Integrand size = 15, antiderivative size = 30 \[ \int \frac {1}{x^6 \sqrt [3]{-x+x^3}} \, dx=\frac {3 \left (-x+x^3\right )^{2/3} \left (5+6 x^2+9 x^4\right )}{80 x^6} \]
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Time = 0.04 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.83, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {2041, 2039} \[ \int \frac {1}{x^6 \sqrt [3]{-x+x^3}} \, dx=\frac {3 \left (x^3-x\right )^{2/3}}{16 x^6}+\frac {9 \left (x^3-x\right )^{2/3}}{40 x^4}+\frac {27 \left (x^3-x\right )^{2/3}}{80 x^2} \]
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Rule 2039
Rule 2041
Rubi steps \begin{align*} \text {integral}& = \frac {3 \left (-x+x^3\right )^{2/3}}{16 x^6}+\frac {3}{4} \int \frac {1}{x^4 \sqrt [3]{-x+x^3}} \, dx \\ & = \frac {3 \left (-x+x^3\right )^{2/3}}{16 x^6}+\frac {9 \left (-x+x^3\right )^{2/3}}{40 x^4}+\frac {9}{20} \int \frac {1}{x^2 \sqrt [3]{-x+x^3}} \, dx \\ & = \frac {3 \left (-x+x^3\right )^{2/3}}{16 x^6}+\frac {9 \left (-x+x^3\right )^{2/3}}{40 x^4}+\frac {27 \left (-x+x^3\right )^{2/3}}{80 x^2} \\ \end{align*}
Time = 4.94 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x^6 \sqrt [3]{-x+x^3}} \, dx=\frac {3 \left (x \left (-1+x^2\right )\right )^{2/3} \left (5+6 x^2+9 x^4\right )}{80 x^6} \]
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Time = 0.84 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.90
method | result | size |
trager | \(\frac {3 \left (x^{3}-x \right )^{\frac {2}{3}} \left (9 x^{4}+6 x^{2}+5\right )}{80 x^{6}}\) | \(27\) |
pseudoelliptic | \(\frac {3 \left (x^{3}-x \right )^{\frac {2}{3}} \left (9 x^{4}+6 x^{2}+5\right )}{80 x^{6}}\) | \(27\) |
risch | \(\frac {-\frac {3}{80} x^{2}-\frac {3}{16}-\frac {9}{80} x^{4}+\frac {27}{80} x^{6}}{x^{5} {\left (x \left (x^{2}-1\right )\right )}^{\frac {1}{3}}}\) | \(32\) |
gosper | \(\frac {3 \left (1+x \right ) \left (x -1\right ) \left (9 x^{4}+6 x^{2}+5\right )}{80 x^{5} \left (x^{3}-x \right )^{\frac {1}{3}}}\) | \(33\) |
meijerg | \(-\frac {3 {\left (-\operatorname {signum}\left (x^{2}-1\right )\right )}^{\frac {1}{3}} \left (\frac {9}{5} x^{4}+\frac {6}{5} x^{2}+1\right ) \left (-x^{2}+1\right )^{\frac {2}{3}}}{16 \operatorname {signum}\left (x^{2}-1\right )^{\frac {1}{3}} x^{\frac {16}{3}}}\) | \(45\) |
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Time = 0.25 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.87 \[ \int \frac {1}{x^6 \sqrt [3]{-x+x^3}} \, dx=\frac {3 \, {\left (9 \, x^{4} + 6 \, x^{2} + 5\right )} {\left (x^{3} - x\right )}^{\frac {2}{3}}}{80 \, x^{6}} \]
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\[ \int \frac {1}{x^6 \sqrt [3]{-x+x^3}} \, dx=\int \frac {1}{x^{6} \sqrt [3]{x \left (x - 1\right ) \left (x + 1\right )}}\, dx \]
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Time = 0.19 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.13 \[ \int \frac {1}{x^6 \sqrt [3]{-x+x^3}} \, dx=\frac {3 \, {\left (9 \, x^{7} - 3 \, x^{5} - x^{3} - 5 \, x\right )}}{80 \, {\left (x + 1\right )}^{\frac {1}{3}} {\left (x - 1\right )}^{\frac {1}{3}} x^{\frac {19}{3}}} \]
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Time = 0.27 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.37 \[ \int \frac {1}{x^6 \sqrt [3]{-x+x^3}} \, dx=\frac {3}{16} \, {\left (\frac {1}{x^{2}} - 1\right )}^{2} {\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {2}{3}} - \frac {3}{5} \, {\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {5}{3}} + \frac {3}{4} \, {\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {2}{3}} \]
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Time = 5.27 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.87 \[ \int \frac {1}{x^6 \sqrt [3]{-x+x^3}} \, dx=\frac {3\,{\left (x^3-x\right )}^{2/3}\,\left (9\,x^4+6\,x^2+5\right )}{80\,x^6} \]
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