Integrand size = 15, antiderivative size = 18 \[ \int \frac {1}{x^3 \sqrt [4]{-x+x^4}} \, dx=\frac {4 \left (-x+x^4\right )^{3/4}}{9 x^3} \]
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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} \[ \int \frac {1}{x^3 \sqrt [4]{-x+x^4}} \, dx=\frac {4 \left (x^4-x\right )^{3/4}}{9 x^3} \]
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Rule 2039
Rubi steps \begin{align*} \text {integral}& = \frac {4 \left (-x+x^4\right )^{3/4}}{9 x^3} \\ \end{align*}
Time = 9.09 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x^3 \sqrt [4]{-x+x^4}} \, dx=\frac {4 \left (x \left (-1+x^3\right )\right )^{3/4}}{9 x^3} \]
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Time = 0.77 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.83
method | result | size |
trager | \(\frac {4 \left (x^{4}-x \right )^{\frac {3}{4}}}{9 x^{3}}\) | \(15\) |
pseudoelliptic | \(\frac {4 \left (x^{4}-x \right )^{\frac {3}{4}}}{9 x^{3}}\) | \(15\) |
risch | \(\frac {\frac {4 x^{3}}{9}-\frac {4}{9}}{x^{2} {\left (x \left (x^{3}-1\right )\right )}^{\frac {1}{4}}}\) | \(20\) |
gosper | \(\frac {4 \left (x -1\right ) \left (x^{2}+x +1\right )}{9 x^{2} \left (x^{4}-x \right )^{\frac {1}{4}}}\) | \(24\) |
meijerg | \(-\frac {4 {\left (-\operatorname {signum}\left (x^{3}-1\right )\right )}^{\frac {1}{4}} \left (-x^{3}+1\right )^{\frac {3}{4}}}{9 \operatorname {signum}\left (x^{3}-1\right )^{\frac {1}{4}} x^{\frac {9}{4}}}\) | \(33\) |
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none
Time = 0.23 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.78 \[ \int \frac {1}{x^3 \sqrt [4]{-x+x^4}} \, dx=\frac {4 \, {\left (x^{4} - x\right )}^{\frac {3}{4}}}{9 \, x^{3}} \]
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\[ \int \frac {1}{x^3 \sqrt [4]{-x+x^4}} \, dx=\int \frac {1}{x^{3} \sqrt [4]{x \left (x - 1\right ) \left (x^{2} + x + 1\right )}}\, dx \]
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none
Time = 0.28 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.39 \[ \int \frac {1}{x^3 \sqrt [4]{-x+x^4}} \, dx=\frac {4 \, {\left (x^{4} - x\right )}}{9 \, {\left (x^{2} + x + 1\right )}^{\frac {1}{4}} {\left (x - 1\right )}^{\frac {1}{4}} x^{\frac {13}{4}}} \]
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none
Time = 0.29 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.61 \[ \int \frac {1}{x^3 \sqrt [4]{-x+x^4}} \, dx=\frac {4}{9} \, {\left (-\frac {1}{x^{3}} + 1\right )}^{\frac {3}{4}} \]
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Time = 4.98 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.78 \[ \int \frac {1}{x^3 \sqrt [4]{-x+x^4}} \, dx=\frac {4\,{\left (x^4-x\right )}^{3/4}}{9\,x^3} \]
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