Integrand size = 28, antiderivative size = 38 \[ \int \frac {\left (-1+x^6\right )^{3/4} \left (2+x^6\right ) \left (-1-x^4+x^6\right )}{x^{12}} \, dx=\frac {2 \left (-1+x^6\right )^{3/4} \left (7+11 x^4-14 x^6-11 x^{10}+7 x^{12}\right )}{77 x^{11}} \]
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Time = 0.06 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.87, number of steps used = 6, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {1847, 1598, 460, 1492} \[ \int \frac {\left (-1+x^6\right )^{3/4} \left (2+x^6\right ) \left (-1-x^4+x^6\right )}{x^{12}} \, dx=\frac {2 \left (x^6-1\right )^{11/4}}{11 x^{11}}-\frac {2 \left (x^6-1\right )^{7/4}}{7 x^7} \]
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Rule 460
Rule 1492
Rule 1598
Rule 1847
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {\left (-1+x^6\right )^{3/4} \left (-2 x^3-x^9\right )}{x^{11}}+\frac {\left (-1+x^6\right )^{3/4} \left (-2+x^6+x^{12}\right )}{x^{12}}\right ) \, dx \\ & = \int \frac {\left (-1+x^6\right )^{3/4} \left (-2 x^3-x^9\right )}{x^{11}} \, dx+\int \frac {\left (-1+x^6\right )^{3/4} \left (-2+x^6+x^{12}\right )}{x^{12}} \, dx \\ & = \int \frac {\left (-2-x^6\right ) \left (-1+x^6\right )^{3/4}}{x^8} \, dx+\int \frac {\left (-1+x^6\right )^{7/4} \left (2+x^6\right )}{x^{12}} \, dx \\ & = -\frac {2 \left (-1+x^6\right )^{7/4}}{7 x^7}+\frac {2 \left (-1+x^6\right )^{11/4}}{11 x^{11}} \\ \end{align*}
Time = 2.22 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.74 \[ \int \frac {\left (-1+x^6\right )^{3/4} \left (2+x^6\right ) \left (-1-x^4+x^6\right )}{x^{12}} \, dx=\frac {2 \left (-1+x^6\right )^{7/4} \left (-7-11 x^4+7 x^6\right )}{77 x^{11}} \]
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Time = 1.02 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.66
method | result | size |
pseudoelliptic | \(\frac {2 \left (x^{6}-1\right )^{\frac {7}{4}} \left (7 x^{6}-11 x^{4}-7\right )}{77 x^{11}}\) | \(25\) |
trager | \(\frac {2 \left (x^{6}-1\right )^{\frac {3}{4}} \left (7 x^{12}-11 x^{10}-14 x^{6}+11 x^{4}+7\right )}{77 x^{11}}\) | \(35\) |
gosper | \(\frac {2 \left (x -1\right ) \left (1+x \right ) \left (x^{2}+x +1\right ) \left (x^{2}-x +1\right ) \left (7 x^{6}-11 x^{4}-7\right ) \left (x^{6}-1\right )^{\frac {3}{4}}}{77 x^{11}}\) | \(45\) |
risch | \(\frac {\frac {2}{11} x^{18}-\frac {6}{11} x^{12}+\frac {6}{11} x^{6}-\frac {2}{11}-\frac {2}{7} x^{16}+\frac {4}{7} x^{10}-\frac {2}{7} x^{4}}{x^{11} \left (x^{6}-1\right )^{\frac {1}{4}}}\) | \(45\) |
meijerg | \(\frac {\operatorname {signum}\left (x^{6}-1\right )^{\frac {3}{4}} x \operatorname {hypergeom}\left (\left [-\frac {3}{4}, \frac {1}{6}\right ], \left [\frac {7}{6}\right ], x^{6}\right )}{{\left (-\operatorname {signum}\left (x^{6}-1\right )\right )}^{\frac {3}{4}}}+\frac {\operatorname {signum}\left (x^{6}-1\right )^{\frac {3}{4}} \operatorname {hypergeom}\left (\left [-\frac {3}{4}, -\frac {1}{6}\right ], \left [\frac {5}{6}\right ], x^{6}\right )}{{\left (-\operatorname {signum}\left (x^{6}-1\right )\right )}^{\frac {3}{4}} x}-\frac {\operatorname {signum}\left (x^{6}-1\right )^{\frac {3}{4}} \operatorname {hypergeom}\left (\left [-\frac {5}{6}, -\frac {3}{4}\right ], \left [\frac {1}{6}\right ], x^{6}\right )}{5 {\left (-\operatorname {signum}\left (x^{6}-1\right )\right )}^{\frac {3}{4}} x^{5}}+\frac {2 \operatorname {signum}\left (x^{6}-1\right )^{\frac {3}{4}} \operatorname {hypergeom}\left (\left [-\frac {7}{6}, -\frac {3}{4}\right ], \left [-\frac {1}{6}\right ], x^{6}\right )}{7 {\left (-\operatorname {signum}\left (x^{6}-1\right )\right )}^{\frac {3}{4}} x^{7}}+\frac {2 \operatorname {signum}\left (x^{6}-1\right )^{\frac {3}{4}} \operatorname {hypergeom}\left (\left [-\frac {11}{6}, -\frac {3}{4}\right ], \left [-\frac {5}{6}\right ], x^{6}\right )}{11 {\left (-\operatorname {signum}\left (x^{6}-1\right )\right )}^{\frac {3}{4}} x^{11}}\) | \(158\) |
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Time = 0.26 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.89 \[ \int \frac {\left (-1+x^6\right )^{3/4} \left (2+x^6\right ) \left (-1-x^4+x^6\right )}{x^{12}} \, dx=\frac {2 \, {\left (7 \, x^{12} - 11 \, x^{10} - 14 \, x^{6} + 11 \, x^{4} + 7\right )} {\left (x^{6} - 1\right )}^{\frac {3}{4}}}{77 \, x^{11}} \]
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Result contains complex when optimal does not.
Time = 3.24 (sec) , antiderivative size = 187, normalized size of antiderivative = 4.92 \[ \int \frac {\left (-1+x^6\right )^{3/4} \left (2+x^6\right ) \left (-1-x^4+x^6\right )}{x^{12}} \, dx=\frac {x e^{\frac {3 i \pi }{4}} \Gamma \left (\frac {1}{6}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {3}{4}, \frac {1}{6} \\ \frac {7}{6} \end {matrix}\middle | {x^{6}} \right )}}{6 \Gamma \left (\frac {7}{6}\right )} + \frac {e^{- \frac {i \pi }{4}} \Gamma \left (- \frac {1}{6}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {3}{4}, - \frac {1}{6} \\ \frac {5}{6} \end {matrix}\middle | {x^{6}} \right )}}{6 x \Gamma \left (\frac {5}{6}\right )} - \frac {e^{- \frac {i \pi }{4}} \Gamma \left (- \frac {5}{6}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {5}{6}, - \frac {3}{4} \\ \frac {1}{6} \end {matrix}\middle | {x^{6}} \right )}}{6 x^{5} \Gamma \left (\frac {1}{6}\right )} + \frac {e^{- \frac {i \pi }{4}} \Gamma \left (- \frac {7}{6}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {7}{6}, - \frac {3}{4} \\ - \frac {1}{6} \end {matrix}\middle | {x^{6}} \right )}}{3 x^{7} \Gamma \left (- \frac {1}{6}\right )} + \frac {e^{- \frac {i \pi }{4}} \Gamma \left (- \frac {11}{6}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {11}{6}, - \frac {3}{4} \\ - \frac {5}{6} \end {matrix}\middle | {x^{6}} \right )}}{3 x^{11} \Gamma \left (- \frac {5}{6}\right )} \]
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Time = 0.32 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.45 \[ \int \frac {\left (-1+x^6\right )^{3/4} \left (2+x^6\right ) \left (-1-x^4+x^6\right )}{x^{12}} \, dx=\frac {2 \, {\left (7 \, x^{12} - 11 \, x^{10} - 14 \, x^{6} + 11 \, x^{4} + 7\right )} {\left (x^{2} + x + 1\right )}^{\frac {3}{4}} {\left (x^{2} - x + 1\right )}^{\frac {3}{4}} {\left (x + 1\right )}^{\frac {3}{4}} {\left (x - 1\right )}^{\frac {3}{4}}}{77 \, x^{11}} \]
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\[ \int \frac {\left (-1+x^6\right )^{3/4} \left (2+x^6\right ) \left (-1-x^4+x^6\right )}{x^{12}} \, dx=\int { \frac {{\left (x^{6} - x^{4} - 1\right )} {\left (x^{6} + 2\right )} {\left (x^{6} - 1\right )}^{\frac {3}{4}}}{x^{12}} \,d x } \]
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Time = 5.46 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.55 \[ \int \frac {\left (-1+x^6\right )^{3/4} \left (2+x^6\right ) \left (-1-x^4+x^6\right )}{x^{12}} \, dx=\frac {2\,x\,{\left (x^6-1\right )}^{3/4}}{11}-\frac {2\,{\left (x^6-1\right )}^{3/4}}{7\,x}-\frac {4\,{\left (x^6-1\right )}^{3/4}}{11\,x^5}+\frac {2\,{\left (x^6-1\right )}^{3/4}}{7\,x^7}+\frac {2\,{\left (x^6-1\right )}^{3/4}}{11\,x^{11}} \]
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