Integrand size = 45, antiderivative size = 217 \[ \int \frac {\left (-1+x+x^3+x^6\right )^{2/3} \left (3-2 x+3 x^6\right )}{\left (-1+x+x^6\right ) \left (-1+x-x^3+x^6\right )} \, dx=\sqrt {3} \arctan \left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{-1+x+x^3+x^6}}\right )-2^{2/3} \sqrt {3} \arctan \left (\frac {\sqrt {3} x}{x+2^{2/3} \sqrt [3]{-1+x+x^3+x^6}}\right )-\log \left (-x+\sqrt [3]{-1+x+x^3+x^6}\right )+2^{2/3} \log \left (-2 x+2^{2/3} \sqrt [3]{-1+x+x^3+x^6}\right )+\frac {1}{2} \log \left (x^2+x \sqrt [3]{-1+x+x^3+x^6}+\left (-1+x+x^3+x^6\right )^{2/3}\right )-\frac {\log \left (2 x^2+2^{2/3} x \sqrt [3]{-1+x+x^3+x^6}+\sqrt [3]{2} \left (-1+x+x^3+x^6\right )^{2/3}\right )}{\sqrt [3]{2}} \]
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\[ \int \frac {\left (-1+x+x^3+x^6\right )^{2/3} \left (3-2 x+3 x^6\right )}{\left (-1+x+x^6\right ) \left (-1+x-x^3+x^6\right )} \, dx=\int \frac {\left (-1+x+x^3+x^6\right )^{2/3} \left (3-2 x+3 x^6\right )}{\left (-1+x+x^6\right ) \left (-1+x-x^3+x^6\right )} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {\left (-1+x+x^3+x^6\right )^{2/3}}{-1+x}+\frac {\left (-1+x+x^3+x^6\right )^{2/3}}{1+x}+\frac {\left (2+3 x+x^2-x^3\right ) \left (-1+x+x^3+x^6\right )^{2/3}}{1-x+x^2+x^4}+\frac {\left (-1-6 x^3-x^4-x^5\right ) \left (-1+x+x^3+x^6\right )^{2/3}}{-1+x+x^6}\right ) \, dx \\ & = \int \frac {\left (-1+x+x^3+x^6\right )^{2/3}}{-1+x} \, dx+\int \frac {\left (-1+x+x^3+x^6\right )^{2/3}}{1+x} \, dx+\int \frac {\left (2+3 x+x^2-x^3\right ) \left (-1+x+x^3+x^6\right )^{2/3}}{1-x+x^2+x^4} \, dx+\int \frac {\left (-1-6 x^3-x^4-x^5\right ) \left (-1+x+x^3+x^6\right )^{2/3}}{-1+x+x^6} \, dx \\ & = \int \frac {\left (-1+x+x^3+x^6\right )^{2/3}}{-1+x} \, dx+\int \frac {\left (-1+x+x^3+x^6\right )^{2/3}}{1+x} \, dx+\int \left (\frac {2 \left (-1+x+x^3+x^6\right )^{2/3}}{1-x+x^2+x^4}+\frac {3 x \left (-1+x+x^3+x^6\right )^{2/3}}{1-x+x^2+x^4}+\frac {x^2 \left (-1+x+x^3+x^6\right )^{2/3}}{1-x+x^2+x^4}-\frac {x^3 \left (-1+x+x^3+x^6\right )^{2/3}}{1-x+x^2+x^4}\right ) \, dx+\int \left (\frac {\left (-1+x+x^3+x^6\right )^{2/3}}{1-x-x^6}-\frac {6 x^3 \left (-1+x+x^3+x^6\right )^{2/3}}{-1+x+x^6}-\frac {x^4 \left (-1+x+x^3+x^6\right )^{2/3}}{-1+x+x^6}-\frac {x^5 \left (-1+x+x^3+x^6\right )^{2/3}}{-1+x+x^6}\right ) \, dx \\ & = 2 \int \frac {\left (-1+x+x^3+x^6\right )^{2/3}}{1-x+x^2+x^4} \, dx+3 \int \frac {x \left (-1+x+x^3+x^6\right )^{2/3}}{1-x+x^2+x^4} \, dx-6 \int \frac {x^3 \left (-1+x+x^3+x^6\right )^{2/3}}{-1+x+x^6} \, dx+\int \frac {\left (-1+x+x^3+x^6\right )^{2/3}}{-1+x} \, dx+\int \frac {\left (-1+x+x^3+x^6\right )^{2/3}}{1+x} \, dx+\int \frac {x^2 \left (-1+x+x^3+x^6\right )^{2/3}}{1-x+x^2+x^4} \, dx-\int \frac {x^3 \left (-1+x+x^3+x^6\right )^{2/3}}{1-x+x^2+x^4} \, dx+\int \frac {\left (-1+x+x^3+x^6\right )^{2/3}}{1-x-x^6} \, dx-\int \frac {x^4 \left (-1+x+x^3+x^6\right )^{2/3}}{-1+x+x^6} \, dx-\int \frac {x^5 \left (-1+x+x^3+x^6\right )^{2/3}}{-1+x+x^6} \, dx \\ \end{align*}
Time = 0.50 (sec) , antiderivative size = 217, normalized size of antiderivative = 1.00 \[ \int \frac {\left (-1+x+x^3+x^6\right )^{2/3} \left (3-2 x+3 x^6\right )}{\left (-1+x+x^6\right ) \left (-1+x-x^3+x^6\right )} \, dx=\sqrt {3} \arctan \left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{-1+x+x^3+x^6}}\right )-2^{2/3} \sqrt {3} \arctan \left (\frac {\sqrt {3} x}{x+2^{2/3} \sqrt [3]{-1+x+x^3+x^6}}\right )-\log \left (-x+\sqrt [3]{-1+x+x^3+x^6}\right )+2^{2/3} \log \left (-2 x+2^{2/3} \sqrt [3]{-1+x+x^3+x^6}\right )+\frac {1}{2} \log \left (x^2+x \sqrt [3]{-1+x+x^3+x^6}+\left (-1+x+x^3+x^6\right )^{2/3}\right )-\frac {\log \left (2 x^2+2^{2/3} x \sqrt [3]{-1+x+x^3+x^6}+\sqrt [3]{2} \left (-1+x+x^3+x^6\right )^{2/3}\right )}{\sqrt [3]{2}} \]
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Time = 36.37 (sec) , antiderivative size = 193, normalized size of antiderivative = 0.89
method | result | size |
pseudoelliptic | \(2^{\frac {2}{3}} \ln \left (\frac {-2^{\frac {1}{3}} x +\left (x^{6}+x^{3}+x -1\right )^{\frac {1}{3}}}{x}\right )-\frac {2^{\frac {2}{3}} \ln \left (\frac {2^{\frac {2}{3}} x^{2}+2^{\frac {1}{3}} \left (x^{6}+x^{3}+x -1\right )^{\frac {1}{3}} x +\left (x^{6}+x^{3}+x -1\right )^{\frac {2}{3}}}{x^{2}}\right )}{2}+\sqrt {3}\, 2^{\frac {2}{3}} \arctan \left (\frac {\sqrt {3}\, \left (x +2^{\frac {2}{3}} \left (x^{6}+x^{3}+x -1\right )^{\frac {1}{3}}\right )}{3 x}\right )-\ln \left (\frac {-x +\left (x^{6}+x^{3}+x -1\right )^{\frac {1}{3}}}{x}\right )+\frac {\ln \left (\frac {x^{2}+x \left (x^{6}+x^{3}+x -1\right )^{\frac {1}{3}}+\left (x^{6}+x^{3}+x -1\right )^{\frac {2}{3}}}{x^{2}}\right )}{2}-\sqrt {3}\, \arctan \left (\frac {\left (x +2 \left (x^{6}+x^{3}+x -1\right )^{\frac {1}{3}}\right ) \sqrt {3}}{3 x}\right )\) | \(193\) |
trager | \(\text {Expression too large to display}\) | \(2023\) |
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Leaf count of result is larger than twice the leaf count of optimal. 615 vs. \(2 (177) = 354\).
Time = 30.92 (sec) , antiderivative size = 615, normalized size of antiderivative = 2.83 \[ \int \frac {\left (-1+x+x^3+x^6\right )^{2/3} \left (3-2 x+3 x^6\right )}{\left (-1+x+x^6\right ) \left (-1+x-x^3+x^6\right )} \, dx=-\frac {1}{3} \cdot 4^{\frac {1}{3}} \sqrt {3} \arctan \left (\frac {3 \cdot 4^{\frac {2}{3}} \sqrt {3} {\left (x^{13} + 4 \, x^{10} + 2 \, x^{8} - 7 \, x^{7} + 4 \, x^{5} - 4 \, x^{4} + x^{3} - 2 \, x^{2} + x\right )} {\left (x^{6} + x^{3} + x - 1\right )}^{\frac {2}{3}} + 6 \cdot 4^{\frac {1}{3}} \sqrt {3} {\left (x^{14} + 16 \, x^{11} + 2 \, x^{9} + 17 \, x^{8} + 16 \, x^{6} - 16 \, x^{5} + x^{4} - 2 \, x^{3} + x^{2}\right )} {\left (x^{6} + x^{3} + x - 1\right )}^{\frac {1}{3}} + \sqrt {3} {\left (x^{18} + 33 \, x^{15} + 3 \, x^{13} + 108 \, x^{12} + 66 \, x^{10} + 5 \, x^{9} + 3 \, x^{8} + 105 \, x^{7} - 108 \, x^{6} + 33 \, x^{5} - 66 \, x^{4} + 34 \, x^{3} - 3 \, x^{2} + 3 \, x - 1\right )}}{3 \, {\left (x^{18} - 3 \, x^{15} + 3 \, x^{13} - 108 \, x^{12} - 6 \, x^{10} - 103 \, x^{9} + 3 \, x^{8} - 111 \, x^{7} + 108 \, x^{6} - 3 \, x^{5} + 6 \, x^{4} - 2 \, x^{3} - 3 \, x^{2} + 3 \, x - 1\right )}}\right ) + \sqrt {3} \arctan \left (-\frac {682 \, \sqrt {3} {\left (x^{6} + x^{3} + x - 1\right )}^{\frac {1}{3}} x^{2} - 248 \, \sqrt {3} {\left (x^{6} + x^{3} + x - 1\right )}^{\frac {2}{3}} x + \sqrt {3} {\left (96 \, x^{6} + 217 \, x^{3} + 96 \, x - 96\right )}}{64 \, x^{6} + 1395 \, x^{3} + 64 \, x - 64}\right ) + \frac {1}{3} \cdot 4^{\frac {1}{3}} \log \left (\frac {3 \cdot 4^{\frac {2}{3}} {\left (x^{6} + x^{3} + x - 1\right )}^{\frac {1}{3}} x^{2} - 6 \, {\left (x^{6} + x^{3} + x - 1\right )}^{\frac {2}{3}} x + 4^{\frac {1}{3}} {\left (x^{6} - x^{3} + x - 1\right )}}{x^{6} - x^{3} + x - 1}\right ) - \frac {1}{6} \cdot 4^{\frac {1}{3}} \log \left (\frac {6 \cdot 4^{\frac {1}{3}} {\left (x^{7} + 5 \, x^{4} + x^{2} - x\right )} {\left (x^{6} + x^{3} + x - 1\right )}^{\frac {2}{3}} + 4^{\frac {2}{3}} {\left (x^{12} + 16 \, x^{9} + 2 \, x^{7} + 17 \, x^{6} + 16 \, x^{4} - 16 \, x^{3} + x^{2} - 2 \, x + 1\right )} + 24 \, {\left (x^{8} + 2 \, x^{5} + x^{3} - x^{2}\right )} {\left (x^{6} + x^{3} + x - 1\right )}^{\frac {1}{3}}}{x^{12} - 2 \, x^{9} + 2 \, x^{7} - x^{6} - 2 \, x^{4} + 2 \, x^{3} + x^{2} - 2 \, x + 1}\right ) - \frac {1}{2} \, \log \left (\frac {x^{6} + 3 \, {\left (x^{6} + x^{3} + x - 1\right )}^{\frac {1}{3}} x^{2} - 3 \, {\left (x^{6} + x^{3} + x - 1\right )}^{\frac {2}{3}} x + x - 1}{x^{6} + x - 1}\right ) \]
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Timed out. \[ \int \frac {\left (-1+x+x^3+x^6\right )^{2/3} \left (3-2 x+3 x^6\right )}{\left (-1+x+x^6\right ) \left (-1+x-x^3+x^6\right )} \, dx=\text {Timed out} \]
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\[ \int \frac {\left (-1+x+x^3+x^6\right )^{2/3} \left (3-2 x+3 x^6\right )}{\left (-1+x+x^6\right ) \left (-1+x-x^3+x^6\right )} \, dx=\int { \frac {{\left (3 \, x^{6} - 2 \, x + 3\right )} {\left (x^{6} + x^{3} + x - 1\right )}^{\frac {2}{3}}}{{\left (x^{6} - x^{3} + x - 1\right )} {\left (x^{6} + x - 1\right )}} \,d x } \]
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\[ \int \frac {\left (-1+x+x^3+x^6\right )^{2/3} \left (3-2 x+3 x^6\right )}{\left (-1+x+x^6\right ) \left (-1+x-x^3+x^6\right )} \, dx=\int { \frac {{\left (3 \, x^{6} - 2 \, x + 3\right )} {\left (x^{6} + x^{3} + x - 1\right )}^{\frac {2}{3}}}{{\left (x^{6} - x^{3} + x - 1\right )} {\left (x^{6} + x - 1\right )}} \,d x } \]
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Timed out. \[ \int \frac {\left (-1+x+x^3+x^6\right )^{2/3} \left (3-2 x+3 x^6\right )}{\left (-1+x+x^6\right ) \left (-1+x-x^3+x^6\right )} \, dx=\int \frac {\left (3\,x^6-2\,x+3\right )\,{\left (x^6+x^3+x-1\right )}^{2/3}}{\left (x^6+x-1\right )\,\left (x^6-x^3+x-1\right )} \,d x \]
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