Integrand size = 15, antiderivative size = 18 \[ \int \frac {1}{x^2 \sqrt [4]{x^2+x^4}} \, dx=-\frac {2 \left (x^2+x^4\right )^{3/4}}{3 x^3} \]
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Time = 0.02 (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} \[ \int \frac {1}{x^2 \sqrt [4]{x^2+x^4}} \, dx=-\frac {2 \left (x^4+x^2\right )^{3/4}}{3 x^3} \]
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Rule 2039
Rubi steps \begin{align*} \text {integral}& = -\frac {2 \left (x^2+x^4\right )^{3/4}}{3 x^3} \\ \end{align*}
Time = 0.13 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x^2 \sqrt [4]{x^2+x^4}} \, dx=-\frac {2 \left (x^2+x^4\right )^{3/4}}{3 x^3} \]
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Time = 0.81 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.72
method | result | size |
meijerg | \(-\frac {2 \left (x^{2}+1\right )^{\frac {3}{4}}}{3 x^{\frac {3}{2}}}\) | \(13\) |
trager | \(-\frac {2 \left (x^{4}+x^{2}\right )^{\frac {3}{4}}}{3 x^{3}}\) | \(15\) |
pseudoelliptic | \(-\frac {2 \left (x^{2} \left (x^{2}+1\right )\right )^{\frac {3}{4}}}{3 x^{3}}\) | \(17\) |
gosper | \(-\frac {2 \left (x^{2}+1\right )}{3 x \left (x^{4}+x^{2}\right )^{\frac {1}{4}}}\) | \(20\) |
risch | \(-\frac {2 \left (x^{2}+1\right )}{3 x \left (x^{2} \left (x^{2}+1\right )\right )^{\frac {1}{4}}}\) | \(22\) |
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none
Time = 0.24 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.78 \[ \int \frac {1}{x^2 \sqrt [4]{x^2+x^4}} \, dx=-\frac {2 \, {\left (x^{4} + x^{2}\right )}^{\frac {3}{4}}}{3 \, x^{3}} \]
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\[ \int \frac {1}{x^2 \sqrt [4]{x^2+x^4}} \, dx=\int \frac {1}{x^{2} \sqrt [4]{x^{2} \left (x^{2} + 1\right )}}\, dx \]
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none
Time = 0.27 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.94 \[ \int \frac {1}{x^2 \sqrt [4]{x^2+x^4}} \, dx=-\frac {2 \, {\left (x^{3} + x\right )}}{3 \, {\left (x^{2} + 1\right )}^{\frac {1}{4}} x^{\frac {5}{2}}} \]
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none
Time = 0.28 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.50 \[ \int \frac {1}{x^2 \sqrt [4]{x^2+x^4}} \, dx=-\frac {2}{3} \, {\left (\frac {1}{x^{2}} + 1\right )}^{\frac {3}{4}} \]
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Time = 5.22 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.78 \[ \int \frac {1}{x^2 \sqrt [4]{x^2+x^4}} \, dx=-\frac {2\,{\left (x^4+x^2\right )}^{3/4}}{3\,x^3} \]
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