Integrand size = 20, antiderivative size = 28 \[ \int \frac {\sqrt [3]{-1+x^6} \left (-1+2 x^6\right )}{x^{15}} \, dx=\frac {\sqrt [3]{-1+x^6} \left (4-15 x^6+11 x^{12}\right )}{56 x^{14}} \]
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Time = 0.01 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.18, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {464, 270} \[ \int \frac {\sqrt [3]{-1+x^6} \left (-1+2 x^6\right )}{x^{15}} \, dx=\frac {11 \left (x^6-1\right )^{4/3}}{56 x^8}-\frac {\left (x^6-1\right )^{4/3}}{14 x^{14}} \]
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Rule 270
Rule 464
Rubi steps \begin{align*} \text {integral}& = -\frac {\left (-1+x^6\right )^{4/3}}{14 x^{14}}+\frac {11}{7} \int \frac {\sqrt [3]{-1+x^6}}{x^9} \, dx \\ & = -\frac {\left (-1+x^6\right )^{4/3}}{14 x^{14}}+\frac {11 \left (-1+x^6\right )^{4/3}}{56 x^8} \\ \end{align*}
Time = 0.95 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.82 \[ \int \frac {\sqrt [3]{-1+x^6} \left (-1+2 x^6\right )}{x^{15}} \, dx=\frac {\left (-1+x^6\right )^{4/3} \left (-4+11 x^6\right )}{56 x^{14}} \]
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Time = 0.90 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.71
method | result | size |
pseudoelliptic | \(\frac {\left (x^{6}-1\right )^{\frac {4}{3}} \left (11 x^{6}-4\right )}{56 x^{14}}\) | \(20\) |
trager | \(\frac {\left (x^{6}-1\right )^{\frac {1}{3}} \left (11 x^{12}-15 x^{6}+4\right )}{56 x^{14}}\) | \(25\) |
risch | \(\frac {11 x^{18}-26 x^{12}+19 x^{6}-4}{56 \left (x^{6}-1\right )^{\frac {2}{3}} x^{14}}\) | \(30\) |
gosper | \(\frac {\left (x -1\right ) \left (1+x \right ) \left (x^{2}+x +1\right ) \left (x^{2}-x +1\right ) \left (11 x^{6}-4\right ) \left (x^{6}-1\right )^{\frac {1}{3}}}{56 x^{14}}\) | \(40\) |
meijerg | \(-\frac {\operatorname {signum}\left (x^{6}-1\right )^{\frac {1}{3}} \left (-x^{6}+1\right )^{\frac {4}{3}}}{4 {\left (-\operatorname {signum}\left (x^{6}-1\right )\right )}^{\frac {1}{3}} x^{8}}+\frac {\operatorname {signum}\left (x^{6}-1\right )^{\frac {1}{3}} \left (-\frac {3}{4} x^{12}-\frac {1}{4} x^{6}+1\right ) \left (-x^{6}+1\right )^{\frac {1}{3}}}{14 {\left (-\operatorname {signum}\left (x^{6}-1\right )\right )}^{\frac {1}{3}} x^{14}}\) | \(78\) |
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Time = 0.24 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.86 \[ \int \frac {\sqrt [3]{-1+x^6} \left (-1+2 x^6\right )}{x^{15}} \, dx=\frac {{\left (11 \, x^{12} - 15 \, x^{6} + 4\right )} {\left (x^{6} - 1\right )}^{\frac {1}{3}}}{56 \, x^{14}} \]
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Result contains complex when optimal does not.
Time = 2.08 (sec) , antiderivative size = 425, normalized size of antiderivative = 15.18 \[ \int \frac {\sqrt [3]{-1+x^6} \left (-1+2 x^6\right )}{x^{15}} \, dx=2 \left (\begin {cases} \frac {\sqrt [3]{-1 + \frac {1}{x^{6}}} e^{- \frac {2 i \pi }{3}} \Gamma \left (- \frac {4}{3}\right )}{6 \Gamma \left (- \frac {1}{3}\right )} - \frac {\sqrt [3]{-1 + \frac {1}{x^{6}}} e^{- \frac {2 i \pi }{3}} \Gamma \left (- \frac {4}{3}\right )}{6 x^{6} \Gamma \left (- \frac {1}{3}\right )} & \text {for}\: \frac {1}{\left |{x^{6}}\right |} > 1 \\- \frac {\sqrt [3]{1 - \frac {1}{x^{6}}} \Gamma \left (- \frac {4}{3}\right )}{6 \Gamma \left (- \frac {1}{3}\right )} + \frac {\sqrt [3]{1 - \frac {1}{x^{6}}} \Gamma \left (- \frac {4}{3}\right )}{6 x^{6} \Gamma \left (- \frac {1}{3}\right )} & \text {otherwise} \end {cases}\right ) - \begin {cases} \frac {3 x^{12} \sqrt [3]{-1 + \frac {1}{x^{6}}} e^{\frac {i \pi }{3}} \Gamma \left (- \frac {7}{3}\right )}{18 x^{12} \Gamma \left (- \frac {1}{3}\right ) - 18 x^{6} \Gamma \left (- \frac {1}{3}\right )} - \frac {2 x^{6} \sqrt [3]{-1 + \frac {1}{x^{6}}} e^{\frac {i \pi }{3}} \Gamma \left (- \frac {7}{3}\right )}{18 x^{12} \Gamma \left (- \frac {1}{3}\right ) - 18 x^{6} \Gamma \left (- \frac {1}{3}\right )} + \frac {4 \sqrt [3]{-1 + \frac {1}{x^{6}}} e^{\frac {i \pi }{3}} \Gamma \left (- \frac {7}{3}\right )}{18 x^{18} \Gamma \left (- \frac {1}{3}\right ) - 18 x^{12} \Gamma \left (- \frac {1}{3}\right )} - \frac {5 \sqrt [3]{-1 + \frac {1}{x^{6}}} e^{\frac {i \pi }{3}} \Gamma \left (- \frac {7}{3}\right )}{18 x^{12} \Gamma \left (- \frac {1}{3}\right ) - 18 x^{6} \Gamma \left (- \frac {1}{3}\right )} & \text {for}\: \frac {1}{\left |{x^{6}}\right |} > 1 \\\frac {\sqrt [3]{1 - \frac {1}{x^{6}}} \Gamma \left (- \frac {7}{3}\right )}{6 \Gamma \left (- \frac {1}{3}\right )} + \frac {\sqrt [3]{1 - \frac {1}{x^{6}}} \Gamma \left (- \frac {7}{3}\right )}{18 x^{6} \Gamma \left (- \frac {1}{3}\right )} - \frac {2 \sqrt [3]{1 - \frac {1}{x^{6}}} \Gamma \left (- \frac {7}{3}\right )}{9 x^{12} \Gamma \left (- \frac {1}{3}\right )} & \text {otherwise} \end {cases} \]
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Time = 0.20 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.89 \[ \int \frac {\sqrt [3]{-1+x^6} \left (-1+2 x^6\right )}{x^{15}} \, dx=\frac {{\left (x^{6} - 1\right )}^{\frac {4}{3}}}{8 \, x^{8}} + \frac {{\left (x^{6} - 1\right )}^{\frac {7}{3}}}{14 \, x^{14}} \]
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\[ \int \frac {\sqrt [3]{-1+x^6} \left (-1+2 x^6\right )}{x^{15}} \, dx=\int { \frac {{\left (2 \, x^{6} - 1\right )} {\left (x^{6} - 1\right )}^{\frac {1}{3}}}{x^{15}} \,d x } \]
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Time = 5.24 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.86 \[ \int \frac {\sqrt [3]{-1+x^6} \left (-1+2 x^6\right )}{x^{15}} \, dx=\frac {7\,{\left (x^6-1\right )}^{4/3}+11\,{\left (x^6-1\right )}^{7/3}}{56\,x^{14}} \]
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