Integrand size = 17, antiderivative size = 20 \[ \int \frac {1}{x^2 \sqrt [4]{-x^2+x^4}} \, dx=\frac {2 \left (-x^2+x^4\right )^{3/4}}{3 x^3} \]
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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} \[ \int \frac {1}{x^2 \sqrt [4]{-x^2+x^4}} \, dx=\frac {2 \left (x^4-x^2\right )^{3/4}}{3 x^3} \]
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Rule 2039
Rubi steps \begin{align*} \text {integral}& = \frac {2 \left (-x^2+x^4\right )^{3/4}}{3 x^3} \\ \end{align*}
Time = 0.15 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x^2 \sqrt [4]{-x^2+x^4}} \, dx=\frac {2 \left (x^2 \left (-1+x^2\right )\right )^{3/4}}{3 x^3} \]
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Time = 0.82 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.85
method | result | size |
trager | \(\frac {2 \left (x^{4}-x^{2}\right )^{\frac {3}{4}}}{3 x^{3}}\) | \(17\) |
pseudoelliptic | \(\frac {2 \left (x^{4}-x^{2}\right )^{\frac {3}{4}}}{3 x^{3}}\) | \(17\) |
risch | \(\frac {\frac {2 x^{2}}{3}-\frac {2}{3}}{x \left (x^{2} \left (x^{2}-1\right )\right )^{\frac {1}{4}}}\) | \(22\) |
gosper | \(\frac {2 \left (1+x \right ) \left (x -1\right )}{3 x \left (x^{4}-x^{2}\right )^{\frac {1}{4}}}\) | \(23\) |
meijerg | \(-\frac {2 {\left (-\operatorname {signum}\left (x^{2}-1\right )\right )}^{\frac {1}{4}} \left (-x^{2}+1\right )^{\frac {3}{4}}}{3 \operatorname {signum}\left (x^{2}-1\right )^{\frac {1}{4}} x^{\frac {3}{2}}}\) | \(33\) |
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none
Time = 0.25 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.80 \[ \int \frac {1}{x^2 \sqrt [4]{-x^2+x^4}} \, dx=\frac {2 \, {\left (x^{4} - x^{2}\right )}^{\frac {3}{4}}}{3 \, x^{3}} \]
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\[ \int \frac {1}{x^2 \sqrt [4]{-x^2+x^4}} \, dx=\int \frac {1}{x^{2} \sqrt [4]{x^{2} \left (x - 1\right ) \left (x + 1\right )}}\, dx \]
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none
Time = 0.20 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {1}{x^2 \sqrt [4]{-x^2+x^4}} \, dx=\frac {2 \, {\left (x^{3} - x\right )}}{3 \, {\left (x + 1\right )}^{\frac {1}{4}} {\left (x - 1\right )}^{\frac {1}{4}} x^{\frac {5}{2}}} \]
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none
Time = 0.28 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.55 \[ \int \frac {1}{x^2 \sqrt [4]{-x^2+x^4}} \, dx=\frac {2}{3} \, {\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {3}{4}} \]
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Time = 5.22 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.80 \[ \int \frac {1}{x^2 \sqrt [4]{-x^2+x^4}} \, dx=\frac {2\,{\left (x^4-x^2\right )}^{3/4}}{3\,x^3} \]
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