Integrand size = 32, antiderivative size = 154 \[ \int \frac {\sqrt [8]{256-256 x^2+96 x^4-16 x^6+x^8}}{-1+x^3} \, dx=\frac {\sqrt [8]{\left (-4+x^2\right )^4} \left (\frac {2 \arctan \left (\frac {\sqrt {-4+x^2}}{\sqrt {3} (-2+x)}\right )}{\sqrt {3}}-\frac {1}{3} \sqrt {2 \left (-3-5 i \sqrt {3}\right )} \arctan \left (\frac {\sqrt {1-\frac {2 i}{\sqrt {3}}} \sqrt {-4+x^2}}{-2+x}\right )-\frac {1}{3} \sqrt {2 \left (-3+5 i \sqrt {3}\right )} \arctan \left (\frac {\sqrt {1+\frac {2 i}{\sqrt {3}}} \sqrt {-4+x^2}}{-2+x}\right )\right )}{\sqrt {-4+x^2}} \]
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Time = 0.44 (sec) , antiderivative size = 225, normalized size of antiderivative = 1.46, number of steps used = 17, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {2099, 1973, 749, 858, 222, 739, 212, 1034, 1094, 1051, 1045, 210} \[ \int \frac {\sqrt [8]{256-256 x^2+96 x^4-16 x^6+x^8}}{-1+x^3} \, dx=-\frac {\sqrt {3+2 \sqrt {21}} \sqrt [8]{\left (x^2-4\right )^4} \arctan \left (\frac {\left (21+4 \sqrt {21}\right ) x-\sqrt {21}+21}{\sqrt {21 \left (3+2 \sqrt {21}\right )} \sqrt {4-x^2}}\right )}{3 \sqrt {4-x^2}}-\frac {\sqrt [8]{\left (x^2-4\right )^4} \text {arctanh}\left (\frac {4-x}{\sqrt {3} \sqrt {4-x^2}}\right )}{\sqrt {3} \sqrt {4-x^2}}+\frac {\sqrt {2 \sqrt {21}-3} \sqrt [8]{\left (x^2-4\right )^4} \text {arctanh}\left (\frac {\left (21-4 \sqrt {21}\right ) x+\sqrt {21}+21}{\sqrt {21 \left (2 \sqrt {21}-3\right )} \sqrt {4-x^2}}\right )}{3 \sqrt {4-x^2}} \]
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Rule 210
Rule 212
Rule 222
Rule 739
Rule 749
Rule 858
Rule 1034
Rule 1045
Rule 1051
Rule 1094
Rule 1973
Rule 2099
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {\sqrt [8]{\left (-4+x^2\right )^4}}{3 (-1+x)}+\frac {(-2-x) \sqrt [8]{\left (-4+x^2\right )^4}}{3 \left (1+x+x^2\right )}\right ) \, dx \\ & = \frac {1}{3} \int \frac {\sqrt [8]{\left (-4+x^2\right )^4}}{-1+x} \, dx+\frac {1}{3} \int \frac {(-2-x) \sqrt [8]{\left (-4+x^2\right )^4}}{1+x+x^2} \, dx \\ & = \frac {\sqrt [8]{\left (-4+x^2\right )^4} \int \frac {\sqrt {1-\frac {x^2}{4}}}{-1+x} \, dx}{3 \sqrt {1-\frac {x^2}{4}}}+\frac {\sqrt [8]{\left (-4+x^2\right )^4} \int \frac {(-2-x) \sqrt {1-\frac {x^2}{4}}}{1+x+x^2} \, dx}{3 \sqrt {1-\frac {x^2}{4}}} \\ & = \frac {\sqrt [8]{\left (-4+x^2\right )^4} \int \frac {1-\frac {x}{4}}{(-1+x) \sqrt {1-\frac {x^2}{4}}} \, dx}{3 \sqrt {1-\frac {x^2}{4}}}+\frac {\sqrt [8]{\left (-4+x^2\right )^4} \int \frac {-2-\frac {5 x}{4}+\frac {x^2}{4}}{\sqrt {1-\frac {x^2}{4}} \left (1+x+x^2\right )} \, dx}{3 \sqrt {1-\frac {x^2}{4}}} \\ & = \frac {\sqrt [8]{\left (-4+x^2\right )^4} \int \frac {1}{(-1+x) \sqrt {1-\frac {x^2}{4}}} \, dx}{4 \sqrt {1-\frac {x^2}{4}}}+\frac {\sqrt [8]{\left (-4+x^2\right )^4} \int \frac {-\frac {9}{4}-\frac {3 x}{2}}{\sqrt {1-\frac {x^2}{4}} \left (1+x+x^2\right )} \, dx}{3 \sqrt {1-\frac {x^2}{4}}} \\ & = -\frac {\sqrt [8]{\left (-4+x^2\right )^4} \text {Subst}\left (\int \frac {1}{\frac {3}{4}-x^2} \, dx,x,\frac {1-\frac {x}{4}}{\sqrt {1-\frac {x^2}{4}}}\right )}{4 \sqrt {1-\frac {x^2}{4}}}-\frac {\left (2 \sqrt [8]{\left (-4+x^2\right )^4}\right ) \int \frac {\frac {3}{16} \left (7+3 \sqrt {21}\right )-\frac {3}{16} \left (7-2 \sqrt {21}\right ) x}{\sqrt {1-\frac {x^2}{4}} \left (1+x+x^2\right )} \, dx}{3 \sqrt {21} \sqrt {1-\frac {x^2}{4}}}+\frac {\left (2 \sqrt [8]{\left (-4+x^2\right )^4}\right ) \int \frac {\frac {3}{16} \left (7-3 \sqrt {21}\right )-\frac {3}{16} \left (7+2 \sqrt {21}\right ) x}{\sqrt {1-\frac {x^2}{4}} \left (1+x+x^2\right )} \, dx}{3 \sqrt {21} \sqrt {1-\frac {x^2}{4}}} \\ & = -\frac {\sqrt [8]{\left (-4+x^2\right )^4} \text {arctanh}\left (\frac {4-x}{\sqrt {3} \sqrt {4-x^2}}\right )}{\sqrt {3} \sqrt {4-x^2}}-\frac {\left (\sqrt {21} \left (3-2 \sqrt {21}\right ) \sqrt [8]{\left (-4+x^2\right )^4}\right ) \text {Subst}\left (\int \frac {1}{-\frac {189}{64} \left (3-2 \sqrt {21}\right )-x^2} \, dx,x,\frac {\frac {3}{16} \left (21+\sqrt {21}\right )+\frac {3}{16} \left (21-4 \sqrt {21}\right ) x}{\sqrt {1-\frac {x^2}{4}}}\right )}{16 \sqrt {1-\frac {x^2}{4}}}+\frac {\left (\sqrt {21} \left (3+2 \sqrt {21}\right ) \sqrt [8]{\left (-4+x^2\right )^4}\right ) \text {Subst}\left (\int \frac {1}{-\frac {189}{64} \left (3+2 \sqrt {21}\right )-x^2} \, dx,x,\frac {\frac {3}{16} \left (21-\sqrt {21}\right )+\frac {3}{16} \left (21+4 \sqrt {21}\right ) x}{\sqrt {1-\frac {x^2}{4}}}\right )}{16 \sqrt {1-\frac {x^2}{4}}} \\ & = -\frac {\sqrt {3+2 \sqrt {21}} \sqrt [8]{\left (-4+x^2\right )^4} \arctan \left (\frac {21-\sqrt {21}+\left (21+4 \sqrt {21}\right ) x}{\sqrt {21 \left (3+2 \sqrt {21}\right )} \sqrt {4-x^2}}\right )}{3 \sqrt {4-x^2}}-\frac {\sqrt [8]{\left (-4+x^2\right )^4} \text {arctanh}\left (\frac {4-x}{\sqrt {3} \sqrt {4-x^2}}\right )}{\sqrt {3} \sqrt {4-x^2}}+\frac {\sqrt {-3+2 \sqrt {21}} \sqrt [8]{\left (-4+x^2\right )^4} \text {arctanh}\left (\frac {21+\sqrt {21}+\left (21-4 \sqrt {21}\right ) x}{\sqrt {21 \left (-3+2 \sqrt {21}\right )} \sqrt {4-x^2}}\right )}{3 \sqrt {4-x^2}} \\ \end{align*}
Time = 0.64 (sec) , antiderivative size = 149, normalized size of antiderivative = 0.97 \[ \int \frac {\sqrt [8]{256-256 x^2+96 x^4-16 x^6+x^8}}{-1+x^3} \, dx=\frac {\sqrt [8]{\left (-4+x^2\right )^4} \left (2 \sqrt {3} \arctan \left (\frac {\sqrt {-4+x^2}}{\sqrt {3} (-2+x)}\right )-\sqrt {-6-10 i \sqrt {3}} \arctan \left (\frac {\sqrt {1-\frac {2 i}{\sqrt {3}}} \sqrt {-4+x^2}}{-2+x}\right )-\sqrt {-6+10 i \sqrt {3}} \arctan \left (\frac {\sqrt {1+\frac {2 i}{\sqrt {3}}} \sqrt {-4+x^2}}{-2+x}\right )\right )}{3 \sqrt {-4+x^2}} \]
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\[\int \frac {\left (x^{8}-16 x^{6}+96 x^{4}-256 x^{2}+256\right )^{\frac {1}{8}}}{x^{3}-1}d x\]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 295 vs. \(2 (112) = 224\).
Time = 0.31 (sec) , antiderivative size = 295, normalized size of antiderivative = 1.92 \[ \int \frac {\sqrt [8]{256-256 x^2+96 x^4-16 x^6+x^8}}{-1+x^3} \, dx=-\frac {2}{3} \, \sqrt {3} \arctan \left (-\frac {1}{3} \, \sqrt {3} {\left (x - 1\right )} + \frac {1}{3} \, \sqrt {3} {\left (x^{8} - 16 \, x^{6} + 96 \, x^{4} - 256 \, x^{2} + 256\right )}^{\frac {1}{8}}\right ) + \frac {1}{6} \, \sqrt {10 i \, \sqrt {3} + 6} \log \left (\sqrt {10 i \, \sqrt {3} + 6} {\left (i \, \sqrt {3} + 1\right )} - 4 \, x + 2 i \, \sqrt {3} + 4 \, {\left (x^{8} - 16 \, x^{6} + 96 \, x^{4} - 256 \, x^{2} + 256\right )}^{\frac {1}{8}} - 2\right ) - \frac {1}{6} \, \sqrt {10 i \, \sqrt {3} + 6} \log \left (\sqrt {10 i \, \sqrt {3} + 6} {\left (-i \, \sqrt {3} - 1\right )} - 4 \, x + 2 i \, \sqrt {3} + 4 \, {\left (x^{8} - 16 \, x^{6} + 96 \, x^{4} - 256 \, x^{2} + 256\right )}^{\frac {1}{8}} - 2\right ) - \frac {1}{6} \, \sqrt {-10 i \, \sqrt {3} + 6} \log \left ({\left (i \, \sqrt {3} - 1\right )} \sqrt {-10 i \, \sqrt {3} + 6} - 4 \, x - 2 i \, \sqrt {3} + 4 \, {\left (x^{8} - 16 \, x^{6} + 96 \, x^{4} - 256 \, x^{2} + 256\right )}^{\frac {1}{8}} - 2\right ) + \frac {1}{6} \, \sqrt {-10 i \, \sqrt {3} + 6} \log \left ({\left (-i \, \sqrt {3} + 1\right )} \sqrt {-10 i \, \sqrt {3} + 6} - 4 \, x - 2 i \, \sqrt {3} + 4 \, {\left (x^{8} - 16 \, x^{6} + 96 \, x^{4} - 256 \, x^{2} + 256\right )}^{\frac {1}{8}} - 2\right ) \]
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\[ \int \frac {\sqrt [8]{256-256 x^2+96 x^4-16 x^6+x^8}}{-1+x^3} \, dx=\int \frac {\sqrt [8]{\left (x - 2\right )^{4} \left (x + 2\right )^{4}}}{\left (x - 1\right ) \left (x^{2} + x + 1\right )}\, dx \]
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\[ \int \frac {\sqrt [8]{256-256 x^2+96 x^4-16 x^6+x^8}}{-1+x^3} \, dx=\int { \frac {{\left (x^{8} - 16 \, x^{6} + 96 \, x^{4} - 256 \, x^{2} + 256\right )}^{\frac {1}{8}}}{x^{3} - 1} \,d x } \]
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\[ \int \frac {\sqrt [8]{256-256 x^2+96 x^4-16 x^6+x^8}}{-1+x^3} \, dx=\int { \frac {{\left (x^{8} - 16 \, x^{6} + 96 \, x^{4} - 256 \, x^{2} + 256\right )}^{\frac {1}{8}}}{x^{3} - 1} \,d x } \]
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Timed out. \[ \int \frac {\sqrt [8]{256-256 x^2+96 x^4-16 x^6+x^8}}{-1+x^3} \, dx=\int \frac {{\left (x^8-16\,x^6+96\,x^4-256\,x^2+256\right )}^{1/8}}{x^3-1} \,d x \]
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