Integrand size = 27, antiderivative size = 149 \[ \int \frac {\left (-1+x^3\right ) \left (1+3 x^3\right )^{2/3}}{x^6 \left (1+x^3\right )} \, dx=\frac {\left (1-2 x^3\right ) \left (1+3 x^3\right )^{2/3}}{5 x^5}+\frac {2\ 2^{2/3} \arctan \left (\frac {\sqrt {3} x}{x+2^{2/3} \sqrt [3]{1+3 x^3}}\right )}{\sqrt {3}}-\frac {2}{3} 2^{2/3} \log \left (-2 x+2^{2/3} \sqrt [3]{1+3 x^3}\right )+\frac {1}{3} 2^{2/3} \log \left (2 x^2+2^{2/3} x \sqrt [3]{1+3 x^3}+\sqrt [3]{2} \left (1+3 x^3\right )^{2/3}\right ) \]
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Time = 0.06 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.81, 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} \[ \int \frac {\left (-1+x^3\right ) \left (1+3 x^3\right )^{2/3}}{x^6 \left (1+x^3\right )} \, dx=\frac {2\ 2^{2/3} \arctan \left (\frac {\frac {2 \sqrt [3]{2} x}{\sqrt [3]{3 x^3+1}}+1}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {1}{3} 2^{2/3} \log \left (x^3+1\right )-2^{2/3} \log \left (\sqrt [3]{2} x-\sqrt [3]{3 x^3+1}\right )+\frac {\left (3 x^3+1\right )^{2/3}}{5 x^5}-\frac {2 \left (3 x^3+1\right )^{2/3}}{5 x^2} \]
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Rule 12
Rule 384
Rule 594
Rule 597
Rubi steps \begin{align*} \text {integral}& = \frac {\left (1+3 x^3\right )^{2/3}}{5 x^5}+\frac {1}{5} \int \frac {4+24 x^3}{x^3 \left (1+x^3\right ) \sqrt [3]{1+3 x^3}} \, dx \\ & = \frac {\left (1+3 x^3\right )^{2/3}}{5 x^5}-\frac {2 \left (1+3 x^3\right )^{2/3}}{5 x^2}-\frac {1}{10} \int -\frac {40}{\left (1+x^3\right ) \sqrt [3]{1+3 x^3}} \, dx \\ & = \frac {\left (1+3 x^3\right )^{2/3}}{5 x^5}-\frac {2 \left (1+3 x^3\right )^{2/3}}{5 x^2}+4 \int \frac {1}{\left (1+x^3\right ) \sqrt [3]{1+3 x^3}} \, dx \\ & = \frac {\left (1+3 x^3\right )^{2/3}}{5 x^5}-\frac {2 \left (1+3 x^3\right )^{2/3}}{5 x^2}+\frac {2\ 2^{2/3} \arctan \left (\frac {1+\frac {2 \sqrt [3]{2} x}{\sqrt [3]{1+3 x^3}}}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {1}{3} 2^{2/3} \log \left (1+x^3\right )-2^{2/3} \log \left (\sqrt [3]{2} x-\sqrt [3]{1+3 x^3}\right ) \\ \end{align*}
Time = 0.34 (sec) , antiderivative size = 149, normalized size of antiderivative = 1.00 \[ \int \frac {\left (-1+x^3\right ) \left (1+3 x^3\right )^{2/3}}{x^6 \left (1+x^3\right )} \, dx=\frac {\left (1-2 x^3\right ) \left (1+3 x^3\right )^{2/3}}{5 x^5}+\frac {2\ 2^{2/3} \arctan \left (\frac {\sqrt {3} x}{x+2^{2/3} \sqrt [3]{1+3 x^3}}\right )}{\sqrt {3}}-\frac {2}{3} 2^{2/3} \log \left (-2 x+2^{2/3} \sqrt [3]{1+3 x^3}\right )+\frac {1}{3} 2^{2/3} \log \left (2 x^2+2^{2/3} x \sqrt [3]{1+3 x^3}+\sqrt [3]{2} \left (1+3 x^3\right )^{2/3}\right ) \]
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Time = 13.80 (sec) , antiderivative size = 140, normalized size of antiderivative = 0.94
method | result | size |
pseudoelliptic | \(\frac {-10 \sqrt {3}\, 2^{\frac {2}{3}} \arctan \left (\frac {\sqrt {3}\, \left (x +2^{\frac {2}{3}} \left (3 x^{3}+1\right )^{\frac {1}{3}}\right )}{3 x}\right ) x^{5}-10 \,2^{\frac {2}{3}} \ln \left (\frac {-2^{\frac {1}{3}} x +\left (3 x^{3}+1\right )^{\frac {1}{3}}}{x}\right ) x^{5}+5 \,2^{\frac {2}{3}} \ln \left (\frac {2^{\frac {2}{3}} x^{2}+2^{\frac {1}{3}} \left (3 x^{3}+1\right )^{\frac {1}{3}} x +\left (3 x^{3}+1\right )^{\frac {2}{3}}}{x^{2}}\right ) x^{5}-6 \left (3 x^{3}+1\right )^{\frac {2}{3}} x^{3}+3 \left (3 x^{3}+1\right )^{\frac {2}{3}}}{15 x^{5}}\) | \(140\) |
risch | \(\text {Expression too large to display}\) | \(958\) |
trager | \(\text {Expression too large to display}\) | \(1163\) |
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Leaf count of result is larger than twice the leaf count of optimal. 279 vs. \(2 (115) = 230\).
Time = 1.87 (sec) , antiderivative size = 279, normalized size of antiderivative = 1.87 \[ \int \frac {\left (-1+x^3\right ) \left (1+3 x^3\right )^{2/3}}{x^6 \left (1+x^3\right )} \, dx=\frac {10 \, \sqrt {3} \left (-4\right )^{\frac {1}{3}} x^{5} \arctan \left (\frac {3 \, \sqrt {3} \left (-4\right )^{\frac {2}{3}} {\left (7 \, x^{7} + 8 \, x^{4} + x\right )} {\left (3 \, x^{3} + 1\right )}^{\frac {2}{3}} - 6 \, \sqrt {3} \left (-4\right )^{\frac {1}{3}} {\left (55 \, x^{8} + 20 \, x^{5} + x^{2}\right )} {\left (3 \, x^{3} + 1\right )}^{\frac {1}{3}} + \sqrt {3} {\left (433 \, x^{9} + 255 \, x^{6} + 39 \, x^{3} + 1\right )}}{3 \, {\left (323 \, x^{9} + 105 \, x^{6} - 3 \, x^{3} - 1\right )}}\right ) + 10 \, \left (-4\right )^{\frac {1}{3}} x^{5} \log \left (-\frac {3 \, \left (-4\right )^{\frac {2}{3}} {\left (3 \, x^{3} + 1\right )}^{\frac {1}{3}} x^{2} - 6 \, {\left (3 \, x^{3} + 1\right )}^{\frac {2}{3}} x - \left (-4\right )^{\frac {1}{3}} {\left (x^{3} + 1\right )}}{x^{3} + 1}\right ) - 5 \, \left (-4\right )^{\frac {1}{3}} x^{5} \log \left (-\frac {6 \, \left (-4\right )^{\frac {1}{3}} {\left (7 \, x^{4} + x\right )} {\left (3 \, x^{3} + 1\right )}^{\frac {2}{3}} - \left (-4\right )^{\frac {2}{3}} {\left (55 \, x^{6} + 20 \, x^{3} + 1\right )} - 24 \, {\left (4 \, x^{5} + x^{2}\right )} {\left (3 \, x^{3} + 1\right )}^{\frac {1}{3}}}{x^{6} + 2 \, x^{3} + 1}\right ) - 9 \, {\left (3 \, x^{3} + 1\right )}^{\frac {2}{3}} {\left (2 \, x^{3} - 1\right )}}{45 \, x^{5}} \]
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\[ \int \frac {\left (-1+x^3\right ) \left (1+3 x^3\right )^{2/3}}{x^6 \left (1+x^3\right )} \, dx=\int \frac {\left (x - 1\right ) \left (3 x^{3} + 1\right )^{\frac {2}{3}} \left (x^{2} + x + 1\right )}{x^{6} \left (x + 1\right ) \left (x^{2} - x + 1\right )}\, dx \]
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\[ \int \frac {\left (-1+x^3\right ) \left (1+3 x^3\right )^{2/3}}{x^6 \left (1+x^3\right )} \, dx=\int { \frac {{\left (3 \, x^{3} + 1\right )}^{\frac {2}{3}} {\left (x^{3} - 1\right )}}{{\left (x^{3} + 1\right )} x^{6}} \,d x } \]
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\[ \int \frac {\left (-1+x^3\right ) \left (1+3 x^3\right )^{2/3}}{x^6 \left (1+x^3\right )} \, dx=\int { \frac {{\left (3 \, x^{3} + 1\right )}^{\frac {2}{3}} {\left (x^{3} - 1\right )}}{{\left (x^{3} + 1\right )} x^{6}} \,d x } \]
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Timed out. \[ \int \frac {\left (-1+x^3\right ) \left (1+3 x^3\right )^{2/3}}{x^6 \left (1+x^3\right )} \, dx=\int \frac {\left (x^3-1\right )\,{\left (3\,x^3+1\right )}^{2/3}}{x^6\,\left (x^3+1\right )} \,d x \]
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