Integrand size = 19, antiderivative size = 15 \[ \int \left (-1+3 x^2\right ) \sqrt [3]{-x+x^3} \, dx=\frac {3}{4} \left (-x+x^3\right )^{4/3} \]
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Time = 0.01 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {1602} \[ \int \left (-1+3 x^2\right ) \sqrt [3]{-x+x^3} \, dx=\frac {3}{4} \left (x^3-x\right )^{4/3} \]
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Rule 1602
Rubi steps \begin{align*} \text {integral}& = \frac {3}{4} \left (-x+x^3\right )^{4/3} \\ \end{align*}
Time = 0.84 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.00 \[ \int \left (-1+3 x^2\right ) \sqrt [3]{-x+x^3} \, dx=\frac {3}{4} \left (x \left (-1+x^2\right )\right )^{4/3} \]
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Time = 0.84 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.80
method | result | size |
derivativedivides | \(\frac {3 \left (x^{3}-x \right )^{\frac {4}{3}}}{4}\) | \(12\) |
default | \(\frac {3 \left (x^{3}-x \right )^{\frac {4}{3}}}{4}\) | \(12\) |
trager | \(\frac {3 x \left (x^{2}-1\right ) \left (x^{3}-x \right )^{\frac {1}{3}}}{4}\) | \(18\) |
risch | \(\frac {3 {\left (x \left (x^{2}-1\right )\right )}^{\frac {1}{3}} x \left (x^{2}-1\right )}{4}\) | \(18\) |
gosper | \(\frac {3 \left (1+x \right ) x \left (x -1\right ) \left (x^{3}-x \right )^{\frac {1}{3}}}{4}\) | \(19\) |
pseudoelliptic | \(\frac {\left (3 x^{3}-3 x \right ) \left (x^{3}-x \right )^{\frac {1}{3}}}{4}\) | \(21\) |
meijerg | \(-\frac {3 \operatorname {signum}\left (x^{2}-1\right )^{\frac {1}{3}} x^{\frac {4}{3}} \operatorname {hypergeom}\left (\left [-\frac {1}{3}, \frac {2}{3}\right ], \left [\frac {5}{3}\right ], x^{2}\right )}{4 {\left (-\operatorname {signum}\left (x^{2}-1\right )\right )}^{\frac {1}{3}}}+\frac {9 \operatorname {signum}\left (x^{2}-1\right )^{\frac {1}{3}} x^{\frac {10}{3}} \operatorname {hypergeom}\left (\left [-\frac {1}{3}, \frac {5}{3}\right ], \left [\frac {8}{3}\right ], x^{2}\right )}{10 {\left (-\operatorname {signum}\left (x^{2}-1\right )\right )}^{\frac {1}{3}}}\) | \(66\) |
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Time = 0.24 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.73 \[ \int \left (-1+3 x^2\right ) \sqrt [3]{-x+x^3} \, dx=\frac {3}{4} \, {\left (x^{3} - x\right )}^{\frac {4}{3}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 27 vs. \(2 (10) = 20\).
Time = 0.07 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.80 \[ \int \left (-1+3 x^2\right ) \sqrt [3]{-x+x^3} \, dx=\frac {3 x^{3} \sqrt [3]{x^{3} - x}}{4} - \frac {3 x \sqrt [3]{x^{3} - x}}{4} \]
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Time = 0.17 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.73 \[ \int \left (-1+3 x^2\right ) \sqrt [3]{-x+x^3} \, dx=\frac {3}{4} \, {\left (x^{3} - x\right )}^{\frac {4}{3}} \]
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Time = 0.29 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.73 \[ \int \left (-1+3 x^2\right ) \sqrt [3]{-x+x^3} \, dx=\frac {3}{4} \, {\left (x^{3} - x\right )}^{\frac {4}{3}} \]
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Time = 5.03 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.73 \[ \int \left (-1+3 x^2\right ) \sqrt [3]{-x+x^3} \, dx=\frac {3\,{\left (x^3-x\right )}^{4/3}}{4} \]
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