Integrand size = 17, antiderivative size = 20 \[ \int \frac {1}{x^3 \sqrt [3]{-x^2+x^6}} \, dx=\frac {3 \left (-x^2+x^6\right )^{2/3}}{8 x^4} \]
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Time = 0.02 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {2039} \[ \int \frac {1}{x^3 \sqrt [3]{-x^2+x^6}} \, dx=\frac {3 \left (x^6-x^2\right )^{2/3}}{8 x^4} \]
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Rule 2039
Rubi steps \begin{align*} \text {integral}& = \frac {3 \left (-x^2+x^6\right )^{2/3}}{8 x^4} \\ \end{align*}
Time = 10.01 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x^3 \sqrt [3]{-x^2+x^6}} \, dx=\frac {3 \left (x^2 \left (-1+x^4\right )\right )^{2/3}}{8 x^4} \]
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Time = 0.83 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.85
method | result | size |
trager | \(\frac {3 \left (x^{6}-x^{2}\right )^{\frac {2}{3}}}{8 x^{4}}\) | \(17\) |
pseudoelliptic | \(\frac {3 \left (x^{6}-x^{2}\right )^{\frac {2}{3}}}{8 x^{4}}\) | \(17\) |
risch | \(\frac {\frac {3 x^{4}}{8}-\frac {3}{8}}{x^{2} \left (x^{2} \left (x^{4}-1\right )\right )^{\frac {1}{3}}}\) | \(22\) |
gosper | \(\frac {3 \left (x^{2}+1\right ) \left (x -1\right ) \left (1+x \right )}{8 x^{2} \left (x^{6}-x^{2}\right )^{\frac {1}{3}}}\) | \(28\) |
meijerg | \(-\frac {3 {\left (-\operatorname {signum}\left (x^{4}-1\right )\right )}^{\frac {1}{3}} \left (-x^{4}+1\right )^{\frac {2}{3}}}{8 \operatorname {signum}\left (x^{4}-1\right )^{\frac {1}{3}} x^{\frac {8}{3}}}\) | \(33\) |
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none
Time = 0.24 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.80 \[ \int \frac {1}{x^3 \sqrt [3]{-x^2+x^6}} \, dx=\frac {3 \, {\left (x^{6} - x^{2}\right )}^{\frac {2}{3}}}{8 \, x^{4}} \]
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\[ \int \frac {1}{x^3 \sqrt [3]{-x^2+x^6}} \, dx=\int \frac {1}{x^{3} \sqrt [3]{x^{2} \left (x - 1\right ) \left (x + 1\right ) \left (x^{2} + 1\right )}}\, dx \]
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none
Time = 0.19 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.50 \[ \int \frac {1}{x^3 \sqrt [3]{-x^2+x^6}} \, dx=\frac {3 \, {\left (x^{6} - x^{2}\right )}}{8 \, {\left (x^{2} + 1\right )}^{\frac {1}{3}} {\left (x^{2} - 1\right )}^{\frac {1}{3}} {\left (x^{2}\right )}^{\frac {7}{3}}} \]
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none
Time = 0.29 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.55 \[ \int \frac {1}{x^3 \sqrt [3]{-x^2+x^6}} \, dx=\frac {3}{8} \, {\left (-\frac {1}{x^{4}} + 1\right )}^{\frac {2}{3}} \]
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Time = 5.22 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.80 \[ \int \frac {1}{x^3 \sqrt [3]{-x^2+x^6}} \, dx=\frac {3\,{\left (x^6-x^2\right )}^{2/3}}{8\,x^4} \]
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