Integrand size = 30, antiderivative size = 38 \[ \int \frac {\left (-1+x^4\right )^{2/3} \left (3+x^4\right ) \left (-2-x^3+2 x^4\right )}{x^9} \, dx=\frac {3 \left (-1+x^4\right )^{2/3} \left (5+4 x^3-10 x^4-4 x^7+5 x^8\right )}{20 x^8} \]
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Time = 0.06 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.87, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {1847, 1598, 460, 1488, 861, 75} \[ \int \frac {\left (-1+x^4\right )^{2/3} \left (3+x^4\right ) \left (-2-x^3+2 x^4\right )}{x^9} \, dx=\frac {3 \left (x^4-1\right )^{8/3}}{4 x^8}-\frac {3 \left (x^4-1\right )^{5/3}}{5 x^5} \]
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Rule 75
Rule 460
Rule 861
Rule 1488
Rule 1598
Rule 1847
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {\left (-1+x^4\right )^{2/3} \left (-3 x^2-x^6\right )}{x^8}+\frac {\left (-1+x^4\right )^{2/3} \left (-6+4 x^4+2 x^8\right )}{x^9}\right ) \, dx \\ & = \int \frac {\left (-1+x^4\right )^{2/3} \left (-3 x^2-x^6\right )}{x^8} \, dx+\int \frac {\left (-1+x^4\right )^{2/3} \left (-6+4 x^4+2 x^8\right )}{x^9} \, dx \\ & = \frac {1}{4} \text {Subst}\left (\int \frac {(-1+x)^{2/3} \left (-6+4 x+2 x^2\right )}{x^3} \, dx,x,x^4\right )+\int \frac {\left (-3-x^4\right ) \left (-1+x^4\right )^{2/3}}{x^6} \, dx \\ & = -\frac {3 \left (-1+x^4\right )^{5/3}}{5 x^5}+\frac {1}{4} \text {Subst}\left (\int \frac {(-1+x)^{5/3} (6+2 x)}{x^3} \, dx,x,x^4\right ) \\ & = -\frac {3 \left (-1+x^4\right )^{5/3}}{5 x^5}+\frac {3 \left (-1+x^4\right )^{8/3}}{4 x^8} \\ \end{align*}
Time = 0.43 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.74 \[ \int \frac {\left (-1+x^4\right )^{2/3} \left (3+x^4\right ) \left (-2-x^3+2 x^4\right )}{x^9} \, dx=\frac {3 \left (-1+x^4\right )^{5/3} \left (-5-4 x^3+5 x^4\right )}{20 x^8} \]
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Time = 1.21 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.66
method | result | size |
pseudoelliptic | \(\frac {3 \left (x^{4}-1\right )^{\frac {5}{3}} \left (5 x^{4}-4 x^{3}-5\right )}{20 x^{8}}\) | \(25\) |
trager | \(\frac {3 \left (x^{4}-1\right )^{\frac {2}{3}} \left (5 x^{8}-4 x^{7}-10 x^{4}+4 x^{3}+5\right )}{20 x^{8}}\) | \(35\) |
gosper | \(\frac {3 \left (x^{2}+1\right ) \left (x -1\right ) \left (1+x \right ) \left (5 x^{4}-4 x^{3}-5\right ) \left (x^{4}-1\right )^{\frac {2}{3}}}{20 x^{8}}\) | \(36\) |
risch | \(\frac {-\frac {3}{5} x^{11}+\frac {6}{5} x^{7}-\frac {3}{5} x^{3}-\frac {9}{4} x^{8}+\frac {9}{4} x^{4}-\frac {3}{4}+\frac {3}{4} x^{12}}{x^{8} \left (x^{4}-1\right )^{\frac {1}{3}}}\) | \(45\) |
meijerg | \(-\frac {\sqrt {3}\, \Gamma \left (\frac {2}{3}\right ) \operatorname {signum}\left (x^{4}-1\right )^{\frac {2}{3}} \left (\frac {2 \pi \sqrt {3}\, x^{4} \operatorname {hypergeom}\left (\left [\frac {1}{3}, 1, 1\right ], \left [2, 2\right ], x^{4}\right )}{3 \Gamma \left (\frac {2}{3}\right )}-\frac {\left (\frac {3}{2}-\frac {\pi \sqrt {3}}{6}-\frac {3 \ln \left (3\right )}{2}+4 \ln \left (x \right )+i \pi \right ) \pi \sqrt {3}}{\Gamma \left (\frac {2}{3}\right )}\right )}{6 \pi {\left (-\operatorname {signum}\left (x^{4}-1\right )\right )}^{\frac {2}{3}}}+\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 {\sqrt {3}\, \Gamma \left (\frac {2}{3}\right ) \operatorname {signum}\left (x^{4}-1\right )^{\frac {2}{3}} \left (-\frac {\pi \sqrt {3}\, x^{4} \operatorname {hypergeom}\left (\left [1, 1, \frac {4}{3}\right ], \left [2, 3\right ], x^{4}\right )}{9 \Gamma \left (\frac {2}{3}\right )}-\frac {2 \left (-\frac {\pi \sqrt {3}}{6}-\frac {3 \ln \left (3\right )}{2}-1+4 \ln \left (x \right )+i \pi \right ) \pi \sqrt {3}}{3 \Gamma \left (\frac {2}{3}\right )}-\frac {\pi \sqrt {3}}{\Gamma \left (\frac {2}{3}\right ) x^{4}}\right )}{3 \pi {\left (-\operatorname {signum}\left (x^{4}-1\right )\right )}^{\frac {2}{3}}}+\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}}+\frac {\sqrt {3}\, \Gamma \left (\frac {2}{3}\right ) \operatorname {signum}\left (x^{4}-1\right )^{\frac {2}{3}} \left (\frac {4 \pi \sqrt {3}\, x^{4} \operatorname {hypergeom}\left (\left [1, 1, \frac {7}{3}\right ], \left [2, 4\right ], x^{4}\right )}{81 \Gamma \left (\frac {2}{3}\right )}+\frac {\left (\frac {3}{2}-\frac {\pi \sqrt {3}}{6}-\frac {3 \ln \left (3\right )}{2}+4 \ln \left (x \right )+i \pi \right ) \pi \sqrt {3}}{9 \Gamma \left (\frac {2}{3}\right )}+\frac {\pi \sqrt {3}}{2 \Gamma \left (\frac {2}{3}\right ) x^{8}}-\frac {2 \pi \sqrt {3}}{3 \Gamma \left (\frac {2}{3}\right ) x^{4}}\right )}{2 \pi {\left (-\operatorname {signum}\left (x^{4}-1\right )\right )}^{\frac {2}{3}}}\) | \(353\) |
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Time = 0.26 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.89 \[ \int \frac {\left (-1+x^4\right )^{2/3} \left (3+x^4\right ) \left (-2-x^3+2 x^4\right )}{x^9} \, dx=\frac {3 \, {\left (5 \, x^{8} - 4 \, x^{7} - 10 \, x^{4} + 4 \, x^{3} + 5\right )} {\left (x^{4} - 1\right )}^{\frac {2}{3}}}{20 \, x^{8}} \]
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Result contains complex when optimal does not.
Time = 3.42 (sec) , antiderivative size = 187, normalized size of antiderivative = 4.92 \[ \int \frac {\left (-1+x^4\right )^{2/3} \left (3+x^4\right ) \left (-2-x^3+2 x^4\right )}{x^9} \, dx=- \frac {x^{\frac {8}{3}} \Gamma \left (- \frac {2}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {2}{3}, - \frac {2}{3} \\ \frac {1}{3} \end {matrix}\middle | {\frac {e^{2 i \pi }}{x^{4}}} \right )}}{2 \Gamma \left (\frac {1}{3}\right )} + \frac {e^{- \frac {i \pi }{3}} \Gamma \left (- \frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {2}{3}, - \frac {1}{4} \\ \frac {3}{4} \end {matrix}\middle | {x^{4}} \right )}}{4 x \Gamma \left (\frac {3}{4}\right )} + \frac {3 e^{- \frac {i \pi }{3}} \Gamma \left (- \frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {5}{4}, - \frac {2}{3} \\ - \frac {1}{4} \end {matrix}\middle | {x^{4}} \right )}}{4 x^{5} \Gamma \left (- \frac {1}{4}\right )} - \frac {\Gamma \left (\frac {1}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {2}{3}, \frac {1}{3} \\ \frac {4}{3} \end {matrix}\middle | {\frac {e^{2 i \pi }}{x^{4}}} \right )}}{x^{\frac {4}{3}} \Gamma \left (\frac {4}{3}\right )} + \frac {3 \Gamma \left (\frac {4}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {2}{3}, \frac {4}{3} \\ \frac {7}{3} \end {matrix}\middle | {\frac {e^{2 i \pi }}{x^{4}}} \right )}}{2 x^{\frac {16}{3}} \Gamma \left (\frac {7}{3}\right )} \]
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\[ \int \frac {\left (-1+x^4\right )^{2/3} \left (3+x^4\right ) \left (-2-x^3+2 x^4\right )}{x^9} \, dx=\int { \frac {{\left (2 \, x^{4} - x^{3} - 2\right )} {\left (x^{4} + 3\right )} {\left (x^{4} - 1\right )}^{\frac {2}{3}}}{x^{9}} \,d x } \]
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\[ \int \frac {\left (-1+x^4\right )^{2/3} \left (3+x^4\right ) \left (-2-x^3+2 x^4\right )}{x^9} \, dx=\int { \frac {{\left (2 \, x^{4} - x^{3} - 2\right )} {\left (x^{4} + 3\right )} {\left (x^{4} - 1\right )}^{\frac {2}{3}}}{x^{9}} \,d x } \]
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Time = 5.25 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.53 \[ \int \frac {\left (-1+x^4\right )^{2/3} \left (3+x^4\right ) \left (-2-x^3+2 x^4\right )}{x^9} \, dx=\frac {3\,{\left (x^4-1\right )}^{2/3}}{4}-\frac {3\,{\left (x^4-1\right )}^{2/3}}{5\,x}-\frac {3\,{\left (x^4-1\right )}^{2/3}}{2\,x^4}+\frac {3\,{\left (x^4-1\right )}^{2/3}}{5\,x^5}+\frac {3\,{\left (x^4-1\right )}^{2/3}}{4\,x^8} \]
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