Integrand size = 27, antiderivative size = 12 \[ \int \frac {-1+x^8}{\sqrt {-1+x^4} \left (1-2 x^4+x^8\right )} \, dx=-\frac {x}{\sqrt {-1+x^4}} \]
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Time = 0.01 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {28, 1418, 391} \[ \int \frac {-1+x^8}{\sqrt {-1+x^4} \left (1-2 x^4+x^8\right )} \, dx=-\frac {x}{\sqrt {x^4-1}} \]
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Rule 28
Rule 391
Rule 1418
Rubi steps \begin{align*} \text {integral}& = \int \frac {-1+x^8}{\left (-1+x^4\right )^{5/2}} \, dx \\ & = \int \frac {1+x^4}{\left (-1+x^4\right )^{3/2}} \, dx \\ & = -\frac {x}{\sqrt {-1+x^4}} \\ \end{align*}
Time = 0.20 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.00 \[ \int \frac {-1+x^8}{\sqrt {-1+x^4} \left (1-2 x^4+x^8\right )} \, dx=-\frac {x}{\sqrt {-1+x^4}} \]
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Time = 0.89 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.92
method | result | size |
gosper | \(-\frac {x}{\sqrt {x^{4}-1}}\) | \(11\) |
default | \(-\frac {x}{\sqrt {x^{4}-1}}\) | \(11\) |
trager | \(-\frac {x}{\sqrt {x^{4}-1}}\) | \(11\) |
risch | \(-\frac {x}{\sqrt {x^{4}-1}}\) | \(11\) |
elliptic | \(-\frac {x}{\sqrt {x^{4}-1}}\) | \(11\) |
pseudoelliptic | \(-\frac {x}{\sqrt {x^{4}-1}}\) | \(11\) |
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none
Time = 0.24 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.83 \[ \int \frac {-1+x^8}{\sqrt {-1+x^4} \left (1-2 x^4+x^8\right )} \, dx=-\frac {x}{\sqrt {x^{4} - 1}} \]
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\[ \int \frac {-1+x^8}{\sqrt {-1+x^4} \left (1-2 x^4+x^8\right )} \, dx=\int \frac {x^{4} + 1}{\sqrt {\left (x - 1\right ) \left (x + 1\right ) \left (x^{2} + 1\right )} \left (x - 1\right ) \left (x + 1\right ) \left (x^{2} + 1\right )}\, dx \]
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Leaf count of result is larger than twice the leaf count of optimal. 27 vs. \(2 (10) = 20\).
Time = 0.32 (sec) , antiderivative size = 27, normalized size of antiderivative = 2.25 \[ \int \frac {-1+x^8}{\sqrt {-1+x^4} \left (1-2 x^4+x^8\right )} \, dx=-\frac {\sqrt {x^{2} + 1} \sqrt {x + 1} \sqrt {x - 1} x}{x^{4} - 1} \]
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none
Time = 0.30 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.83 \[ \int \frac {-1+x^8}{\sqrt {-1+x^4} \left (1-2 x^4+x^8\right )} \, dx=-\frac {x}{\sqrt {x^{4} - 1}} \]
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Time = 5.03 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.83 \[ \int \frac {-1+x^8}{\sqrt {-1+x^4} \left (1-2 x^4+x^8\right )} \, dx=-\frac {x}{\sqrt {x^4-1}} \]
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