Integrand size = 13, antiderivative size = 13 \[ \int \frac {1}{x^2 \left (-1+x^4\right )^{3/4}} \, dx=\frac {\sqrt [4]{-1+x^4}}{x} \]
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Time = 0.00 (sec) , antiderivative size = 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 = {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 = 0.14 (sec) , antiderivative size = 13, 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 = 0.78 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.92
method | result | size |
trager | \(\frac {\left (x^{4}-1\right )^{\frac {1}{4}}}{x}\) | \(12\) |
risch | \(\frac {\left (x^{4}-1\right )^{\frac {1}{4}}}{x}\) | \(12\) |
pseudoelliptic | \(\frac {\left (x^{4}-1\right )^{\frac {1}{4}}}{x}\) | \(12\) |
gosper | \(\frac {\left (x -1\right ) \left (1+x \right ) \left (x^{2}+1\right )}{x \left (x^{4}-1\right )^{\frac {3}{4}}}\) | \(23\) |
meijerg | \(-\frac {{\left (-\operatorname {signum}\left (x^{4}-1\right )\right )}^{\frac {3}{4}} \left (-x^{4}+1\right )^{\frac {1}{4}}}{\operatorname {signum}\left (x^{4}-1\right )^{\frac {3}{4}} x}\) | \(33\) |
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none
Time = 0.25 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.85 \[ \int \frac {1}{x^2 \left (-1+x^4\right )^{3/4}} \, dx=\frac {{\left (x^{4} - 1\right )}^{\frac {1}{4}}}{x} \]
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Result contains complex when optimal does not.
Time = 0.36 (sec) , antiderivative size = 61, normalized size of antiderivative = 4.69 \[ \int \frac {1}{x^2 \left (-1+x^4\right )^{3/4}} \, dx=\begin {cases} - \frac {\sqrt [4]{-1 + \frac {1}{x^{4}}} e^{\frac {i \pi }{4}} \Gamma \left (- \frac {1}{4}\right )}{4 \Gamma \left (\frac {3}{4}\right )} & \text {for}\: \frac {1}{\left |{x^{4}}\right |} > 1 \\- \frac {\sqrt [4]{1 - \frac {1}{x^{4}}} \Gamma \left (- \frac {1}{4}\right )}{4 \Gamma \left (\frac {3}{4}\right )} & \text {otherwise} \end {cases} \]
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none
Time = 0.20 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.85 \[ \int \frac {1}{x^2 \left (-1+x^4\right )^{3/4}} \, dx=\frac {{\left (x^{4} - 1\right )}^{\frac {1}{4}}}{x} \]
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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 = 5.05 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.85 \[ \int \frac {1}{x^2 \left (-1+x^4\right )^{3/4}} \, dx=\frac {{\left (x^4-1\right )}^{1/4}}{x} \]
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