Integrand size = 27, antiderivative size = 174 \[ \int \frac {\left (-2+x^3\right ) \left (-1+x^3\right )^{2/3}}{x^3 \left (-1+2 x^3\right )} \, dx=-\frac {\left (-1+x^3\right )^{2/3}}{x^2}-\frac {1}{2} \sqrt {3} \arctan \left (\frac {\sqrt {3} x}{-x+2 \sqrt [3]{-1+x^3}}\right )+\frac {\arctan \left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{-1+x^3}}\right )}{2 \sqrt {3}}-\frac {1}{6} \log \left (-x+\sqrt [3]{-1+x^3}\right )-\frac {1}{2} \log \left (x+\sqrt [3]{-1+x^3}\right )+\frac {1}{4} \log \left (x^2-x \sqrt [3]{-1+x^3}+\left (-1+x^3\right )^{2/3}\right )+\frac {1}{12} \log \left (x^2+x \sqrt [3]{-1+x^3}+\left (-1+x^3\right )^{2/3}\right ) \]
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Time = 0.04 (sec) , antiderivative size = 125, normalized size of antiderivative = 0.72, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {594, 544, 245, 384} \[ \int \frac {\left (-2+x^3\right ) \left (-1+x^3\right )^{2/3}}{x^3 \left (-1+2 x^3\right )} \, dx=\frac {1}{2} \sqrt {3} \arctan \left (\frac {1-\frac {2 x}{\sqrt [3]{x^3-1}}}{\sqrt {3}}\right )+\frac {\arctan \left (\frac {\frac {2 x}{\sqrt [3]{x^3-1}}+1}{\sqrt {3}}\right )}{2 \sqrt {3}}+\frac {1}{4} \log \left (2 x^3-1\right )-\frac {3}{4} \log \left (-\sqrt [3]{x^3-1}-x\right )-\frac {1}{4} \log \left (\sqrt [3]{x^3-1}-x\right )-\frac {\left (x^3-1\right )^{2/3}}{x^2} \]
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Rule 245
Rule 384
Rule 544
Rule 594
Rubi steps \begin{align*} \text {integral}& = -\frac {\left (-1+x^3\right )^{2/3}}{x^2}-\frac {1}{2} \int \frac {-2-2 x^3}{\sqrt [3]{-1+x^3} \left (-1+2 x^3\right )} \, dx \\ & = -\frac {\left (-1+x^3\right )^{2/3}}{x^2}+\frac {1}{2} \int \frac {1}{\sqrt [3]{-1+x^3}} \, dx+\frac {3}{2} \int \frac {1}{\sqrt [3]{-1+x^3} \left (-1+2 x^3\right )} \, dx \\ & = -\frac {\left (-1+x^3\right )^{2/3}}{x^2}+\frac {1}{2} \sqrt {3} \arctan \left (\frac {1-\frac {2 x}{\sqrt [3]{-1+x^3}}}{\sqrt {3}}\right )+\frac {\arctan \left (\frac {1+\frac {2 x}{\sqrt [3]{-1+x^3}}}{\sqrt {3}}\right )}{2 \sqrt {3}}+\frac {1}{4} \log \left (-1+2 x^3\right )-\frac {3}{4} \log \left (-x-\sqrt [3]{-1+x^3}\right )-\frac {1}{4} \log \left (-x+\sqrt [3]{-1+x^3}\right ) \\ \end{align*}
Time = 0.26 (sec) , antiderivative size = 162, normalized size of antiderivative = 0.93 \[ \int \frac {\left (-2+x^3\right ) \left (-1+x^3\right )^{2/3}}{x^3 \left (-1+2 x^3\right )} \, dx=\frac {1}{12} \left (-\frac {12 \left (-1+x^3\right )^{2/3}}{x^2}+6 \sqrt {3} \arctan \left (\frac {\sqrt {3} x}{x-2 \sqrt [3]{-1+x^3}}\right )+2 \sqrt {3} \arctan \left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{-1+x^3}}\right )-2 \log \left (-x+\sqrt [3]{-1+x^3}\right )-6 \log \left (x+\sqrt [3]{-1+x^3}\right )+3 \log \left (x^2-x \sqrt [3]{-1+x^3}+\left (-1+x^3\right )^{2/3}\right )+\log \left (x^2+x \sqrt [3]{-1+x^3}+\left (-1+x^3\right )^{2/3}\right )\right ) \]
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Time = 16.86 (sec) , antiderivative size = 172, normalized size of antiderivative = 0.99
method | result | size |
pseudoelliptic | \(\frac {-2 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (x +2 \left (x^{3}-1\right )^{\frac {1}{3}}\right )}{3 x}\right ) x^{2}-6 \sqrt {3}\, \arctan \left (\frac {\left (x -2 \left (x^{3}-1\right )^{\frac {1}{3}}\right ) \sqrt {3}}{3 x}\right ) x^{2}+\ln \left (\frac {x^{2}+x \left (x^{3}-1\right )^{\frac {1}{3}}+\left (x^{3}-1\right )^{\frac {2}{3}}}{x^{2}}\right ) x^{2}-2 \ln \left (\frac {-x +\left (x^{3}-1\right )^{\frac {1}{3}}}{x}\right ) x^{2}+3 \ln \left (\frac {x^{2}-x \left (x^{3}-1\right )^{\frac {1}{3}}+\left (x^{3}-1\right )^{\frac {2}{3}}}{x^{2}}\right ) x^{2}-6 \ln \left (\frac {x +\left (x^{3}-1\right )^{\frac {1}{3}}}{x}\right ) x^{2}-12 \left (x^{3}-1\right )^{\frac {2}{3}}}{12 x^{2}}\) | \(172\) |
risch | \(\text {Expression too large to display}\) | \(1191\) |
trager | \(\text {Expression too large to display}\) | \(1241\) |
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Time = 4.19 (sec) , antiderivative size = 214, normalized size of antiderivative = 1.23 \[ \int \frac {\left (-2+x^3\right ) \left (-1+x^3\right )^{2/3}}{x^3 \left (-1+2 x^3\right )} \, dx=\frac {2 \, \sqrt {3} x^{2} \arctan \left (\frac {383838 \, \sqrt {3} {\left (x^{10} - 3 \, x^{4} - 2 \, x\right )} {\left (x^{3} - 1\right )}^{\frac {2}{3}} + 13468 \, \sqrt {3} {\left (x^{11} - 3 \, x^{8} + 4 \, x^{2}\right )} {\left (x^{3} - 1\right )}^{\frac {1}{3}} + \sqrt {3} {\left (198653 \, x^{12} + 393594 \, x^{9} + 5568 \, x^{6} - 400090 \, x^{3} - 198189\right )}}{3 \, {\left (185185 \, x^{12} + 370434 \, x^{9} - 96 \, x^{6} - 370322 \, x^{3} - 185193\right )}}\right ) - x^{2} \log \left (\frac {8 \, x^{9} - 12 \, x^{6} + 6 \, x^{3} - 3 \, {\left (x^{10} - 3 \, x^{4} - 2 \, x\right )} {\left (x^{3} - 1\right )}^{\frac {2}{3}} + 3 \, {\left (x^{11} - 3 \, x^{8} + 4 \, x^{2}\right )} {\left (x^{3} - 1\right )}^{\frac {1}{3}} - 1}{8 \, x^{9} - 12 \, x^{6} + 6 \, x^{3} - 1}\right ) - 12 \, {\left (x^{3} - 1\right )}^{\frac {2}{3}}}{12 \, x^{2}} \]
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\[ \int \frac {\left (-2+x^3\right ) \left (-1+x^3\right )^{2/3}}{x^3 \left (-1+2 x^3\right )} \, dx=\int \frac {\left (\left (x - 1\right ) \left (x^{2} + x + 1\right )\right )^{\frac {2}{3}} \left (x^{3} - 2\right )}{x^{3} \cdot \left (2 x^{3} - 1\right )}\, dx \]
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\[ \int \frac {\left (-2+x^3\right ) \left (-1+x^3\right )^{2/3}}{x^3 \left (-1+2 x^3\right )} \, dx=\int { \frac {{\left (x^{3} - 1\right )}^{\frac {2}{3}} {\left (x^{3} - 2\right )}}{{\left (2 \, x^{3} - 1\right )} x^{3}} \,d x } \]
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\[ \int \frac {\left (-2+x^3\right ) \left (-1+x^3\right )^{2/3}}{x^3 \left (-1+2 x^3\right )} \, dx=\int { \frac {{\left (x^{3} - 1\right )}^{\frac {2}{3}} {\left (x^{3} - 2\right )}}{{\left (2 \, x^{3} - 1\right )} x^{3}} \,d x } \]
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Timed out. \[ \int \frac {\left (-2+x^3\right ) \left (-1+x^3\right )^{2/3}}{x^3 \left (-1+2 x^3\right )} \, dx=\int \frac {{\left (x^3-1\right )}^{2/3}\,\left (x^3-2\right )}{x^3\,\left (2\,x^3-1\right )} \,d x \]
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