Integrand size = 17, antiderivative size = 27 \[ \int \frac {1}{x^7 \sqrt [3]{-x^2+x^6}} \, dx=\frac {3 \left (2+3 x^4\right ) \left (-x^2+x^6\right )^{2/3}}{40 x^8} \]
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Time = 0.03 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.52, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {2041, 2039} \[ \int \frac {1}{x^7 \sqrt [3]{-x^2+x^6}} \, dx=\frac {3 \left (x^6-x^2\right )^{2/3}}{20 x^8}+\frac {9 \left (x^6-x^2\right )^{2/3}}{40 x^4} \]
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Rule 2039
Rule 2041
Rubi steps \begin{align*} \text {integral}& = \frac {3 \left (-x^2+x^6\right )^{2/3}}{20 x^8}+\frac {3}{5} \int \frac {1}{x^3 \sqrt [3]{-x^2+x^6}} \, dx \\ & = \frac {3 \left (-x^2+x^6\right )^{2/3}}{20 x^8}+\frac {9 \left (-x^2+x^6\right )^{2/3}}{40 x^4} \\ \end{align*}
Time = 10.02 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x^7 \sqrt [3]{-x^2+x^6}} \, dx=\frac {3 \left (x^2 \left (-1+x^4\right )\right )^{2/3} \left (2+3 x^4\right )}{40 x^8} \]
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Time = 0.89 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.89
method | result | size |
trager | \(\frac {3 \left (3 x^{4}+2\right ) \left (x^{6}-x^{2}\right )^{\frac {2}{3}}}{40 x^{8}}\) | \(24\) |
pseudoelliptic | \(\frac {3 \left (3 x^{4}+2\right ) \left (x^{6}-x^{2}\right )^{\frac {2}{3}}}{40 x^{8}}\) | \(24\) |
risch | \(\frac {-\frac {3}{40} x^{4}-\frac {3}{20}+\frac {9}{40} x^{8}}{x^{6} \left (x^{2} \left (x^{4}-1\right )\right )^{\frac {1}{3}}}\) | \(29\) |
gosper | \(\frac {3 \left (x^{2}+1\right ) \left (x -1\right ) \left (1+x \right ) \left (3 x^{4}+2\right )}{40 x^{6} \left (x^{6}-x^{2}\right )^{\frac {1}{3}}}\) | \(35\) |
meijerg | \(-\frac {3 {\left (-\operatorname {signum}\left (x^{4}-1\right )\right )}^{\frac {1}{3}} \left (1+\frac {3 x^{4}}{2}\right ) \left (-x^{4}+1\right )^{\frac {2}{3}}}{20 \operatorname {signum}\left (x^{4}-1\right )^{\frac {1}{3}} x^{\frac {20}{3}}}\) | \(40\) |
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Time = 0.25 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.85 \[ \int \frac {1}{x^7 \sqrt [3]{-x^2+x^6}} \, dx=\frac {3 \, {\left (x^{6} - x^{2}\right )}^{\frac {2}{3}} {\left (3 \, x^{4} + 2\right )}}{40 \, x^{8}} \]
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\[ \int \frac {1}{x^7 \sqrt [3]{-x^2+x^6}} \, dx=\int \frac {1}{x^{7} \sqrt [3]{x^{2} \left (x - 1\right ) \left (x + 1\right ) \left (x^{2} + 1\right )}}\, dx \]
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Time = 0.19 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.37 \[ \int \frac {1}{x^7 \sqrt [3]{-x^2+x^6}} \, dx=\frac {3 \, {\left (3 \, x^{10} - x^{6} - 2 \, x^{2}\right )}}{40 \, {\left (x^{2} + 1\right )}^{\frac {1}{3}} {\left (x^{2} - 1\right )}^{\frac {1}{3}} {\left (x^{2}\right )}^{\frac {13}{3}}} \]
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Time = 0.27 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.85 \[ \int \frac {1}{x^7 \sqrt [3]{-x^2+x^6}} \, dx=-\frac {3}{20} \, {\left (-\frac {1}{x^{4}} + 1\right )}^{\frac {5}{3}} + \frac {3}{8} \, {\left (-\frac {1}{x^{4}} + 1\right )}^{\frac {2}{3}} \]
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Time = 6.04 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.30 \[ \int \frac {1}{x^7 \sqrt [3]{-x^2+x^6}} \, dx=\frac {9\,x^4\,{\left (x^6-x^2\right )}^{2/3}+6\,{\left (x^6-x^2\right )}^{2/3}}{40\,x^8} \]
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