Integrand size = 13, antiderivative size = 13 \[ \int x^3 \sqrt [3]{-1+x^4} \, dx=\frac {3}{16} \left (-1+x^4\right )^{4/3} \]
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Time = 0.00 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {267} \[ \int x^3 \sqrt [3]{-1+x^4} \, dx=\frac {3}{16} \left (x^4-1\right )^{4/3} \]
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Rule 267
Rubi steps \begin{align*} \text {integral}& = \frac {3}{16} \left (-1+x^4\right )^{4/3} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.00 \[ \int x^3 \sqrt [3]{-1+x^4} \, dx=\frac {3}{16} \left (-1+x^4\right )^{4/3} \]
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Time = 0.78 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.77
method | result | size |
derivativedivides | \(\frac {3 \left (x^{4}-1\right )^{\frac {4}{3}}}{16}\) | \(10\) |
default | \(\frac {3 \left (x^{4}-1\right )^{\frac {4}{3}}}{16}\) | \(10\) |
risch | \(\frac {3 \left (x^{4}-1\right )^{\frac {4}{3}}}{16}\) | \(10\) |
pseudoelliptic | \(\frac {3 \left (x^{4}-1\right )^{\frac {4}{3}}}{16}\) | \(10\) |
trager | \(\left (\frac {3 x^{4}}{16}-\frac {3}{16}\right ) \left (x^{4}-1\right )^{\frac {1}{3}}\) | \(16\) |
gosper | \(\frac {3 \left (x -1\right ) \left (1+x \right ) \left (x^{2}+1\right ) \left (x^{4}-1\right )^{\frac {1}{3}}}{16}\) | \(21\) |
meijerg | \(\frac {\operatorname {signum}\left (x^{4}-1\right )^{\frac {1}{3}} x^{4} \operatorname {hypergeom}\left (\left [-\frac {1}{3}, 1\right ], \left [2\right ], x^{4}\right )}{4 {\left (-\operatorname {signum}\left (x^{4}-1\right )\right )}^{\frac {1}{3}}}\) | \(33\) |
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Time = 0.23 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.69 \[ \int x^3 \sqrt [3]{-1+x^4} \, dx=\frac {3}{16} \, {\left (x^{4} - 1\right )}^{\frac {4}{3}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 26 vs. \(2 (10) = 20\).
Time = 0.07 (sec) , antiderivative size = 26, normalized size of antiderivative = 2.00 \[ \int x^3 \sqrt [3]{-1+x^4} \, dx=\frac {3 x^{4} \sqrt [3]{x^{4} - 1}}{16} - \frac {3 \sqrt [3]{x^{4} - 1}}{16} \]
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Time = 0.18 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.69 \[ \int x^3 \sqrt [3]{-1+x^4} \, dx=\frac {3}{16} \, {\left (x^{4} - 1\right )}^{\frac {4}{3}} \]
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Time = 0.28 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.69 \[ \int x^3 \sqrt [3]{-1+x^4} \, dx=\frac {3}{16} \, {\left (x^{4} - 1\right )}^{\frac {4}{3}} \]
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Time = 4.93 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.69 \[ \int x^3 \sqrt [3]{-1+x^4} \, dx=\frac {3\,{\left (x^4-1\right )}^{4/3}}{16} \]
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