Integrand size = 18, antiderivative size = 33 \[ \int \frac {\left (-4+x^3\right ) \left (-1+x^3\right )^{2/3}}{x^{12}} \, dx=\frac {\left (-1+x^3\right )^{2/3} \left (160-95 x^3-26 x^6-39 x^9\right )}{440 x^{11}} \]
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Time = 0.01 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.48, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {464, 277, 270} \[ \int \frac {\left (-4+x^3\right ) \left (-1+x^3\right )^{2/3}}{x^{12}} \, dx=-\frac {4 \left (x^3-1\right )^{5/3}}{11 x^{11}}-\frac {13 \left (x^3-1\right )^{5/3}}{88 x^8}-\frac {39 \left (x^3-1\right )^{5/3}}{440 x^5} \]
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Rule 270
Rule 277
Rule 464
Rubi steps \begin{align*} \text {integral}& = -\frac {4 \left (-1+x^3\right )^{5/3}}{11 x^{11}}-\frac {13}{11} \int \frac {\left (-1+x^3\right )^{2/3}}{x^9} \, dx \\ & = -\frac {4 \left (-1+x^3\right )^{5/3}}{11 x^{11}}-\frac {13 \left (-1+x^3\right )^{5/3}}{88 x^8}-\frac {39}{88} \int \frac {\left (-1+x^3\right )^{2/3}}{x^6} \, dx \\ & = -\frac {4 \left (-1+x^3\right )^{5/3}}{11 x^{11}}-\frac {13 \left (-1+x^3\right )^{5/3}}{88 x^8}-\frac {39 \left (-1+x^3\right )^{5/3}}{440 x^5} \\ \end{align*}
Time = 0.11 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.00 \[ \int \frac {\left (-4+x^3\right ) \left (-1+x^3\right )^{2/3}}{x^{12}} \, dx=\frac {\left (-1+x^3\right )^{2/3} \left (160-95 x^3-26 x^6-39 x^9\right )}{440 x^{11}} \]
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Time = 0.87 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.76
method | result | size |
pseudoelliptic | \(-\frac {\left (39 x^{6}+65 x^{3}+160\right ) \left (x^{3}-1\right )^{\frac {5}{3}}}{440 x^{11}}\) | \(25\) |
trager | \(-\frac {\left (39 x^{9}+26 x^{6}+95 x^{3}-160\right ) \left (x^{3}-1\right )^{\frac {2}{3}}}{440 x^{11}}\) | \(30\) |
gosper | \(-\frac {\left (x -1\right ) \left (x^{2}+x +1\right ) \left (39 x^{6}+65 x^{3}+160\right ) \left (x^{3}-1\right )^{\frac {2}{3}}}{440 x^{11}}\) | \(34\) |
risch | \(-\frac {39 x^{12}-13 x^{9}+69 x^{6}-255 x^{3}+160}{440 x^{11} \left (x^{3}-1\right )^{\frac {1}{3}}}\) | \(35\) |
meijerg | \(\frac {4 \operatorname {signum}\left (x^{3}-1\right )^{\frac {2}{3}} \left (-\frac {9}{20} x^{9}-\frac {3}{10} x^{6}-\frac {1}{4} x^{3}+1\right ) \left (-x^{3}+1\right )^{\frac {2}{3}}}{11 {\left (-\operatorname {signum}\left (x^{3}-1\right )\right )}^{\frac {2}{3}} x^{11}}-\frac {\operatorname {signum}\left (x^{3}-1\right )^{\frac {2}{3}} \left (-\frac {3}{5} x^{6}-\frac {2}{5} x^{3}+1\right ) \left (-x^{3}+1\right )^{\frac {2}{3}}}{8 {\left (-\operatorname {signum}\left (x^{3}-1\right )\right )}^{\frac {2}{3}} x^{8}}\) | \(95\) |
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Time = 0.24 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.88 \[ \int \frac {\left (-4+x^3\right ) \left (-1+x^3\right )^{2/3}}{x^{12}} \, dx=-\frac {{\left (39 \, x^{9} + 26 \, x^{6} + 95 \, x^{3} - 160\right )} {\left (x^{3} - 1\right )}^{\frac {2}{3}}}{440 \, x^{11}} \]
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Result contains complex when optimal does not.
Time = 1.68 (sec) , antiderivative size = 571, normalized size of antiderivative = 17.30 \[ \int \frac {\left (-4+x^3\right ) \left (-1+x^3\right )^{2/3}}{x^{12}} \, dx=- 4 \left (\begin {cases} \frac {2 \left (-1 + \frac {1}{x^{3}}\right )^{\frac {2}{3}} e^{- \frac {i \pi }{3}} \Gamma \left (- \frac {11}{3}\right )}{3 \Gamma \left (- \frac {2}{3}\right )} + \frac {4 \left (-1 + \frac {1}{x^{3}}\right )^{\frac {2}{3}} e^{- \frac {i \pi }{3}} \Gamma \left (- \frac {11}{3}\right )}{9 x^{3} \Gamma \left (- \frac {2}{3}\right )} + \frac {10 \left (-1 + \frac {1}{x^{3}}\right )^{\frac {2}{3}} e^{- \frac {i \pi }{3}} \Gamma \left (- \frac {11}{3}\right )}{27 x^{6} \Gamma \left (- \frac {2}{3}\right )} - \frac {40 \left (-1 + \frac {1}{x^{3}}\right )^{\frac {2}{3}} e^{- \frac {i \pi }{3}} \Gamma \left (- \frac {11}{3}\right )}{27 x^{9} \Gamma \left (- \frac {2}{3}\right )} & \text {for}\: \frac {1}{\left |{x^{3}}\right |} > 1 \\- \frac {2 \left (1 - \frac {1}{x^{3}}\right )^{\frac {2}{3}} \Gamma \left (- \frac {11}{3}\right )}{3 \Gamma \left (- \frac {2}{3}\right )} - \frac {4 \left (1 - \frac {1}{x^{3}}\right )^{\frac {2}{3}} \Gamma \left (- \frac {11}{3}\right )}{9 x^{3} \Gamma \left (- \frac {2}{3}\right )} - \frac {10 \left (1 - \frac {1}{x^{3}}\right )^{\frac {2}{3}} \Gamma \left (- \frac {11}{3}\right )}{27 x^{6} \Gamma \left (- \frac {2}{3}\right )} + \frac {40 \left (1 - \frac {1}{x^{3}}\right )^{\frac {2}{3}} \Gamma \left (- \frac {11}{3}\right )}{27 x^{9} \Gamma \left (- \frac {2}{3}\right )} & \text {otherwise} \end {cases}\right ) + \begin {cases} \frac {3 x^{6} \left (-1 + \frac {1}{x^{3}}\right )^{\frac {2}{3}} e^{\frac {2 i \pi }{3}} \Gamma \left (- \frac {8}{3}\right )}{9 x^{6} \Gamma \left (- \frac {2}{3}\right ) - 9 x^{3} \Gamma \left (- \frac {2}{3}\right )} - \frac {x^{3} \left (-1 + \frac {1}{x^{3}}\right )^{\frac {2}{3}} e^{\frac {2 i \pi }{3}} \Gamma \left (- \frac {8}{3}\right )}{9 x^{6} \Gamma \left (- \frac {2}{3}\right ) - 9 x^{3} \Gamma \left (- \frac {2}{3}\right )} + \frac {5 \left (-1 + \frac {1}{x^{3}}\right )^{\frac {2}{3}} e^{\frac {2 i \pi }{3}} \Gamma \left (- \frac {8}{3}\right )}{9 x^{9} \Gamma \left (- \frac {2}{3}\right ) - 9 x^{6} \Gamma \left (- \frac {2}{3}\right )} - \frac {7 \left (-1 + \frac {1}{x^{3}}\right )^{\frac {2}{3}} e^{\frac {2 i \pi }{3}} \Gamma \left (- \frac {8}{3}\right )}{9 x^{6} \Gamma \left (- \frac {2}{3}\right ) - 9 x^{3} \Gamma \left (- \frac {2}{3}\right )} & \text {for}\: \frac {1}{\left |{x^{3}}\right |} > 1 \\\frac {\left (1 - \frac {1}{x^{3}}\right )^{\frac {2}{3}} \Gamma \left (- \frac {8}{3}\right )}{3 \Gamma \left (- \frac {2}{3}\right )} + \frac {2 \left (1 - \frac {1}{x^{3}}\right )^{\frac {2}{3}} \Gamma \left (- \frac {8}{3}\right )}{9 x^{3} \Gamma \left (- \frac {2}{3}\right )} - \frac {5 \left (1 - \frac {1}{x^{3}}\right )^{\frac {2}{3}} \Gamma \left (- \frac {8}{3}\right )}{9 x^{6} \Gamma \left (- \frac {2}{3}\right )} & \text {otherwise} \end {cases} \]
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Time = 0.20 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.12 \[ \int \frac {\left (-4+x^3\right ) \left (-1+x^3\right )^{2/3}}{x^{12}} \, dx=-\frac {3 \, {\left (x^{3} - 1\right )}^{\frac {5}{3}}}{5 \, x^{5}} + \frac {7 \, {\left (x^{3} - 1\right )}^{\frac {8}{3}}}{8 \, x^{8}} - \frac {4 \, {\left (x^{3} - 1\right )}^{\frac {11}{3}}}{11 \, x^{11}} \]
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\[ \int \frac {\left (-4+x^3\right ) \left (-1+x^3\right )^{2/3}}{x^{12}} \, dx=\int { \frac {{\left (x^{3} - 1\right )}^{\frac {2}{3}} {\left (x^{3} - 4\right )}}{x^{12}} \,d x } \]
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Time = 5.38 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.48 \[ \int \frac {\left (-4+x^3\right ) \left (-1+x^3\right )^{2/3}}{x^{12}} \, dx=\frac {4\,{\left (x^3-1\right )}^{2/3}}{11\,x^{11}}-\frac {13\,{\left (x^3-1\right )}^{2/3}}{220\,x^5}-\frac {19\,{\left (x^3-1\right )}^{2/3}}{88\,x^8}-\frac {39\,{\left (x^3-1\right )}^{2/3}}{440\,x^2} \]
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