Integrand size = 18, antiderivative size = 16 \[ \int \frac {-1+x^8}{x^4 \sqrt {-1+x^4}} \, dx=\frac {\left (-1+x^4\right )^{3/2}}{3 x^3} \]
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Time = 0.01 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {1493, 460} \[ \int \frac {-1+x^8}{x^4 \sqrt {-1+x^4}} \, dx=\frac {\left (x^4-1\right )^{3/2}}{3 x^3} \]
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Rule 460
Rule 1493
Rubi steps \begin{align*} \text {integral}& = \int \frac {\sqrt {-1+x^4} \left (1+x^4\right )}{x^4} \, dx \\ & = \frac {\left (-1+x^4\right )^{3/2}}{3 x^3} \\ \end{align*}
Time = 0.15 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00 \[ \int \frac {-1+x^8}{x^4 \sqrt {-1+x^4}} \, dx=\frac {\left (-1+x^4\right )^{3/2}}{3 x^3} \]
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Time = 0.94 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.81
method | result | size |
trager | \(\frac {\left (x^{4}-1\right )^{\frac {3}{2}}}{3 x^{3}}\) | \(13\) |
elliptic | \(\frac {\left (x^{4}-1\right )^{\frac {3}{2}}}{3 x^{3}}\) | \(13\) |
risch | \(\frac {x^{8}-2 x^{4}+1}{3 x^{3} \sqrt {x^{4}-1}}\) | \(23\) |
gosper | \(\frac {\sqrt {x^{4}-1}\, \left (x -1\right ) \left (1+x \right ) \left (x^{2}+1\right )}{3 x^{3}}\) | \(24\) |
default | \(\frac {x \sqrt {x^{4}-1}}{3}-\frac {\sqrt {x^{4}-1}}{3 x^{3}}\) | \(24\) |
pseudoelliptic | \(-\frac {\left (1+x \right ) \left (x -1\right ) \left (i-x \right ) \sqrt {x^{4}-1}\, \left (i+x \right )}{3 x^{3}}\) | \(29\) |
meijerg | \(\frac {\sqrt {-\operatorname {signum}\left (x^{4}-1\right )}\, x^{5} \operatorname {hypergeom}\left (\left [\frac {1}{2}, \frac {5}{4}\right ], \left [\frac {9}{4}\right ], x^{4}\right )}{5 \sqrt {\operatorname {signum}\left (x^{4}-1\right )}}+\frac {\sqrt {-\operatorname {signum}\left (x^{4}-1\right )}\, \operatorname {hypergeom}\left (\left [-\frac {3}{4}, \frac {1}{2}\right ], \left [\frac {1}{4}\right ], x^{4}\right )}{3 \sqrt {\operatorname {signum}\left (x^{4}-1\right )}\, x^{3}}\) | \(66\) |
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none
Time = 0.24 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.75 \[ \int \frac {-1+x^8}{x^4 \sqrt {-1+x^4}} \, dx=\frac {{\left (x^{4} - 1\right )}^{\frac {3}{2}}}{3 \, x^{3}} \]
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Result contains complex when optimal does not.
Time = 0.94 (sec) , antiderivative size = 56, normalized size of antiderivative = 3.50 \[ \int \frac {-1+x^8}{x^4 \sqrt {-1+x^4}} \, dx=- \frac {i x^{5} \Gamma \left (\frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {5}{4} \\ \frac {9}{4} \end {matrix}\middle | {x^{4}} \right )}}{4 \Gamma \left (\frac {9}{4}\right )} + \frac {i \Gamma \left (- \frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {3}{4}, \frac {1}{2} \\ \frac {1}{4} \end {matrix}\middle | {x^{4}} \right )}}{4 x^{3} \Gamma \left (\frac {1}{4}\right )} \]
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Leaf count of result is larger than twice the leaf count of optimal. 27 vs. \(2 (12) = 24\).
Time = 0.31 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.69 \[ \int \frac {-1+x^8}{x^4 \sqrt {-1+x^4}} \, dx=\frac {{\left (x^{4} - 1\right )} \sqrt {x^{2} + 1} \sqrt {x + 1} \sqrt {x - 1}}{3 \, x^{3}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 25 vs. \(2 (12) = 24\).
Time = 0.30 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.56 \[ \int \frac {-1+x^8}{x^4 \sqrt {-1+x^4}} \, dx=\frac {1}{3} \, \sqrt {x^{4} - 1} x - \frac {\sqrt {-\frac {1}{x^{4}} + 1}}{3 \, x} \]
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Time = 5.10 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.75 \[ \int \frac {-1+x^8}{x^4 \sqrt {-1+x^4}} \, dx=\frac {{\left (x^4-1\right )}^{3/2}}{3\,x^3} \]
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