Integrand size = 30, antiderivative size = 143 \[ \int \frac {\left (-1+x^3\right )^{2/3} \left (4+4 x^3+x^6\right )}{x^9 \left (1+x^3\right )} \, dx=\frac {\left (-1+x^3\right )^{2/3} \left (-5+2 x^3-2 x^6\right )}{10 x^8}+\frac {2^{2/3} \arctan \left (\frac {\sqrt {3} x}{x+2^{2/3} \sqrt [3]{-1+x^3}}\right )}{\sqrt {3}}-\frac {1}{3} 2^{2/3} \log \left (-2 x+2^{2/3} \sqrt [3]{-1+x^3}\right )+\frac {\log \left (2 x^2+2^{2/3} x \sqrt [3]{-1+x^3}+\sqrt [3]{2} \left (-1+x^3\right )^{2/3}\right )}{3 \sqrt [3]{2}} \]
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Time = 0.19 (sec) , antiderivative size = 213, normalized size of antiderivative = 1.49, number of steps used = 11, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {28, 600, 594, 597, 12, 384} \[ \int \frac {\left (-1+x^3\right )^{2/3} \left (4+4 x^3+x^6\right )}{x^9 \left (1+x^3\right )} \, dx=\frac {2\ 2^{2/3} \arctan \left (\frac {\frac {2 \sqrt [3]{2} x}{\sqrt [3]{x^3-1}}+1}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {\arctan \left (\frac {\frac {2 \sqrt [3]{2} x}{\sqrt [3]{x^3-1}}+1}{\sqrt {3}}\right )}{4 \sqrt [3]{2} \sqrt {3}}+\frac {1}{3} 2^{2/3} \log \left (x^3+1\right )-\frac {\log \left (x^3+1\right )}{24 \sqrt [3]{2}}-2^{2/3} \log \left (\sqrt [3]{2} x-\sqrt [3]{x^3-1}\right )+\frac {\log \left (\sqrt [3]{2} x-\sqrt [3]{x^3-1}\right )}{8 \sqrt [3]{2}}-\frac {\left (x^3-1\right )^{2/3}}{2 x^8}+\frac {11 \left (x^3-1\right )^{2/3}}{20 x^5}-\frac {79 \left (x^3-1\right )^{2/3}}{80 x^2} \]
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Rule 12
Rule 28
Rule 384
Rule 594
Rule 597
Rule 600
Rubi steps \begin{align*} \text {integral}& = \int \frac {\left (-1+x^3\right )^{2/3} \left (2+x^3\right )^2}{x^9 \left (1+x^3\right )} \, dx \\ & = \frac {1}{8} \int \frac {\left (-1+x^3\right )^{2/3} \left (2+x^3\right )}{x^6 \left (1+x^3\right )} \, dx+2 \int \frac {\left (-1+x^3\right )^{2/3} \left (2+x^3\right )}{x^9 \left (1+x^3\right )} \, dx \\ & = -\frac {\left (-1+x^3\right )^{2/3}}{2 x^8}-\frac {\left (-1+x^3\right )^{2/3}}{20 x^5}+\frac {1}{40} \int \frac {9-x^3}{x^3 \sqrt [3]{-1+x^3} \left (1+x^3\right )} \, dx+\frac {1}{4} \int \frac {12-4 x^3}{x^6 \sqrt [3]{-1+x^3} \left (1+x^3\right )} \, dx \\ & = -\frac {\left (-1+x^3\right )^{2/3}}{2 x^8}+\frac {11 \left (-1+x^3\right )^{2/3}}{20 x^5}+\frac {9 \left (-1+x^3\right )^{2/3}}{80 x^2}+\frac {1}{80} \int -\frac {20}{\sqrt [3]{-1+x^3} \left (1+x^3\right )} \, dx+\frac {1}{20} \int \frac {-44+36 x^3}{x^3 \sqrt [3]{-1+x^3} \left (1+x^3\right )} \, dx \\ & = -\frac {\left (-1+x^3\right )^{2/3}}{2 x^8}+\frac {11 \left (-1+x^3\right )^{2/3}}{20 x^5}-\frac {79 \left (-1+x^3\right )^{2/3}}{80 x^2}+\frac {1}{40} \int \frac {160}{\sqrt [3]{-1+x^3} \left (1+x^3\right )} \, dx-\frac {1}{4} \int \frac {1}{\sqrt [3]{-1+x^3} \left (1+x^3\right )} \, dx \\ & = -\frac {\left (-1+x^3\right )^{2/3}}{2 x^8}+\frac {11 \left (-1+x^3\right )^{2/3}}{20 x^5}-\frac {79 \left (-1+x^3\right )^{2/3}}{80 x^2}-\frac {\arctan \left (\frac {1+\frac {2 \sqrt [3]{2} x}{\sqrt [3]{-1+x^3}}}{\sqrt {3}}\right )}{4 \sqrt [3]{2} \sqrt {3}}-\frac {\log \left (1+x^3\right )}{24 \sqrt [3]{2}}+\frac {\log \left (\sqrt [3]{2} x-\sqrt [3]{-1+x^3}\right )}{8 \sqrt [3]{2}}+4 \int \frac {1}{\sqrt [3]{-1+x^3} \left (1+x^3\right )} \, dx \\ & = -\frac {\left (-1+x^3\right )^{2/3}}{2 x^8}+\frac {11 \left (-1+x^3\right )^{2/3}}{20 x^5}-\frac {79 \left (-1+x^3\right )^{2/3}}{80 x^2}-\frac {\arctan \left (\frac {1+\frac {2 \sqrt [3]{2} x}{\sqrt [3]{-1+x^3}}}{\sqrt {3}}\right )}{4 \sqrt [3]{2} \sqrt {3}}+\frac {2\ 2^{2/3} \arctan \left (\frac {1+\frac {2 \sqrt [3]{2} x}{\sqrt [3]{-1+x^3}}}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {\log \left (1+x^3\right )}{24 \sqrt [3]{2}}+\frac {1}{3} 2^{2/3} \log \left (1+x^3\right )+\frac {\log \left (\sqrt [3]{2} x-\sqrt [3]{-1+x^3}\right )}{8 \sqrt [3]{2}}-2^{2/3} \log \left (\sqrt [3]{2} x-\sqrt [3]{-1+x^3}\right ) \\ \end{align*}
Time = 0.35 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.00 \[ \int \frac {\left (-1+x^3\right )^{2/3} \left (4+4 x^3+x^6\right )}{x^9 \left (1+x^3\right )} \, dx=\frac {\left (-1+x^3\right )^{2/3} \left (-5+2 x^3-2 x^6\right )}{10 x^8}+\frac {2^{2/3} \arctan \left (\frac {\sqrt {3} x}{x+2^{2/3} \sqrt [3]{-1+x^3}}\right )}{\sqrt {3}}-\frac {1}{3} 2^{2/3} \log \left (-2 x+2^{2/3} \sqrt [3]{-1+x^3}\right )+\frac {\log \left (2 x^2+2^{2/3} x \sqrt [3]{-1+x^3}+\sqrt [3]{2} \left (-1+x^3\right )^{2/3}\right )}{3 \sqrt [3]{2}} \]
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Time = 13.60 (sec) , antiderivative size = 119, normalized size of antiderivative = 0.83
method | result | size |
pseudoelliptic | \(\frac {-5 x^{8} \left (2 \arctan \left (\frac {\sqrt {3}\, \left (x +2^{\frac {2}{3}} \left (x^{3}-1\right )^{\frac {1}{3}}\right )}{3 x}\right ) \sqrt {3}+2 \ln \left (\frac {-2^{\frac {1}{3}} x +\left (x^{3}-1\right )^{\frac {1}{3}}}{x}\right )-\ln \left (\frac {2^{\frac {2}{3}} x^{2}+2^{\frac {1}{3}} x \left (x^{3}-1\right )^{\frac {1}{3}}+\left (x^{3}-1\right )^{\frac {2}{3}}}{x^{2}}\right )\right ) 2^{\frac {2}{3}}-3 \left (x^{3}-1\right )^{\frac {2}{3}} \left (2 x^{6}-2 x^{3}+5\right )}{30 x^{8}}\) | \(119\) |
risch | \(\text {Expression too large to display}\) | \(913\) |
trager | \(\text {Expression too large to display}\) | \(1125\) |
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Leaf count of result is larger than twice the leaf count of optimal. 276 vs. \(2 (110) = 220\).
Time = 1.74 (sec) , antiderivative size = 276, normalized size of antiderivative = 1.93 \[ \int \frac {\left (-1+x^3\right )^{2/3} \left (4+4 x^3+x^6\right )}{x^9 \left (1+x^3\right )} \, dx=-\frac {10 \, \sqrt {3} \left (-4\right )^{\frac {1}{3}} x^{8} \arctan \left (\frac {3 \, \sqrt {3} \left (-4\right )^{\frac {2}{3}} {\left (5 \, x^{7} + 4 \, x^{4} - x\right )} {\left (x^{3} - 1\right )}^{\frac {2}{3}} + 6 \, \sqrt {3} \left (-4\right )^{\frac {1}{3}} {\left (19 \, x^{8} - 16 \, x^{5} + x^{2}\right )} {\left (x^{3} - 1\right )}^{\frac {1}{3}} - \sqrt {3} {\left (71 \, x^{9} - 111 \, x^{6} + 33 \, x^{3} - 1\right )}}{3 \, {\left (109 \, x^{9} - 105 \, x^{6} + 3 \, x^{3} + 1\right )}}\right ) - 10 \, \left (-4\right )^{\frac {1}{3}} x^{8} \log \left (-\frac {3 \, \left (-4\right )^{\frac {2}{3}} {\left (x^{3} - 1\right )}^{\frac {1}{3}} x^{2} - 6 \, {\left (x^{3} - 1\right )}^{\frac {2}{3}} x + \left (-4\right )^{\frac {1}{3}} {\left (x^{3} + 1\right )}}{x^{3} + 1}\right ) + 5 \, \left (-4\right )^{\frac {1}{3}} x^{8} \log \left (-\frac {6 \, \left (-4\right )^{\frac {1}{3}} {\left (5 \, x^{4} - x\right )} {\left (x^{3} - 1\right )}^{\frac {2}{3}} - \left (-4\right )^{\frac {2}{3}} {\left (19 \, x^{6} - 16 \, x^{3} + 1\right )} - 24 \, {\left (2 \, x^{5} - x^{2}\right )} {\left (x^{3} - 1\right )}^{\frac {1}{3}}}{x^{6} + 2 \, x^{3} + 1}\right ) + 9 \, {\left (2 \, x^{6} - 2 \, x^{3} + 5\right )} {\left (x^{3} - 1\right )}^{\frac {2}{3}}}{90 \, x^{8}} \]
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\[ \int \frac {\left (-1+x^3\right )^{2/3} \left (4+4 x^3+x^6\right )}{x^9 \left (1+x^3\right )} \, dx=\int \frac {\left (\left (x - 1\right ) \left (x^{2} + x + 1\right )\right )^{\frac {2}{3}} \left (x^{3} + 2\right )^{2}}{x^{9} \left (x + 1\right ) \left (x^{2} - x + 1\right )}\, dx \]
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\[ \int \frac {\left (-1+x^3\right )^{2/3} \left (4+4 x^3+x^6\right )}{x^9 \left (1+x^3\right )} \, dx=\int { \frac {{\left (x^{6} + 4 \, x^{3} + 4\right )} {\left (x^{3} - 1\right )}^{\frac {2}{3}}}{{\left (x^{3} + 1\right )} x^{9}} \,d x } \]
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\[ \int \frac {\left (-1+x^3\right )^{2/3} \left (4+4 x^3+x^6\right )}{x^9 \left (1+x^3\right )} \, dx=\int { \frac {{\left (x^{6} + 4 \, x^{3} + 4\right )} {\left (x^{3} - 1\right )}^{\frac {2}{3}}}{{\left (x^{3} + 1\right )} x^{9}} \,d x } \]
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Timed out. \[ \int \frac {\left (-1+x^3\right )^{2/3} \left (4+4 x^3+x^6\right )}{x^9 \left (1+x^3\right )} \, dx=\int \frac {{\left (x^3-1\right )}^{2/3}\,\left (x^6+4\,x^3+4\right )}{x^9\,\left (x^3+1\right )} \,d x \]
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