Integrand size = 15, antiderivative size = 18 \[ \int \frac {1}{x^3 \sqrt [3]{x^2+x^6}} \, dx=-\frac {3 \left (x^2+x^6\right )^{2/3}}{8 x^4} \]
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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} \[ \int \frac {1}{x^3 \sqrt [3]{x^2+x^6}} \, dx=-\frac {3 \left (x^6+x^2\right )^{2/3}}{8 x^4} \]
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Rule 2039
Rubi steps \begin{align*} \text {integral}& = -\frac {3 \left (x^2+x^6\right )^{2/3}}{8 x^4} \\ \end{align*}
Time = 10.01 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x^3 \sqrt [3]{x^2+x^6}} \, dx=-\frac {3 \left (x^2+x^6\right )^{2/3}}{8 x^4} \]
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Time = 0.81 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.72
method | result | size |
meijerg | \(-\frac {3 \left (x^{4}+1\right )^{\frac {2}{3}}}{8 x^{\frac {8}{3}}}\) | \(13\) |
trager | \(-\frac {3 \left (x^{6}+x^{2}\right )^{\frac {2}{3}}}{8 x^{4}}\) | \(15\) |
pseudoelliptic | \(-\frac {3 \left (x^{2} \left (x^{4}+1\right )\right )^{\frac {2}{3}}}{8 x^{4}}\) | \(17\) |
gosper | \(-\frac {3 \left (x^{4}+1\right )}{8 x^{2} \left (x^{6}+x^{2}\right )^{\frac {1}{3}}}\) | \(20\) |
risch | \(-\frac {3 \left (x^{4}+1\right )}{8 x^{2} \left (x^{2} \left (x^{4}+1\right )\right )^{\frac {1}{3}}}\) | \(22\) |
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none
Time = 0.24 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.78 \[ \int \frac {1}{x^3 \sqrt [3]{x^2+x^6}} \, dx=-\frac {3 \, {\left (x^{6} + x^{2}\right )}^{\frac {2}{3}}}{8 \, x^{4}} \]
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\[ \int \frac {1}{x^3 \sqrt [3]{x^2+x^6}} \, dx=\int \frac {1}{x^{3} \sqrt [3]{x^{2} \left (x^{4} + 1\right )}}\, dx \]
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none
Time = 0.26 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.17 \[ \int \frac {1}{x^3 \sqrt [3]{x^2+x^6}} \, dx=-\frac {3 \, {\left (x^{6} + x^{2}\right )}}{8 \, {\left (x^{4} + 1\right )}^{\frac {1}{3}} {\left (x^{2}\right )}^{\frac {7}{3}}} \]
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none
Time = 0.29 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.50 \[ \int \frac {1}{x^3 \sqrt [3]{x^2+x^6}} \, dx=-\frac {3}{8} \, {\left (\frac {1}{x^{4}} + 1\right )}^{\frac {2}{3}} \]
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Time = 5.08 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.78 \[ \int \frac {1}{x^3 \sqrt [3]{x^2+x^6}} \, dx=-\frac {3\,{\left (x^6+x^2\right )}^{2/3}}{8\,x^4} \]
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