Integrand size = 15, antiderivative size = 18 \[ \int \frac {1}{x^2 \sqrt {-x+x^4}} \, dx=\frac {2 \sqrt {-x+x^4}}{3 x^2} \]
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Time = 0.01 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {2039} \[ \int \frac {1}{x^2 \sqrt {-x+x^4}} \, dx=\frac {2 \sqrt {x^4-x}}{3 x^2} \]
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Rule 2039
Rubi steps \begin{align*} \text {integral}& = \frac {2 \sqrt {-x+x^4}}{3 x^2} \\ \end{align*}
Time = 0.13 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x^2 \sqrt {-x+x^4}} \, dx=\frac {2 \sqrt {x \left (-1+x^3\right )}}{3 x^2} \]
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Time = 2.94 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.83
method | result | size |
default | \(\frac {2 \sqrt {x^{4}-x}}{3 x^{2}}\) | \(15\) |
trager | \(\frac {2 \sqrt {x^{4}-x}}{3 x^{2}}\) | \(15\) |
elliptic | \(\frac {2 \sqrt {x^{4}-x}}{3 x^{2}}\) | \(15\) |
pseudoelliptic | \(\frac {2 \sqrt {x^{4}-x}}{3 x^{2}}\) | \(15\) |
risch | \(\frac {\frac {2 x^{3}}{3}-\frac {2}{3}}{x \sqrt {x \left (x^{3}-1\right )}}\) | \(20\) |
gosper | \(\frac {2 \left (x -1\right ) \left (x^{2}+x +1\right )}{3 x \sqrt {x^{4}-x}}\) | \(24\) |
meijerg | \(-\frac {2 \sqrt {-\operatorname {signum}\left (x^{3}-1\right )}\, \sqrt {-x^{3}+1}}{3 \sqrt {\operatorname {signum}\left (x^{3}-1\right )}\, x^{\frac {3}{2}}}\) | \(33\) |
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none
Time = 0.24 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.78 \[ \int \frac {1}{x^2 \sqrt {-x+x^4}} \, dx=\frac {2 \, \sqrt {x^{4} - x}}{3 \, x^{2}} \]
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\[ \int \frac {1}{x^2 \sqrt {-x+x^4}} \, dx=\int \frac {1}{x^{2} \sqrt {x \left (x - 1\right ) \left (x^{2} + x + 1\right )}}\, dx \]
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none
Time = 0.29 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.39 \[ \int \frac {1}{x^2 \sqrt {-x+x^4}} \, dx=\frac {2 \, {\left (x^{4} - x\right )}}{3 \, \sqrt {x^{2} + x + 1} \sqrt {x - 1} x^{\frac {5}{2}}} \]
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none
Time = 0.28 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.61 \[ \int \frac {1}{x^2 \sqrt {-x+x^4}} \, dx=\frac {2}{3} \, \sqrt {-\frac {1}{x^{3}} + 1} \]
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Time = 4.96 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.78 \[ \int \frac {1}{x^2 \sqrt {-x+x^4}} \, dx=\frac {2\,\sqrt {x^4-x}}{3\,x^2} \]
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