Integrand size = 18, antiderivative size = 16 \[ \int \frac {\sqrt [3]{-1+x^6} \left (1+x^6\right )}{x^5} \, dx=\frac {\left (-1+x^6\right )^{4/3}}{4 x^4} \]
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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} \[ \int \frac {\sqrt [3]{-1+x^6} \left (1+x^6\right )}{x^5} \, dx=\frac {\left (x^6-1\right )^{4/3}}{4 x^4} \]
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Rule 460
Rubi steps \begin{align*} \text {integral}& = \frac {\left (-1+x^6\right )^{4/3}}{4 x^4} \\ \end{align*}
Time = 0.14 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt [3]{-1+x^6} \left (1+x^6\right )}{x^5} \, dx=\frac {\left (-1+x^6\right )^{4/3}}{4 x^4} \]
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Time = 0.82 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.81
method | result | size |
trager | \(\frac {\left (x^{6}-1\right )^{\frac {4}{3}}}{4 x^{4}}\) | \(13\) |
pseudoelliptic | \(\frac {\left (x^{6}-1\right )^{\frac {4}{3}}}{4 x^{4}}\) | \(13\) |
risch | \(\frac {x^{12}-2 x^{6}+1}{4 \left (x^{6}-1\right )^{\frac {2}{3}} x^{4}}\) | \(23\) |
gosper | \(\frac {\left (x^{6}-1\right )^{\frac {1}{3}} \left (x -1\right ) \left (1+x \right ) \left (x^{2}+x +1\right ) \left (x^{2}-x +1\right )}{4 x^{4}}\) | \(33\) |
meijerg | \(\frac {\operatorname {signum}\left (x^{6}-1\right )^{\frac {1}{3}} x^{2} \operatorname {hypergeom}\left (\left [-\frac {1}{3}, \frac {1}{3}\right ], \left [\frac {4}{3}\right ], x^{6}\right )}{2 {\left (-\operatorname {signum}\left (x^{6}-1\right )\right )}^{\frac {1}{3}}}-\frac {\operatorname {signum}\left (x^{6}-1\right )^{\frac {1}{3}} \operatorname {hypergeom}\left (\left [-\frac {2}{3}, -\frac {1}{3}\right ], \left [\frac {1}{3}\right ], x^{6}\right )}{4 {\left (-\operatorname {signum}\left (x^{6}-1\right )\right )}^{\frac {1}{3}} x^{4}}\) | \(66\) |
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none
Time = 0.26 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.75 \[ \int \frac {\sqrt [3]{-1+x^6} \left (1+x^6\right )}{x^5} \, dx=\frac {{\left (x^{6} - 1\right )}^{\frac {4}{3}}}{4 \, x^{4}} \]
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Result contains complex when optimal does not.
Time = 1.50 (sec) , antiderivative size = 71, normalized size of antiderivative = 4.44 \[ \int \frac {\sqrt [3]{-1+x^6} \left (1+x^6\right )}{x^5} \, dx=\frac {x^{2} e^{\frac {i \pi }{3}} \Gamma \left (\frac {1}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{3}, \frac {1}{3} \\ \frac {4}{3} \end {matrix}\middle | {x^{6}} \right )}}{6 \Gamma \left (\frac {4}{3}\right )} - \frac {e^{- \frac {2 i \pi }{3}} \Gamma \left (- \frac {2}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {2}{3}, - \frac {1}{3} \\ \frac {1}{3} \end {matrix}\middle | {x^{6}} \right )}}{6 x^{4} \Gamma \left (\frac {1}{3}\right )} \]
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Leaf count of result is larger than twice the leaf count of optimal. 38 vs. \(2 (12) = 24\).
Time = 0.33 (sec) , antiderivative size = 38, normalized size of antiderivative = 2.38 \[ \int \frac {\sqrt [3]{-1+x^6} \left (1+x^6\right )}{x^5} \, dx=\frac {{\left (x^{6} - 1\right )} {\left (x^{2} + x + 1\right )}^{\frac {1}{3}} {\left (x^{2} - x + 1\right )}^{\frac {1}{3}} {\left (x + 1\right )}^{\frac {1}{3}} {\left (x - 1\right )}^{\frac {1}{3}}}{4 \, x^{4}} \]
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\[ \int \frac {\sqrt [3]{-1+x^6} \left (1+x^6\right )}{x^5} \, dx=\int { \frac {{\left (x^{6} + 1\right )} {\left (x^{6} - 1\right )}^{\frac {1}{3}}}{x^{5}} \,d x } \]
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Time = 5.11 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.75 \[ \int \frac {\sqrt [3]{-1+x^6} \left (1+x^6\right )}{x^5} \, dx=\frac {{\left (x^6-1\right )}^{4/3}}{4\,x^4} \]
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