Integrand size = 22, antiderivative size = 16 \[ \int \frac {\sqrt [4]{x^2+x^4}}{x^2 \left (1+x^2\right )} \, dx=-\frac {2 \sqrt [4]{x^2+x^4}}{x} \]
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Time = 0.02 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {1325, 2025, 1160, 277, 270} \[ \int \frac {\sqrt [4]{x^2+x^4}}{x^2 \left (1+x^2\right )} \, dx=-\frac {2 \sqrt [4]{x^4+x^2}}{x} \]
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Rule 270
Rule 277
Rule 1160
Rule 1325
Rule 2025
Rubi steps \begin{align*} \text {integral}& = \int \frac {1}{\left (x^2+x^4\right )^{3/4}} \, dx \\ & = -\frac {2 \sqrt [4]{x^2+x^4}}{x} \\ \end{align*}
Time = 0.16 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt [4]{x^2+x^4}}{x^2 \left (1+x^2\right )} \, dx=-\frac {2 \sqrt [4]{x^2+x^4}}{x} \]
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Time = 0.98 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.81
method | result | size |
meijerg | \(-\frac {2 \left (x^{2}+1\right )^{\frac {1}{4}}}{\sqrt {x}}\) | \(13\) |
gosper | \(-\frac {2 \left (x^{4}+x^{2}\right )^{\frac {1}{4}}}{x}\) | \(15\) |
trager | \(-\frac {2 \left (x^{4}+x^{2}\right )^{\frac {1}{4}}}{x}\) | \(15\) |
risch | \(-\frac {2 \left (x^{2} \left (x^{2}+1\right )\right )^{\frac {1}{4}}}{x}\) | \(17\) |
pseudoelliptic | \(-\frac {2 \left (x^{2} \left (x^{2}+1\right )\right )^{\frac {1}{4}}}{x}\) | \(17\) |
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none
Time = 0.25 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.88 \[ \int \frac {\sqrt [4]{x^2+x^4}}{x^2 \left (1+x^2\right )} \, dx=-\frac {2 \, {\left (x^{4} + x^{2}\right )}^{\frac {1}{4}}}{x} \]
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\[ \int \frac {\sqrt [4]{x^2+x^4}}{x^2 \left (1+x^2\right )} \, dx=\int \frac {\sqrt [4]{x^{2} \left (x^{2} + 1\right )}}{x^{2} \left (x^{2} + 1\right )}\, dx \]
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none
Time = 0.28 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.06 \[ \int \frac {\sqrt [4]{x^2+x^4}}{x^2 \left (1+x^2\right )} \, dx=-\frac {2 \, {\left (x^{3} + x\right )}}{{\left (x^{2} + 1\right )}^{\frac {3}{4}} x^{\frac {3}{2}}} \]
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none
Time = 0.30 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.56 \[ \int \frac {\sqrt [4]{x^2+x^4}}{x^2 \left (1+x^2\right )} \, dx=-2 \, {\left (\frac {1}{x^{2}} + 1\right )}^{\frac {1}{4}} \]
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Time = 4.97 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.88 \[ \int \frac {\sqrt [4]{x^2+x^4}}{x^2 \left (1+x^2\right )} \, dx=-\frac {2\,{\left (x^4+x^2\right )}^{1/4}}{x} \]
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