Integrand size = 27, antiderivative size = 134 \[ \int \frac {\left (-2+x^3\right ) \left (1+x^3\right )^{2/3}}{x^6 \left (-1+2 x^3\right )} \, dx=\frac {\left (-4-19 x^3\right ) \left (1+x^3\right )^{2/3}}{10 x^5}+3 \sqrt [6]{3} \arctan \left (\frac {3^{5/6} x}{\sqrt [3]{3} x+2 \sqrt [3]{1+x^3}}\right )-3^{2/3} \log \left (-3 x+3^{2/3} \sqrt [3]{1+x^3}\right )+\frac {1}{2} 3^{2/3} \log \left (3 x^2+3^{2/3} x \sqrt [3]{1+x^3}+\sqrt [3]{3} \left (1+x^3\right )^{2/3}\right ) \]
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Time = 0.06 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.84, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {594, 597, 12, 384} \[ \int \frac {\left (-2+x^3\right ) \left (1+x^3\right )^{2/3}}{x^6 \left (-1+2 x^3\right )} \, dx=3 \sqrt [6]{3} \arctan \left (\frac {\frac {2 \sqrt [3]{3} x}{\sqrt [3]{x^3+1}}+1}{\sqrt {3}}\right )+\frac {1}{2} 3^{2/3} \log \left (2 x^3-1\right )-\frac {3}{2} 3^{2/3} \log \left (\sqrt [3]{3} x-\sqrt [3]{x^3+1}\right )-\frac {2 \left (x^3+1\right )^{2/3}}{5 x^5}-\frac {19 \left (x^3+1\right )^{2/3}}{10 x^2} \]
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Rule 12
Rule 384
Rule 594
Rule 597
Rubi steps \begin{align*} \text {integral}& = -\frac {2 \left (1+x^3\right )^{2/3}}{5 x^5}-\frac {1}{5} \int \frac {19+7 x^3}{x^3 \sqrt [3]{1+x^3} \left (-1+2 x^3\right )} \, dx \\ & = -\frac {2 \left (1+x^3\right )^{2/3}}{5 x^5}-\frac {19 \left (1+x^3\right )^{2/3}}{10 x^2}-\frac {1}{10} \int \frac {90}{\sqrt [3]{1+x^3} \left (-1+2 x^3\right )} \, dx \\ & = -\frac {2 \left (1+x^3\right )^{2/3}}{5 x^5}-\frac {19 \left (1+x^3\right )^{2/3}}{10 x^2}-9 \int \frac {1}{\sqrt [3]{1+x^3} \left (-1+2 x^3\right )} \, dx \\ & = -\frac {2 \left (1+x^3\right )^{2/3}}{5 x^5}-\frac {19 \left (1+x^3\right )^{2/3}}{10 x^2}+3 \sqrt [6]{3} \arctan \left (\frac {1+\frac {2 \sqrt [3]{3} x}{\sqrt [3]{1+x^3}}}{\sqrt {3}}\right )+\frac {1}{2} 3^{2/3} \log \left (-1+2 x^3\right )-\frac {3}{2} 3^{2/3} \log \left (\sqrt [3]{3} x-\sqrt [3]{1+x^3}\right ) \\ \end{align*}
Time = 0.32 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.00 \[ \int \frac {\left (-2+x^3\right ) \left (1+x^3\right )^{2/3}}{x^6 \left (-1+2 x^3\right )} \, dx=\frac {\left (-4-19 x^3\right ) \left (1+x^3\right )^{2/3}}{10 x^5}+3 \sqrt [6]{3} \arctan \left (\frac {3^{5/6} x}{\sqrt [3]{3} x+2 \sqrt [3]{1+x^3}}\right )-3^{2/3} \log \left (-3 x+3^{2/3} \sqrt [3]{1+x^3}\right )+\frac {1}{2} 3^{2/3} \log \left (3 x^2+3^{2/3} x \sqrt [3]{1+x^3}+\sqrt [3]{3} \left (1+x^3\right )^{2/3}\right ) \]
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Time = 13.49 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.96
method | result | size |
pseudoelliptic | \(\frac {-10 \,3^{\frac {2}{3}} \ln \left (\frac {-3^{\frac {1}{3}} x +\left (x^{3}+1\right )^{\frac {1}{3}}}{x}\right ) x^{5}+5 \,3^{\frac {2}{3}} \ln \left (\frac {3^{\frac {2}{3}} x^{2}+3^{\frac {1}{3}} \left (x^{3}+1\right )^{\frac {1}{3}} x +\left (x^{3}+1\right )^{\frac {2}{3}}}{x^{2}}\right ) x^{5}-30 \,3^{\frac {1}{6}} \arctan \left (\frac {\sqrt {3}\, \left (2 \,3^{\frac {2}{3}} \left (x^{3}+1\right )^{\frac {1}{3}}+3 x \right )}{9 x}\right ) x^{5}-19 x^{3} \left (x^{3}+1\right )^{\frac {2}{3}}-4 \left (x^{3}+1\right )^{\frac {2}{3}}}{10 x^{5}}\) | \(128\) |
risch | \(\text {Expression too large to display}\) | \(612\) |
trager | \(\text {Expression too large to display}\) | \(747\) |
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Leaf count of result is larger than twice the leaf count of optimal. 274 vs. \(2 (104) = 208\).
Time = 1.55 (sec) , antiderivative size = 274, normalized size of antiderivative = 2.04 \[ \int \frac {\left (-2+x^3\right ) \left (1+x^3\right )^{2/3}}{x^6 \left (-1+2 x^3\right )} \, dx=-\frac {10 \, \sqrt {3} \left (-9\right )^{\frac {1}{3}} x^{5} \arctan \left (\frac {2 \, \sqrt {3} \left (-9\right )^{\frac {2}{3}} {\left (14 \, x^{7} - 5 \, x^{4} - x\right )} {\left (x^{3} + 1\right )}^{\frac {2}{3}} + 6 \, \sqrt {3} \left (-9\right )^{\frac {1}{3}} {\left (31 \, x^{8} + 23 \, x^{5} + x^{2}\right )} {\left (x^{3} + 1\right )}^{\frac {1}{3}} - \sqrt {3} {\left (127 \, x^{9} + 201 \, x^{6} + 48 \, x^{3} + 1\right )}}{3 \, {\left (251 \, x^{9} + 231 \, x^{6} + 6 \, x^{3} - 1\right )}}\right ) - 10 \, \left (-9\right )^{\frac {1}{3}} x^{5} \log \left (\frac {3 \, \left (-9\right )^{\frac {2}{3}} {\left (x^{3} + 1\right )}^{\frac {1}{3}} x^{2} - 9 \, {\left (x^{3} + 1\right )}^{\frac {2}{3}} x + \left (-9\right )^{\frac {1}{3}} {\left (2 \, x^{3} - 1\right )}}{2 \, x^{3} - 1}\right ) + 5 \, \left (-9\right )^{\frac {1}{3}} x^{5} \log \left (-\frac {9 \, \left (-9\right )^{\frac {1}{3}} {\left (7 \, x^{4} + x\right )} {\left (x^{3} + 1\right )}^{\frac {2}{3}} - \left (-9\right )^{\frac {2}{3}} {\left (31 \, x^{6} + 23 \, x^{3} + 1\right )} - 27 \, {\left (5 \, x^{5} + 2 \, x^{2}\right )} {\left (x^{3} + 1\right )}^{\frac {1}{3}}}{4 \, x^{6} - 4 \, x^{3} + 1}\right ) + 3 \, {\left (19 \, x^{3} + 4\right )} {\left (x^{3} + 1\right )}^{\frac {2}{3}}}{30 \, x^{5}} \]
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\[ \int \frac {\left (-2+x^3\right ) \left (1+x^3\right )^{2/3}}{x^6 \left (-1+2 x^3\right )} \, dx=\int \frac {\left (\left (x + 1\right ) \left (x^{2} - x + 1\right )\right )^{\frac {2}{3}} \left (x^{3} - 2\right )}{x^{6} \cdot \left (2 x^{3} - 1\right )}\, dx \]
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\[ \int \frac {\left (-2+x^3\right ) \left (1+x^3\right )^{2/3}}{x^6 \left (-1+2 x^3\right )} \, dx=\int { \frac {{\left (x^{3} + 1\right )}^{\frac {2}{3}} {\left (x^{3} - 2\right )}}{{\left (2 \, x^{3} - 1\right )} x^{6}} \,d x } \]
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\[ \int \frac {\left (-2+x^3\right ) \left (1+x^3\right )^{2/3}}{x^6 \left (-1+2 x^3\right )} \, dx=\int { \frac {{\left (x^{3} + 1\right )}^{\frac {2}{3}} {\left (x^{3} - 2\right )}}{{\left (2 \, x^{3} - 1\right )} x^{6}} \,d x } \]
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Timed out. \[ \int \frac {\left (-2+x^3\right ) \left (1+x^3\right )^{2/3}}{x^6 \left (-1+2 x^3\right )} \, dx=\int \frac {{\left (x^3+1\right )}^{2/3}\,\left (x^3-2\right )}{x^6\,\left (2\,x^3-1\right )} \,d x \]
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