Integrand size = 18, antiderivative size = 16 \[ \int \frac {\left (-1+x^4\right )^{2/3} \left (3+x^4\right )}{x^6} \, dx=\frac {3 \left (-1+x^4\right )^{5/3}}{5 x^5} \]
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Time = 0.00 (sec) , antiderivative size = 16, 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} \[ \int \frac {\left (-1+x^4\right )^{2/3} \left (3+x^4\right )}{x^6} \, dx=\frac {3 \left (x^4-1\right )^{5/3}}{5 x^5} \]
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Rule 460
Rubi steps \begin{align*} \text {integral}& = \frac {3 \left (-1+x^4\right )^{5/3}}{5 x^5} \\ \end{align*}
Time = 0.18 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00 \[ \int \frac {\left (-1+x^4\right )^{2/3} \left (3+x^4\right )}{x^6} \, dx=\frac {3 \left (-1+x^4\right )^{5/3}}{5 x^5} \]
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Time = 0.87 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.81
method | result | size |
trager | \(\frac {3 \left (x^{4}-1\right )^{\frac {5}{3}}}{5 x^{5}}\) | \(13\) |
pseudoelliptic | \(\frac {3 \left (x^{4}-1\right )^{\frac {5}{3}}}{5 x^{5}}\) | \(13\) |
risch | \(\frac {\frac {3}{5} x^{8}-\frac {6}{5} x^{4}+\frac {3}{5}}{x^{5} \left (x^{4}-1\right )^{\frac {1}{3}}}\) | \(23\) |
gosper | \(\frac {3 \left (x -1\right ) \left (1+x \right ) \left (x^{2}+1\right ) \left (x^{4}-1\right )^{\frac {2}{3}}}{5 x^{5}}\) | \(24\) |
meijerg | \(-\frac {\operatorname {signum}\left (x^{4}-1\right )^{\frac {2}{3}} \operatorname {hypergeom}\left (\left [-\frac {2}{3}, -\frac {1}{4}\right ], \left [\frac {3}{4}\right ], x^{4}\right )}{{\left (-\operatorname {signum}\left (x^{4}-1\right )\right )}^{\frac {2}{3}} x}-\frac {3 \operatorname {signum}\left (x^{4}-1\right )^{\frac {2}{3}} \operatorname {hypergeom}\left (\left [-\frac {5}{4}, -\frac {2}{3}\right ], \left [-\frac {1}{4}\right ], x^{4}\right )}{5 {\left (-\operatorname {signum}\left (x^{4}-1\right )\right )}^{\frac {2}{3}} x^{5}}\) | \(66\) |
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none
Time = 0.24 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.75 \[ \int \frac {\left (-1+x^4\right )^{2/3} \left (3+x^4\right )}{x^6} \, dx=\frac {3 \, {\left (x^{4} - 1\right )}^{\frac {5}{3}}}{5 \, x^{5}} \]
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Result contains complex when optimal does not.
Time = 1.14 (sec) , antiderivative size = 78, normalized size of antiderivative = 4.88 \[ \int \frac {\left (-1+x^4\right )^{2/3} \left (3+x^4\right )}{x^6} \, dx=- \frac {e^{- \frac {i \pi }{3}} \Gamma \left (- \frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {2}{3}, - \frac {1}{4} \\ \frac {3}{4} \end {matrix}\middle | {x^{4}} \right )}}{4 x \Gamma \left (\frac {3}{4}\right )} - \frac {3 e^{- \frac {i \pi }{3}} \Gamma \left (- \frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {5}{4}, - \frac {2}{3} \\ - \frac {1}{4} \end {matrix}\middle | {x^{4}} \right )}}{4 x^{5} \Gamma \left (- \frac {1}{4}\right )} \]
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Leaf count of result is larger than twice the leaf count of optimal. 27 vs. \(2 (12) = 24\).
Time = 0.34 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.69 \[ \int \frac {\left (-1+x^4\right )^{2/3} \left (3+x^4\right )}{x^6} \, dx=\frac {3 \, {\left (x^{4} - 1\right )} {\left (x^{2} + 1\right )}^{\frac {2}{3}} {\left (x + 1\right )}^{\frac {2}{3}} {\left (x - 1\right )}^{\frac {2}{3}}}{5 \, x^{5}} \]
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\[ \int \frac {\left (-1+x^4\right )^{2/3} \left (3+x^4\right )}{x^6} \, dx=\int { \frac {{\left (x^{4} + 3\right )} {\left (x^{4} - 1\right )}^{\frac {2}{3}}}{x^{6}} \,d x } \]
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Time = 5.10 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.69 \[ \int \frac {\left (-1+x^4\right )^{2/3} \left (3+x^4\right )}{x^6} \, dx=-\frac {3\,{\left (x^4-1\right )}^{2/3}-3\,x^4\,{\left (x^4-1\right )}^{2/3}}{5\,x^5} \]
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