Integrand size = 25, antiderivative size = 173 \[ \int \frac {\left (-1+x^3\right )^{2/3} \left (1+x^3\right )}{x^6 \left (2+x^3\right )} \, dx=\frac {\left (-4-x^3\right ) \left (-1+x^3\right )^{2/3}}{40 x^5}+\frac {\sqrt [6]{3} \arctan \left (\frac {3^{5/6} x}{\sqrt [3]{3} x+2 \sqrt [3]{2} \sqrt [3]{-1+x^3}}\right )}{4\ 2^{2/3}}-\frac {\log \left (-3 x+\sqrt [3]{2} 3^{2/3} \sqrt [3]{-1+x^3}\right )}{4\ 2^{2/3} \sqrt [3]{3}}+\frac {\log \left (3 x^2+\sqrt [3]{2} 3^{2/3} x \sqrt [3]{-1+x^3}+2^{2/3} \sqrt [3]{3} \left (-1+x^3\right )^{2/3}\right )}{8\ 2^{2/3} \sqrt [3]{3}} \]
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Time = 0.06 (sec) , antiderivative size = 130, normalized size of antiderivative = 0.75, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {594, 597, 12, 384} \[ \int \frac {\left (-1+x^3\right )^{2/3} \left (1+x^3\right )}{x^6 \left (2+x^3\right )} \, dx=\frac {\sqrt [6]{3} \arctan \left (\frac {\frac {2^{2/3} \sqrt [3]{3} x}{\sqrt [3]{x^3-1}}+1}{\sqrt {3}}\right )}{4\ 2^{2/3}}+\frac {\log \left (x^3+2\right )}{8\ 2^{2/3} \sqrt [3]{3}}-\frac {1}{8} \left (\frac {3}{2}\right )^{2/3} \log \left (\sqrt [3]{\frac {3}{2}} x-\sqrt [3]{x^3-1}\right )-\frac {\left (x^3-1\right )^{2/3}}{10 x^5}-\frac {\left (x^3-1\right )^{2/3}}{40 x^2} \]
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Rule 12
Rule 384
Rule 594
Rule 597
Rubi steps \begin{align*} \text {integral}& = -\frac {\left (-1+x^3\right )^{2/3}}{10 x^5}+\frac {1}{10} \int \frac {-1+7 x^3}{x^3 \sqrt [3]{-1+x^3} \left (2+x^3\right )} \, dx \\ & = -\frac {\left (-1+x^3\right )^{2/3}}{10 x^5}-\frac {\left (-1+x^3\right )^{2/3}}{40 x^2}+\frac {1}{40} \int \frac {30}{\sqrt [3]{-1+x^3} \left (2+x^3\right )} \, dx \\ & = -\frac {\left (-1+x^3\right )^{2/3}}{10 x^5}-\frac {\left (-1+x^3\right )^{2/3}}{40 x^2}+\frac {3}{4} \int \frac {1}{\sqrt [3]{-1+x^3} \left (2+x^3\right )} \, dx \\ & = -\frac {\left (-1+x^3\right )^{2/3}}{10 x^5}-\frac {\left (-1+x^3\right )^{2/3}}{40 x^2}+\frac {\sqrt [6]{3} \arctan \left (\frac {1+\frac {2^{2/3} \sqrt [3]{3} x}{\sqrt [3]{-1+x^3}}}{\sqrt {3}}\right )}{4\ 2^{2/3}}+\frac {\log \left (2+x^3\right )}{8\ 2^{2/3} \sqrt [3]{3}}-\frac {1}{8} \left (\frac {3}{2}\right )^{2/3} \log \left (\sqrt [3]{\frac {3}{2}} x-\sqrt [3]{-1+x^3}\right ) \\ \end{align*}
Time = 0.38 (sec) , antiderivative size = 167, normalized size of antiderivative = 0.97 \[ \int \frac {\left (-1+x^3\right )^{2/3} \left (1+x^3\right )}{x^6 \left (2+x^3\right )} \, dx=\frac {1}{240} \left (-\frac {6 \left (-1+x^3\right )^{2/3} \left (4+x^3\right )}{x^5}+30 \sqrt [3]{2} \sqrt [6]{3} \arctan \left (\frac {3^{5/6} x}{\sqrt [3]{3} x+2 \sqrt [3]{2} \sqrt [3]{-1+x^3}}\right )-10 \sqrt [3]{2} 3^{2/3} \log \left (-3 x+\sqrt [3]{2} 3^{2/3} \sqrt [3]{-1+x^3}\right )+5 \sqrt [3]{2} 3^{2/3} \log \left (3 x^2+\sqrt [3]{2} 3^{2/3} x \sqrt [3]{-1+x^3}+2^{2/3} \sqrt [3]{3} \left (-1+x^3\right )^{2/3}\right )\right ) \]
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Time = 14.58 (sec) , antiderivative size = 152, normalized size of antiderivative = 0.88
method | result | size |
pseudoelliptic | \(\frac {\left (-6 x^{3}-24\right ) \left (x^{3}-1\right )^{\frac {2}{3}}+5 \left (\left (\ln \left (\frac {2^{\frac {1}{3}} 3^{\frac {2}{3}} x^{2}+2^{\frac {2}{3}} 3^{\frac {1}{3}} {\left (\left (-1+x \right ) \left (x^{2}+x +1\right )\right )}^{\frac {1}{3}} x +2 {\left (\left (-1+x \right ) \left (x^{2}+x +1\right )\right )}^{\frac {2}{3}}}{x^{2}}\right )-2 \ln \left (\frac {-2^{\frac {2}{3}} 3^{\frac {1}{3}} x +2 {\left (\left (-1+x \right ) \left (x^{2}+x +1\right )\right )}^{\frac {1}{3}}}{x}\right )+\ln \left (2\right )\right ) 3^{\frac {2}{3}}-6 \arctan \left (\frac {\sqrt {3}\, \left (2 \,2^{\frac {1}{3}} 3^{\frac {2}{3}} \left (x^{3}-1\right )^{\frac {1}{3}}+3 x \right )}{9 x}\right ) 3^{\frac {1}{6}}\right ) 2^{\frac {1}{3}} x^{5}}{240 x^{5}}\) | \(152\) |
risch | \(\text {Expression too large to display}\) | \(607\) |
trager | \(\text {Expression too large to display}\) | \(1096\) |
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Leaf count of result is larger than twice the leaf count of optimal. 289 vs. \(2 (123) = 246\).
Time = 2.03 (sec) , antiderivative size = 289, normalized size of antiderivative = 1.67 \[ \int \frac {\left (-1+x^3\right )^{2/3} \left (1+x^3\right )}{x^6 \left (2+x^3\right )} \, dx=\frac {10 \cdot 12^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} x^{5} \log \left (-\frac {18 \cdot 12^{\frac {1}{3}} \left (-1\right )^{\frac {2}{3}} {\left (x^{3} - 1\right )}^{\frac {1}{3}} x^{2} + 12^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} {\left (x^{3} + 2\right )} - 36 \, {\left (x^{3} - 1\right )}^{\frac {2}{3}} x}{x^{3} + 2}\right ) - 5 \cdot 12^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} x^{5} \log \left (-\frac {6 \cdot 12^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} {\left (4 \, x^{4} - x\right )} {\left (x^{3} - 1\right )}^{\frac {2}{3}} - 12^{\frac {1}{3}} \left (-1\right )^{\frac {2}{3}} {\left (55 \, x^{6} - 50 \, x^{3} + 4\right )} - 18 \, {\left (7 \, x^{5} - 4 \, x^{2}\right )} {\left (x^{3} - 1\right )}^{\frac {1}{3}}}{x^{6} + 4 \, x^{3} + 4}\right ) - 60 \cdot 12^{\frac {1}{6}} \left (-1\right )^{\frac {1}{3}} x^{5} \arctan \left (\frac {12^{\frac {1}{6}} {\left (12 \cdot 12^{\frac {2}{3}} \left (-1\right )^{\frac {2}{3}} {\left (4 \, x^{7} + 7 \, x^{4} - 2 \, x\right )} {\left (x^{3} - 1\right )}^{\frac {2}{3}} + 36 \, \left (-1\right )^{\frac {1}{3}} {\left (55 \, x^{8} - 50 \, x^{5} + 4 \, x^{2}\right )} {\left (x^{3} - 1\right )}^{\frac {1}{3}} - 12^{\frac {1}{3}} {\left (377 \, x^{9} - 600 \, x^{6} + 204 \, x^{3} - 8\right )}\right )}}{6 \, {\left (487 \, x^{9} - 480 \, x^{6} + 12 \, x^{3} + 8\right )}}\right ) - 36 \, {\left (x^{3} + 4\right )} {\left (x^{3} - 1\right )}^{\frac {2}{3}}}{1440 \, x^{5}} \]
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\[ \int \frac {\left (-1+x^3\right )^{2/3} \left (1+x^3\right )}{x^6 \left (2+x^3\right )} \, dx=\int \frac {\left (\left (x - 1\right ) \left (x^{2} + x + 1\right )\right )^{\frac {2}{3}} \left (x + 1\right ) \left (x^{2} - x + 1\right )}{x^{6} \left (x^{3} + 2\right )}\, dx \]
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\[ \int \frac {\left (-1+x^3\right )^{2/3} \left (1+x^3\right )}{x^6 \left (2+x^3\right )} \, dx=\int { \frac {{\left (x^{3} + 1\right )} {\left (x^{3} - 1\right )}^{\frac {2}{3}}}{{\left (x^{3} + 2\right )} x^{6}} \,d x } \]
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\[ \int \frac {\left (-1+x^3\right )^{2/3} \left (1+x^3\right )}{x^6 \left (2+x^3\right )} \, dx=\int { \frac {{\left (x^{3} + 1\right )} {\left (x^{3} - 1\right )}^{\frac {2}{3}}}{{\left (x^{3} + 2\right )} x^{6}} \,d x } \]
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Timed out. \[ \int \frac {\left (-1+x^3\right )^{2/3} \left (1+x^3\right )}{x^6 \left (2+x^3\right )} \, dx=\int \frac {{\left (x^3-1\right )}^{2/3}\,\left (x^3+1\right )}{x^6\,\left (x^3+2\right )} \,d x \]
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