Integrand size = 18, antiderivative size = 14 \[ \int \frac {2+x^6}{x^2 \left (-1+x^6\right )^{3/4}} \, dx=\frac {2 \sqrt [4]{-1+x^6}}{x} \]
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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, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {460} \[ \int \frac {2+x^6}{x^2 \left (-1+x^6\right )^{3/4}} \, dx=\frac {2 \sqrt [4]{x^6-1}}{x} \]
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Rule 460
Rubi steps \begin{align*} \text {integral}& = \frac {2 \sqrt [4]{-1+x^6}}{x} \\ \end{align*}
Time = 0.37 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00 \[ \int \frac {2+x^6}{x^2 \left (-1+x^6\right )^{3/4}} \, dx=\frac {2 \sqrt [4]{-1+x^6}}{x} \]
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Time = 0.87 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.93
method | result | size |
trager | \(\frac {2 \left (x^{6}-1\right )^{\frac {1}{4}}}{x}\) | \(13\) |
risch | \(\frac {2 \left (x^{6}-1\right )^{\frac {1}{4}}}{x}\) | \(13\) |
pseudoelliptic | \(\frac {2 \left (x^{6}-1\right )^{\frac {1}{4}}}{x}\) | \(13\) |
gosper | \(\frac {2 \left (x -1\right ) \left (1+x \right ) \left (x^{2}+x +1\right ) \left (x^{2}-x +1\right )}{x \left (x^{6}-1\right )^{\frac {3}{4}}}\) | \(33\) |
meijerg | \(\frac {{\left (-\operatorname {signum}\left (x^{6}-1\right )\right )}^{\frac {3}{4}} x^{5} \operatorname {hypergeom}\left (\left [\frac {3}{4}, \frac {5}{6}\right ], \left [\frac {11}{6}\right ], x^{6}\right )}{5 \operatorname {signum}\left (x^{6}-1\right )^{\frac {3}{4}}}-\frac {2 {\left (-\operatorname {signum}\left (x^{6}-1\right )\right )}^{\frac {3}{4}} \operatorname {hypergeom}\left (\left [-\frac {1}{6}, \frac {3}{4}\right ], \left [\frac {5}{6}\right ], x^{6}\right )}{\operatorname {signum}\left (x^{6}-1\right )^{\frac {3}{4}} x}\) | \(66\) |
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none
Time = 0.24 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.86 \[ \int \frac {2+x^6}{x^2 \left (-1+x^6\right )^{3/4}} \, dx=\frac {2 \, {\left (x^{6} - 1\right )}^{\frac {1}{4}}}{x} \]
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Result contains complex when optimal does not.
Time = 0.90 (sec) , antiderivative size = 66, normalized size of antiderivative = 4.71 \[ \int \frac {2+x^6}{x^2 \left (-1+x^6\right )^{3/4}} \, dx=\frac {x^{5} e^{- \frac {3 i \pi }{4}} \Gamma \left (\frac {5}{6}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {3}{4}, \frac {5}{6} \\ \frac {11}{6} \end {matrix}\middle | {x^{6}} \right )}}{6 \Gamma \left (\frac {11}{6}\right )} - \frac {e^{\frac {i \pi }{4}} \Gamma \left (- \frac {1}{6}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{6}, \frac {3}{4} \\ \frac {5}{6} \end {matrix}\middle | {x^{6}} \right )}}{3 x \Gamma \left (\frac {5}{6}\right )} \]
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Leaf count of result is larger than twice the leaf count of optimal. 33 vs. \(2 (12) = 24\).
Time = 0.32 (sec) , antiderivative size = 33, normalized size of antiderivative = 2.36 \[ \int \frac {2+x^6}{x^2 \left (-1+x^6\right )^{3/4}} \, dx=\frac {2 \, {\left (x^{2} + x + 1\right )}^{\frac {1}{4}} {\left (x^{2} - x + 1\right )}^{\frac {1}{4}} {\left (x + 1\right )}^{\frac {1}{4}} {\left (x - 1\right )}^{\frac {1}{4}}}{x} \]
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\[ \int \frac {2+x^6}{x^2 \left (-1+x^6\right )^{3/4}} \, dx=\int { \frac {x^{6} + 2}{{\left (x^{6} - 1\right )}^{\frac {3}{4}} x^{2}} \,d x } \]
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Time = 4.91 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.86 \[ \int \frac {2+x^6}{x^2 \left (-1+x^6\right )^{3/4}} \, dx=\frac {2\,{\left (x^6-1\right )}^{1/4}}{x} \]
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