Integrand size = 13, antiderivative size = 14 \[ \int \frac {1}{x^2 \left (1+x^4\right )^{3/4}} \, dx=-\frac {\sqrt [4]{1+x^4}}{x} \]
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Time = 0.00 (sec) , antiderivative size = 14, 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 = {270} \[ \int \frac {1}{x^2 \left (1+x^4\right )^{3/4}} \, dx=-\frac {\sqrt [4]{x^4+1}}{x} \]
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Rule 270
Rubi steps \begin{align*} \text {integral}& = -\frac {\sqrt [4]{1+x^4}}{x} \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x^2 \left (1+x^4\right )^{3/4}} \, dx=-\frac {\sqrt [4]{1+x^4}}{x} \]
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Time = 0.18 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.93
method | result | size |
gosper | \(-\frac {\left (x^{4}+1\right )^{\frac {1}{4}}}{x}\) | \(13\) |
trager | \(-\frac {\left (x^{4}+1\right )^{\frac {1}{4}}}{x}\) | \(13\) |
meijerg | \(-\frac {\left (x^{4}+1\right )^{\frac {1}{4}}}{x}\) | \(13\) |
risch | \(-\frac {\left (x^{4}+1\right )^{\frac {1}{4}}}{x}\) | \(13\) |
pseudoelliptic | \(-\frac {\left (x^{4}+1\right )^{\frac {1}{4}}}{x}\) | \(13\) |
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none
Time = 0.27 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.86 \[ \int \frac {1}{x^2 \left (1+x^4\right )^{3/4}} \, dx=-\frac {{\left (x^{4} + 1\right )}^{\frac {1}{4}}}{x} \]
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Leaf count of result is larger than twice the leaf count of optimal. 22 vs. \(2 (10) = 20\).
Time = 0.32 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.57 \[ \int \frac {1}{x^2 \left (1+x^4\right )^{3/4}} \, dx=\frac {\sqrt [4]{1 + \frac {1}{x^{4}}} \Gamma \left (- \frac {1}{4}\right )}{4 \Gamma \left (\frac {3}{4}\right )} \]
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none
Time = 0.20 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.86 \[ \int \frac {1}{x^2 \left (1+x^4\right )^{3/4}} \, dx=-\frac {{\left (x^{4} + 1\right )}^{\frac {1}{4}}}{x} \]
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\[ \int \frac {1}{x^2 \left (1+x^4\right )^{3/4}} \, dx=\int { \frac {1}{{\left (x^{4} + 1\right )}^{\frac {3}{4}} x^{2}} \,d x } \]
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Time = 15.02 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.86 \[ \int \frac {1}{x^2 \left (1+x^4\right )^{3/4}} \, dx=-\frac {{\left (x^4+1\right )}^{1/4}}{x} \]
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