Integrand size = 15, antiderivative size = 18 \[ \int \frac {\sqrt [3]{-x+x^3}}{x^4} \, dx=\frac {3 \left (-x+x^3\right )^{4/3}}{8 x^4} \]
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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} \[ \int \frac {\sqrt [3]{-x+x^3}}{x^4} \, dx=\frac {3 \left (x^3-x\right )^{4/3}}{8 x^4} \]
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Rule 2039
Rubi steps \begin{align*} \text {integral}& = \frac {3 \left (-x+x^3\right )^{4/3}}{8 x^4} \\ \end{align*}
Time = 0.45 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt [3]{-x+x^3}}{x^4} \, dx=\frac {3 \left (x \left (-1+x^2\right )\right )^{4/3}}{8 x^4} \]
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Time = 0.80 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.11
method | result | size |
trager | \(\frac {3 \left (x^{2}-1\right ) \left (x^{3}-x \right )^{\frac {1}{3}}}{8 x^{3}}\) | \(20\) |
pseudoelliptic | \(\frac {3 \left (x^{2}-1\right ) \left (x^{3}-x \right )^{\frac {1}{3}}}{8 x^{3}}\) | \(20\) |
gosper | \(\frac {3 \left (1+x \right ) \left (x -1\right ) \left (x^{3}-x \right )^{\frac {1}{3}}}{8 x^{3}}\) | \(21\) |
risch | \(\frac {3 {\left (x \left (x^{2}-1\right )\right )}^{\frac {1}{3}} \left (x^{4}-2 x^{2}+1\right )}{8 x^{3} \left (x^{2}-1\right )}\) | \(32\) |
meijerg | \(-\frac {3 \operatorname {signum}\left (x^{2}-1\right )^{\frac {1}{3}} \left (-x^{2}+1\right )^{\frac {4}{3}}}{8 {\left (-\operatorname {signum}\left (x^{2}-1\right )\right )}^{\frac {1}{3}} x^{\frac {8}{3}}}\) | \(33\) |
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none
Time = 0.24 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.06 \[ \int \frac {\sqrt [3]{-x+x^3}}{x^4} \, dx=\frac {3 \, {\left (x^{3} - x\right )}^{\frac {1}{3}} {\left (x^{2} - 1\right )}}{8 \, x^{3}} \]
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\[ \int \frac {\sqrt [3]{-x+x^3}}{x^4} \, dx=\int \frac {\sqrt [3]{x \left (x - 1\right ) \left (x + 1\right )}}{x^{4}}\, dx \]
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none
Time = 0.19 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.22 \[ \int \frac {\sqrt [3]{-x+x^3}}{x^4} \, dx=\frac {3 \, {\left (x^{3} - x\right )} {\left (x + 1\right )}^{\frac {1}{3}} {\left (x - 1\right )}^{\frac {1}{3}}}{8 \, x^{\frac {11}{3}}} \]
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none
Time = 0.29 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.61 \[ \int \frac {\sqrt [3]{-x+x^3}}{x^4} \, dx=\frac {3}{8} \, {\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {4}{3}} \]
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Time = 4.97 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.72 \[ \int \frac {\sqrt [3]{-x+x^3}}{x^4} \, dx=-\frac {3\,{\left (x^3-x\right )}^{1/3}-3\,x^2\,{\left (x^3-x\right )}^{1/3}}{8\,x^3} \]
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