Integrand size = 11, antiderivative size = 11 \[ \int \frac {x}{\left (-1+x^2\right )^{3/4}} \, dx=2 \sqrt [4]{-1+x^2} \]
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Time = 0.00 (sec) , antiderivative size = 11, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {267} \[ \int \frac {x}{\left (-1+x^2\right )^{3/4}} \, dx=2 \sqrt [4]{x^2-1} \]
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Rule 267
Rubi steps \begin{align*} \text {integral}& = 2 \sqrt [4]{-1+x^2} \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 11, normalized size of antiderivative = 1.00 \[ \int \frac {x}{\left (-1+x^2\right )^{3/4}} \, dx=2 \sqrt [4]{-1+x^2} \]
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Time = 0.77 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.91
method | result | size |
derivativedivides | \(2 \left (x^{2}-1\right )^{\frac {1}{4}}\) | \(10\) |
default | \(2 \left (x^{2}-1\right )^{\frac {1}{4}}\) | \(10\) |
trager | \(2 \left (x^{2}-1\right )^{\frac {1}{4}}\) | \(10\) |
risch | \(2 \left (x^{2}-1\right )^{\frac {1}{4}}\) | \(10\) |
pseudoelliptic | \(2 \left (x^{2}-1\right )^{\frac {1}{4}}\) | \(10\) |
gosper | \(\frac {2 \left (x -1\right ) \left (1+x \right )}{\left (x^{2}-1\right )^{\frac {3}{4}}}\) | \(16\) |
meijerg | \(\frac {{\left (-\operatorname {signum}\left (x^{2}-1\right )\right )}^{\frac {3}{4}} x^{2} \operatorname {hypergeom}\left (\left [\frac {3}{4}, 1\right ], \left [2\right ], x^{2}\right )}{2 \operatorname {signum}\left (x^{2}-1\right )^{\frac {3}{4}}}\) | \(33\) |
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none
Time = 0.23 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.82 \[ \int \frac {x}{\left (-1+x^2\right )^{3/4}} \, dx=2 \, {\left (x^{2} - 1\right )}^{\frac {1}{4}} \]
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Time = 0.07 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.73 \[ \int \frac {x}{\left (-1+x^2\right )^{3/4}} \, dx=2 \sqrt [4]{x^{2} - 1} \]
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none
Time = 0.20 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.82 \[ \int \frac {x}{\left (-1+x^2\right )^{3/4}} \, dx=2 \, {\left (x^{2} - 1\right )}^{\frac {1}{4}} \]
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none
Time = 0.29 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.82 \[ \int \frac {x}{\left (-1+x^2\right )^{3/4}} \, dx=2 \, {\left (x^{2} - 1\right )}^{\frac {1}{4}} \]
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Time = 5.07 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.82 \[ \int \frac {x}{\left (-1+x^2\right )^{3/4}} \, dx=2\,{\left (x^2-1\right )}^{1/4} \]
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