Integrand size = 25, antiderivative size = 289 \[ \int \frac {\left (-1+x^3\right )^{2/3} \left (4+x^6\right )}{x^6 \left (-4+x^6\right )} \, dx=\frac {\left (1-x^3\right ) \left (-1+x^3\right )^{2/3}}{5 x^5}-\frac {\arctan \left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{2} \sqrt [3]{-1+x^3}}\right )}{2\ 2^{2/3} \sqrt {3}}+\frac {\sqrt [6]{3} \arctan \left (\frac {3^{5/6} x}{\sqrt [3]{3} x+2 \sqrt [3]{2} \sqrt [3]{-1+x^3}}\right )}{2\ 2^{2/3}}+\frac {\log \left (-x+\sqrt [3]{2} \sqrt [3]{-1+x^3}\right )}{6\ 2^{2/3}}-\frac {\log \left (-3 x+\sqrt [3]{2} 3^{2/3} \sqrt [3]{-1+x^3}\right )}{2\ 2^{2/3} \sqrt [3]{3}}-\frac {\log \left (x^2+\sqrt [3]{2} x \sqrt [3]{-1+x^3}+2^{2/3} \left (-1+x^3\right )^{2/3}\right )}{12\ 2^{2/3}}+\frac {\log \left (3 x^2+\sqrt [3]{2} 3^{2/3} x \sqrt [3]{-1+x^3}+2^{2/3} \sqrt [3]{3} \left (-1+x^3\right )^{2/3}\right )}{4\ 2^{2/3} \sqrt [3]{3}} \]
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Time = 0.39 (sec) , antiderivative size = 197, normalized size of antiderivative = 0.68, number of steps used = 9, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {6857, 270, 399, 245, 384} \[ \int \frac {\left (-1+x^3\right )^{2/3} \left (4+x^6\right )}{x^6 \left (-4+x^6\right )} \, dx=-\frac {\arctan \left (\frac {\frac {2^{2/3} x}{\sqrt [3]{x^3-1}}+1}{\sqrt {3}}\right )}{2\ 2^{2/3} \sqrt {3}}+\frac {\sqrt [6]{3} \arctan \left (\frac {\frac {2^{2/3} \sqrt [3]{3} x}{\sqrt [3]{x^3-1}}+1}{\sqrt {3}}\right )}{2\ 2^{2/3}}-\frac {\log \left (x^3-2\right )}{12\ 2^{2/3}}+\frac {\log \left (x^3+2\right )}{4\ 2^{2/3} \sqrt [3]{3}}-\frac {1}{4} \left (\frac {3}{2}\right )^{2/3} \log \left (\sqrt [3]{\frac {3}{2}} x-\sqrt [3]{x^3-1}\right )+\frac {\log \left (\frac {x}{\sqrt [3]{2}}-\sqrt [3]{x^3-1}\right )}{4\ 2^{2/3}}-\frac {\left (x^3-1\right )^{5/3}}{5 x^5} \]
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Rule 245
Rule 270
Rule 384
Rule 399
Rule 6857
Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {\left (-1+x^3\right )^{2/3}}{x^6}+\frac {\left (-1+x^3\right )^{2/3}}{2 \left (-2+x^3\right )}-\frac {\left (-1+x^3\right )^{2/3}}{2 \left (2+x^3\right )}\right ) \, dx \\ & = \frac {1}{2} \int \frac {\left (-1+x^3\right )^{2/3}}{-2+x^3} \, dx-\frac {1}{2} \int \frac {\left (-1+x^3\right )^{2/3}}{2+x^3} \, dx-\int \frac {\left (-1+x^3\right )^{2/3}}{x^6} \, dx \\ & = -\frac {\left (-1+x^3\right )^{5/3}}{5 x^5}+\frac {1}{2} \int \frac {1}{\left (-2+x^3\right ) \sqrt [3]{-1+x^3}} \, dx+\frac {3}{2} \int \frac {1}{\sqrt [3]{-1+x^3} \left (2+x^3\right )} \, dx \\ & = -\frac {\left (-1+x^3\right )^{5/3}}{5 x^5}-\frac {\arctan \left (\frac {1+\frac {2^{2/3} x}{\sqrt [3]{-1+x^3}}}{\sqrt {3}}\right )}{2\ 2^{2/3} \sqrt {3}}+\frac {\sqrt [6]{3} \arctan \left (\frac {1+\frac {2^{2/3} \sqrt [3]{3} x}{\sqrt [3]{-1+x^3}}}{\sqrt {3}}\right )}{2\ 2^{2/3}}-\frac {\log \left (-2+x^3\right )}{12\ 2^{2/3}}+\frac {\log \left (2+x^3\right )}{4\ 2^{2/3} \sqrt [3]{3}}-\frac {1}{4} \left (\frac {3}{2}\right )^{2/3} \log \left (\sqrt [3]{\frac {3}{2}} x-\sqrt [3]{-1+x^3}\right )+\frac {\log \left (\frac {x}{\sqrt [3]{2}}-\sqrt [3]{-1+x^3}\right )}{4\ 2^{2/3}} \\ \end{align*}
Time = 0.62 (sec) , antiderivative size = 272, normalized size of antiderivative = 0.94 \[ \int \frac {\left (-1+x^3\right )^{2/3} \left (4+x^6\right )}{x^6 \left (-4+x^6\right )} \, dx=\frac {1}{120} \left (-\frac {24 \left (-1+x^3\right )^{5/3}}{x^5}-10 \sqrt [3]{2} \sqrt {3} \arctan \left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{2} \sqrt [3]{-1+x^3}}\right )+30 \sqrt [3]{2} \sqrt [6]{3} \arctan \left (\frac {3^{5/6} x}{\sqrt [3]{3} x+2 \sqrt [3]{2} \sqrt [3]{-1+x^3}}\right )+10 \sqrt [3]{2} \log \left (-x+\sqrt [3]{2} \sqrt [3]{-1+x^3}\right )-10 \sqrt [3]{2} 3^{2/3} \log \left (-3 x+\sqrt [3]{2} 3^{2/3} \sqrt [3]{-1+x^3}\right )-5 \sqrt [3]{2} \log \left (x^2+\sqrt [3]{2} x \sqrt [3]{-1+x^3}+2^{2/3} \left (-1+x^3\right )^{2/3}\right )+5 \sqrt [3]{2} 3^{2/3} \log \left (3 x^2+\sqrt [3]{2} 3^{2/3} x \sqrt [3]{-1+x^3}+2^{2/3} \sqrt [3]{3} \left (-1+x^3\right )^{2/3}\right )\right ) \]
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Time = 4.47 (sec) , antiderivative size = 258, normalized size of antiderivative = 0.89
method | result | size |
pseudoelliptic | \(\frac {\left (-24 x^{3}+24\right ) \left (x^{3}-1\right )^{\frac {2}{3}}-5 \,2^{\frac {1}{3}} x^{5} \left (\left (2 \ln \left (\frac {-2^{\frac {2}{3}} 3^{\frac {1}{3}} x +2 {\left (\left (-1+x \right ) \left (x^{2}+x +1\right )\right )}^{\frac {1}{3}}}{x}\right )-\ln \left (\frac {2^{\frac {2}{3}} 3^{\frac {1}{3}} {\left (\left (-1+x \right ) \left (x^{2}+x +1\right )\right )}^{\frac {1}{3}} x +2^{\frac {1}{3}} 3^{\frac {2}{3}} x^{2}+2 {\left (\left (-1+x \right ) \left (x^{2}+x +1\right )\right )}^{\frac {2}{3}}}{x^{2}}\right )-\ln \left (2\right )\right ) 3^{\frac {2}{3}}+6 \arctan \left (\frac {\sqrt {3}\, \left (2 \,2^{\frac {1}{3}} 3^{\frac {2}{3}} \left (x^{3}-1\right )^{\frac {1}{3}}+3 x \right )}{9 x}\right ) 3^{\frac {1}{6}}-2 \arctan \left (\frac {\sqrt {3}\, \left (x +2 \,2^{\frac {1}{3}} \left (x^{3}-1\right )^{\frac {1}{3}}\right )}{3 x}\right ) \sqrt {3}+\ln \left (\frac {2^{\frac {2}{3}} x {\left (\left (-1+x \right ) \left (x^{2}+x +1\right )\right )}^{\frac {1}{3}}+2^{\frac {1}{3}} x^{2}+2 {\left (\left (-1+x \right ) \left (x^{2}+x +1\right )\right )}^{\frac {2}{3}}}{x^{2}}\right )-2 \ln \left (\frac {-2^{\frac {2}{3}} x +2 {\left (\left (-1+x \right ) \left (x^{2}+x +1\right )\right )}^{\frac {1}{3}}}{x}\right )+\ln \left (2\right )\right )}{120 x^{5}}\) | \(258\) |
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Leaf count of result is larger than twice the leaf count of optimal. 534 vs. \(2 (202) = 404\).
Time = 3.66 (sec) , antiderivative size = 534, normalized size of antiderivative = 1.85 \[ \int \frac {\left (-1+x^3\right )^{2/3} \left (4+x^6\right )}{x^6 \left (-4+x^6\right )} \, dx=\frac {10 \cdot 12^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} x^{5} \log \left (-\frac {18 \cdot 12^{\frac {1}{3}} \left (-1\right )^{\frac {2}{3}} {\left (x^{3} - 1\right )}^{\frac {1}{3}} x^{2} + 12^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} {\left (x^{3} + 2\right )} - 36 \, {\left (x^{3} - 1\right )}^{\frac {2}{3}} x}{x^{3} + 2}\right ) - 5 \cdot 12^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} x^{5} \log \left (-\frac {6 \cdot 12^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} {\left (4 \, x^{4} - x\right )} {\left (x^{3} - 1\right )}^{\frac {2}{3}} - 12^{\frac {1}{3}} \left (-1\right )^{\frac {2}{3}} {\left (55 \, x^{6} - 50 \, x^{3} + 4\right )} - 18 \, {\left (7 \, x^{5} - 4 \, x^{2}\right )} {\left (x^{3} - 1\right )}^{\frac {1}{3}}}{x^{6} + 4 \, x^{3} + 4}\right ) + 20 \cdot 4^{\frac {1}{6}} \sqrt {3} x^{5} \arctan \left (\frac {4^{\frac {1}{6}} {\left (12 \cdot 4^{\frac {2}{3}} \sqrt {3} {\left (2 \, x^{7} - 5 \, x^{4} + 2 \, x\right )} {\left (x^{3} - 1\right )}^{\frac {2}{3}} + 4^{\frac {1}{3}} \sqrt {3} {\left (91 \, x^{9} - 168 \, x^{6} + 84 \, x^{3} - 8\right )} + 12 \, \sqrt {3} {\left (19 \, x^{8} - 22 \, x^{5} + 4 \, x^{2}\right )} {\left (x^{3} - 1\right )}^{\frac {1}{3}}\right )}}{6 \, {\left (53 \, x^{9} - 48 \, x^{6} - 12 \, x^{3} + 8\right )}}\right ) + 10 \cdot 4^{\frac {2}{3}} x^{5} \log \left (\frac {6 \cdot 4^{\frac {1}{3}} {\left (x^{3} - 1\right )}^{\frac {1}{3}} x^{2} + 4^{\frac {2}{3}} {\left (x^{3} - 2\right )} - 12 \, {\left (x^{3} - 1\right )}^{\frac {2}{3}} x}{x^{3} - 2}\right ) - 5 \cdot 4^{\frac {2}{3}} x^{5} \log \left (\frac {6 \cdot 4^{\frac {2}{3}} {\left (2 \, x^{4} - x\right )} {\left (x^{3} - 1\right )}^{\frac {2}{3}} + 4^{\frac {1}{3}} {\left (19 \, x^{6} - 22 \, x^{3} + 4\right )} + 6 \, {\left (5 \, x^{5} - 4 \, x^{2}\right )} {\left (x^{3} - 1\right )}^{\frac {1}{3}}}{x^{6} - 4 \, x^{3} + 4}\right ) - 60 \cdot 12^{\frac {1}{6}} \left (-1\right )^{\frac {1}{3}} x^{5} \arctan \left (\frac {12^{\frac {1}{6}} {\left (12 \cdot 12^{\frac {2}{3}} \left (-1\right )^{\frac {2}{3}} {\left (4 \, x^{7} + 7 \, x^{4} - 2 \, x\right )} {\left (x^{3} - 1\right )}^{\frac {2}{3}} + 36 \, \left (-1\right )^{\frac {1}{3}} {\left (55 \, x^{8} - 50 \, x^{5} + 4 \, x^{2}\right )} {\left (x^{3} - 1\right )}^{\frac {1}{3}} - 12^{\frac {1}{3}} {\left (377 \, x^{9} - 600 \, x^{6} + 204 \, x^{3} - 8\right )}\right )}}{6 \, {\left (487 \, x^{9} - 480 \, x^{6} + 12 \, x^{3} + 8\right )}}\right ) - 144 \, {\left (x^{3} - 1\right )}^{\frac {5}{3}}}{720 \, x^{5}} \]
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\[ \int \frac {\left (-1+x^3\right )^{2/3} \left (4+x^6\right )}{x^6 \left (-4+x^6\right )} \, dx=\int \frac {\left (\left (x - 1\right ) \left (x^{2} + x + 1\right )\right )^{\frac {2}{3}} \left (x^{6} + 4\right )}{x^{6} \left (x^{3} - 2\right ) \left (x^{3} + 2\right )}\, dx \]
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\[ \int \frac {\left (-1+x^3\right )^{2/3} \left (4+x^6\right )}{x^6 \left (-4+x^6\right )} \, dx=\int { \frac {{\left (x^{6} + 4\right )} {\left (x^{3} - 1\right )}^{\frac {2}{3}}}{{\left (x^{6} - 4\right )} x^{6}} \,d x } \]
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\[ \int \frac {\left (-1+x^3\right )^{2/3} \left (4+x^6\right )}{x^6 \left (-4+x^6\right )} \, dx=\int { \frac {{\left (x^{6} + 4\right )} {\left (x^{3} - 1\right )}^{\frac {2}{3}}}{{\left (x^{6} - 4\right )} x^{6}} \,d x } \]
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Timed out. \[ \int \frac {\left (-1+x^3\right )^{2/3} \left (4+x^6\right )}{x^6 \left (-4+x^6\right )} \, dx=\int \frac {{\left (x^3-1\right )}^{2/3}\,\left (x^6+4\right )}{x^6\,\left (x^6-4\right )} \,d x \]
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