Integrand size = 28, antiderivative size = 128 \[ \int \frac {\left (-1+x^6\right ) \left (1+x^3+x^6\right )^{2/3}}{1+x^6+x^{12}} \, dx=-\frac {\arctan \left (\frac {\sqrt {3} x}{x+2^{2/3} \sqrt [3]{1+x^3+x^6}}\right )}{\sqrt [3]{2} \sqrt {3}}+\frac {\log \left (-2 x+2^{2/3} \sqrt [3]{1+x^3+x^6}\right )}{3 \sqrt [3]{2}}-\frac {\log \left (2 x^2+2^{2/3} x \sqrt [3]{1+x^3+x^6}+\sqrt [3]{2} \left (1+x^3+x^6\right )^{2/3}\right )}{6 \sqrt [3]{2}} \]
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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 \frac {-1+x^6}{\left (1-x^3+x^6\right ) \sqrt [3]{1+x^3+x^6}} \, dx \\ & = \int \left (\frac {1}{\sqrt [3]{1+x^3+x^6}}-\frac {2-x^3}{\left (1-x^3+x^6\right ) \sqrt [3]{1+x^3+x^6}}\right ) \, dx \\ & = \int \frac {1}{\sqrt [3]{1+x^3+x^6}} \, dx-\int \frac {2-x^3}{\left (1-x^3+x^6\right ) \sqrt [3]{1+x^3+x^6}} \, dx \\ & = \frac {\left (\sqrt [3]{1+\frac {2 x^3}{1-i \sqrt {3}}} \sqrt [3]{1+\frac {2 x^3}{1+i \sqrt {3}}}\right ) \int \frac {1}{\sqrt [3]{1+\frac {2 x^3}{1-i \sqrt {3}}} \sqrt [3]{1+\frac {2 x^3}{1+i \sqrt {3}}}} \, dx}{\sqrt [3]{1+x^3+x^6}}-\int \left (\frac {-1-i \sqrt {3}}{\left (-1-i \sqrt {3}+2 x^3\right ) \sqrt [3]{1+x^3+x^6}}+\frac {-1+i \sqrt {3}}{\left (-1+i \sqrt {3}+2 x^3\right ) \sqrt [3]{1+x^3+x^6}}\right ) \, dx \\ & = \frac {x \sqrt [3]{1+\frac {2 x^3}{1-i \sqrt {3}}} \sqrt [3]{1+\frac {2 x^3}{1+i \sqrt {3}}} \operatorname {AppellF1}\left (\frac {1}{3},\frac {1}{3},\frac {1}{3},\frac {4}{3},-\frac {2 x^3}{1-i \sqrt {3}},-\frac {2 x^3}{1+i \sqrt {3}}\right )}{\sqrt [3]{1+x^3+x^6}}-\left (-1-i \sqrt {3}\right ) \int \frac {1}{\left (-1-i \sqrt {3}+2 x^3\right ) \sqrt [3]{1+x^3+x^6}} \, dx-\left (-1+i \sqrt {3}\right ) \int \frac {1}{\left (-1+i \sqrt {3}+2 x^3\right ) \sqrt [3]{1+x^3+x^6}} \, dx \\ \end{align*}
Time = 0.42 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.91 \[ \int \frac {\left (-1+x^6\right ) \left (1+x^3+x^6\right )^{2/3}}{1+x^6+x^{12}} \, dx=-\frac {2 \sqrt {3} \arctan \left (\frac {\sqrt {3} x}{x+2^{2/3} \sqrt [3]{1+x^3+x^6}}\right )-2 \log \left (-2 x+2^{2/3} \sqrt [3]{1+x^3+x^6}\right )+\log \left (2 x^2+2^{2/3} x \sqrt [3]{1+x^3+x^6}+\sqrt [3]{2} \left (1+x^3+x^6\right )^{2/3}\right )}{6 \sqrt [3]{2}} \]
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Time = 6.50 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.79
method | result | size |
pseudoelliptic | \(\frac {2^{\frac {2}{3}} \left (2 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (x +2^{\frac {2}{3}} \left (x^{6}+x^{3}+1\right )^{\frac {1}{3}}\right )}{3 x}\right )+2 \ln \left (\frac {-2^{\frac {1}{3}} x +\left (x^{6}+x^{3}+1\right )^{\frac {1}{3}}}{x}\right )-\ln \left (\frac {2^{\frac {2}{3}} x^{2}+2^{\frac {1}{3}} \left (x^{6}+x^{3}+1\right )^{\frac {1}{3}} x +\left (x^{6}+x^{3}+1\right )^{\frac {2}{3}}}{x^{2}}\right )\right )}{12}\) | \(101\) |
trager | \(\text {Expression too large to display}\) | \(699\) |
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Leaf count of result is larger than twice the leaf count of optimal. 322 vs. \(2 (98) = 196\).
Time = 6.30 (sec) , antiderivative size = 322, normalized size of antiderivative = 2.52 \[ \int \frac {\left (-1+x^6\right ) \left (1+x^3+x^6\right )^{2/3}}{1+x^6+x^{12}} \, dx=-\frac {1}{18} \, \sqrt {6} 2^{\frac {1}{6}} \arctan \left (\frac {2^{\frac {1}{6}} {\left (6 \, \sqrt {6} 2^{\frac {2}{3}} {\left (x^{13} + 16 \, x^{10} + 21 \, x^{7} + 16 \, x^{4} + x\right )} {\left (x^{6} + x^{3} + 1\right )}^{\frac {2}{3}} - \sqrt {6} 2^{\frac {1}{3}} {\left (x^{18} - 21 \, x^{15} - 102 \, x^{12} - 133 \, x^{9} - 102 \, x^{6} - 21 \, x^{3} + 1\right )} - 24 \, \sqrt {6} {\left (x^{14} + x^{11} + x^{5} + x^{2}\right )} {\left (x^{6} + x^{3} + 1\right )}^{\frac {1}{3}}\right )}}{6 \, {\left (x^{18} + 51 \, x^{15} + 114 \, x^{12} + 155 \, x^{9} + 114 \, x^{6} + 51 \, x^{3} + 1\right )}}\right ) + \frac {1}{18} \cdot 2^{\frac {2}{3}} \log \left (-\frac {3 \cdot 2^{\frac {2}{3}} {\left (x^{6} + x^{3} + 1\right )}^{\frac {2}{3}} x - 6 \, {\left (x^{6} + x^{3} + 1\right )}^{\frac {1}{3}} x^{2} - 2^{\frac {1}{3}} {\left (x^{6} - x^{3} + 1\right )}}{x^{6} - x^{3} + 1}\right ) - \frac {1}{36} \cdot 2^{\frac {2}{3}} \log \left (\frac {2^{\frac {2}{3}} {\left (x^{12} + 16 \, x^{9} + 21 \, x^{6} + 16 \, x^{3} + 1\right )} + 12 \cdot 2^{\frac {1}{3}} {\left (x^{8} + 2 \, x^{5} + x^{2}\right )} {\left (x^{6} + x^{3} + 1\right )}^{\frac {1}{3}} + 6 \, {\left (x^{7} + 5 \, x^{4} + x\right )} {\left (x^{6} + x^{3} + 1\right )}^{\frac {2}{3}}}{x^{12} - 2 \, x^{9} + 3 \, x^{6} - 2 \, x^{3} + 1}\right ) \]
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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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\[ \int \frac {\left (-1+x^6\right ) \left (1+x^3+x^6\right )^{2/3}}{1+x^6+x^{12}} \, dx=\int { \frac {{\left (x^{6} + x^{3} + 1\right )}^{\frac {2}{3}} {\left (x^{6} - 1\right )}}{x^{12} + x^{6} + 1} \,d x } \]
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\[ \int \frac {\left (-1+x^6\right ) \left (1+x^3+x^6\right )^{2/3}}{1+x^6+x^{12}} \, dx=\int { \frac {{\left (x^{6} + x^{3} + 1\right )}^{\frac {2}{3}} {\left (x^{6} - 1\right )}}{x^{12} + x^{6} + 1} \,d x } \]
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Timed out. \[ \int \frac {\left (-1+x^6\right ) \left (1+x^3+x^6\right )^{2/3}}{1+x^6+x^{12}} \, dx=\int \frac {\left (x^6-1\right )\,{\left (x^6+x^3+1\right )}^{2/3}}{x^{12}+x^6+1} \,d x \]
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