left (1+i \sqrt {3}\right ) \left (-1+x^3+x^6\right )^{2/3}}{-1+i \sqrt {3}+2 x^6}\right ) \, dx \\ & = \left (1-i \sqrt {3}\right ) \int \frac {\left (-1+x^3+x^6\right )^{2/3}}{-1-i \sqrt {3}+2 x^6} \, dx+\left (1+i \sqrt {3}\right ) \int \frac {\left (-1+x^3+x^6\right )^{2/3}}{-1+i \sqrt {3}+2 x^6} \, dx \\ & = \left (1-i \sqrt {3}\right ) \int \left (\frac {\sqrt {1+i \sqrt {3}} \left (-1+x^3+x^6\right )^{2/3}}{2 \left (-1-i \sqrt {3}\right ) \left (\sqrt {1+i \sqrt {3}}-\sqrt {2} x^3\right )}+\frac {\sqrt {1+i \sqrt {3}} \left (-1+x^3+x^6\right )^{2/3}}{2 \left (-1-i \sqrt {3}\right ) \left (\sqrt {1+i \sqrt {3}}+\sqrt {2} x^3\right )}\right ) \, dx+\left (1+i \sqrt {3}\right ) \int \left (\frac {\sqrt {1-i \sqrt {3}} \left (-1+x^3+x^6\right )^{2/3}}{2 \left (-1+i \sqrt {3}\right ) \left (\sqrt {1-i \sqrt {3}}-\sqrt {2} x^3\right )}+\frac {\sqrt {1-i \sqrt {3}} \left (-1+x^3+x^6\right )^{2/3}}{2 \left (-1+i \sqrt {3}\right ) \left (\sqrt {1-i \sqrt {3}}+\sqrt {2} x^3\right )}\right ) \, dx \\ & = -\frac {\left (1-i \sqrt {3}\right ) \int \frac {\left (-1+x^3+x^6\right )^{2/3}}{\sqrt {1+i \sqrt {3}}-\sqrt {2} x^3} \, dx}{2 \sqrt {1+i \sqrt {3}}}-\frac {\left (1-i \sqrt {3}\right ) \int \frac {\left (-1+x^3+x^6\right )^{2/3}}{\sqrt {1+i \sqrt {3}}+\sqrt {2} x^3} \, dx}{2 \sqrt {1+i \sqrt {3}}}-\frac {\left (1+i \sqrt {3}\right ) \int \frac {\left (-1+x^3+x^6\right )^{2/3}}{\sqrt {1-i \sqrt {3}}-\sqrt {2} x^3} \, dx}{2 \sqrt {1-i \sqrt {3}}}-\frac {\left (1+i \sqrt {3}\right ) \int \frac {\left (-1+x^3+x^6\right )^{2/3}}{\sqrt {1-i \sqrt {3}}+\sqrt {2} x^3} \, dx}{2 \sqrt {1-i \sqrt {3}}} \\ \end{align*}
Time = 0.38 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.00 \[ \int \frac {\left (1+x^6\right ) \left (-1+x^3+x^6\right )^{2/3}}{1-x^6+x^{12}} \, dx=\frac {1}{6} \text {RootSum}\left [2-2 \text {$\#$1}^3+\text {$\#$1}^6\&,\frac {-\log (x) \text {$\#$1}^2+\log \left (\sqrt [3]{-1+x^3+x^6}-x \text {$\#$1}\right ) \text {$\#$1}^2}{-1+\text {$\#$1}^3}\&\right ] \]
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Time = 21.25 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.89
method | result | size |
pseudoelliptic | \(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (4 \textit {\_Z}^{3}+1\right )}{\sum }\textit {\_R} \ln \left (\frac {4 \left (x^{6}+x^{3}-1\right )^{\frac {1}{3}} \textit {\_R}^{2} x -2 \textit {\_R} \,x^{2}+\left (x^{6}+x^{3}-1\right )^{\frac {2}{3}}}{x^{2}}\right )\right )}{3}\) | \(54\) |
trager | \(\text {Expression too large to display}\) | \(2936\) |
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Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 19.06 (sec) , antiderivative size = 583, normalized size of antiderivative = 9.56 \[ \int \frac {\left (1+x^6\right ) \left (-1+x^3+x^6\right )^{2/3}}{1-x^6+x^{12}} \, dx=\frac {1}{18} \cdot 4^{\frac {1}{6}} \sqrt {3} \left (-1\right )^{\frac {1}{3}} \arctan \left (\frac {4^{\frac {1}{6}} {\left (6 \cdot 4^{\frac {2}{3}} \sqrt {3} \left (-1\right )^{\frac {2}{3}} {\left (x^{31} - 14 \, x^{28} - 75 \, x^{25} + 82 \, x^{22} + 293 \, x^{19} - 132 \, x^{16} - 293 \, x^{13} + 82 \, x^{10} + 75 \, x^{7} - 14 \, x^{4} - x\right )} {\left (x^{6} + x^{3} - 1\right )}^{\frac {2}{3}} - 12 \, \sqrt {3} \left (-1\right )^{\frac {1}{3}} {\left (3 \, x^{32} + 49 \, x^{29} - 51 \, x^{26} - 344 \, x^{23} + 99 \, x^{20} + 609 \, x^{17} - 99 \, x^{14} - 344 \, x^{11} + 51 \, x^{8} + 49 \, x^{5} - 3 \, x^{2}\right )} {\left (x^{6} + x^{3} - 1\right )}^{\frac {1}{3}} + 4^{\frac {1}{3}} \sqrt {3} {\left (x^{36} + 54 \, x^{33} - 129 \, x^{30} - 846 \, x^{27} + 258 \, x^{24} + 2502 \, x^{21} - 169 \, x^{18} - 2502 \, x^{15} + 258 \, x^{12} + 846 \, x^{9} - 129 \, x^{6} - 54 \, x^{3} + 1\right )}\right )}}{6 \, {\left (x^{36} - 54 \, x^{33} - 489 \, x^{30} + 270 \, x^{27} + 2922 \, x^{24} - 54 \, x^{21} - 4921 \, x^{18} + 54 \, x^{15} + 2922 \, x^{12} - 270 \, x^{9} - 489 \, x^{6} + 54 \, x^{3} + 1\right )}}\right ) + \frac {1}{36} \cdot 4^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} \log \left (\frac {3 \cdot 4^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} {\left (x^{7} + 4 \, x^{4} - x\right )} {\left (x^{6} + x^{3} - 1\right )}^{\frac {2}{3}} - 4^{\frac {1}{3}} \left (-1\right )^{\frac {2}{3}} {\left (x^{12} + 18 \, x^{9} + 17 \, x^{6} - 18 \, x^{3} + 1\right )} - 6 \, {\left (3 \, x^{8} + 5 \, x^{5} - 3 \, x^{2}\right )} {\left (x^{6} + x^{3} - 1\right )}^{\frac {1}{3}}}{x^{12} - x^{6} + 1}\right ) - \frac {1}{72} \cdot 4^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} \log \left (\frac {6 \cdot 4^{\frac {1}{3}} \left (-1\right )^{\frac {2}{3}} {\left (3 \, x^{20} - 5 \, x^{17} - 12 \, x^{14} + 11 \, x^{11} + 12 \, x^{8} - 5 \, x^{5} - 3 \, x^{2}\right )} {\left (x^{6} + x^{3} - 1\right )}^{\frac {1}{3}} - 4^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} {\left (x^{24} - 18 \, x^{21} - 2 \, x^{18} + 72 \, x^{15} + 3 \, x^{12} - 72 \, x^{9} - 2 \, x^{6} + 18 \, x^{3} + 1\right )} - 12 \, {\left (x^{19} - 5 \, x^{16} - 2 \, x^{13} + 11 \, x^{10} + 2 \, x^{7} - 5 \, x^{4} - x\right )} {\left (x^{6} + x^{3} - 1\right )}^{\frac {2}{3}}}{x^{24} - 2 \, x^{18} + 3 \, x^{12} - 2 \, x^{6} + 1}\right ) \]
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Timed out. \[ \int \frac {\left (1+x^6\right ) \left (-1+x^3+x^6\right )^{2/3}}{1-x^6+x^{12}} \, dx=\text {Timed out} \]
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Not integrable
Time = 0.24 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.49 \[ \int \frac {\left (1+x^6\right ) \left (-1+x^3+x^6\right )^{2/3}}{1-x^6+x^{12}} \, dx=\int { \frac {{\left (x^{6} + x^{3} - 1\right )}^{\frac {2}{3}} {\left (x^{6} + 1\right )}}{x^{12} - x^{6} + 1} \,d x } \]
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Not integrable
Time = 0.30 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.49 \[ \int \frac {\left (1+x^6\right ) \left (-1+x^3+x^6\right )^{2/3}}{1-x^6+x^{12}} \, dx=\int { \frac {{\left (x^{6} + x^{3} - 1\right )}^{\frac {2}{3}} {\left (x^{6} + 1\right )}}{x^{12} - x^{6} + 1} \,d x } \]
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Not integrable
Time = 6.05 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.49 \[ \int \frac {\left (1+x^6\right ) \left (-1+x^3+x^6\right )^{2/3}}{1-x^6+x^{12}} \, dx=\int \frac {\left (x^6+1\right )\,{\left (x^6+x^3-1\right )}^{2/3}}{x^{12}-x^6+1} \,d x \]
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