Integrand size = 26, antiderivative size = 38 \[ \int \frac {\sqrt [3]{-1+x^6} \left (1+x^6\right ) \left (-1+x^3+x^6\right )}{x^8} \, dx=\frac {\sqrt [3]{-1+x^6} \left (4-7 x^3-8 x^6+7 x^9+4 x^{12}\right )}{28 x^7} \]
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Time = 0.08 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.29, number of steps used = 6, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {1849, 1600, 1598, 460} \[ \int \frac {\sqrt [3]{-1+x^6} \left (1+x^6\right ) \left (-1+x^3+x^6\right )}{x^8} \, dx=\frac {\left (x^6-1\right )^{4/3}}{7 x}-\frac {\left (x^6-1\right )^{4/3}}{7 x^7}+\frac {\left (x^6-1\right )^{4/3}}{4 x^4} \]
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Rule 460
Rule 1598
Rule 1600
Rule 1849
Rubi steps \begin{align*} \text {integral}& = -\frac {\left (-1+x^6\right )^{4/3}}{7 x^7}+\frac {1}{14} \int \frac {\sqrt [3]{-1+x^6} \left (14 x^2+2 x^5+14 x^8+14 x^{11}\right )}{x^7} \, dx \\ & = -\frac {\left (-1+x^6\right )^{4/3}}{7 x^7}+\frac {1}{14} \int \frac {\sqrt [3]{-1+x^6} \left (14 x+2 x^4+14 x^7+14 x^{10}\right )}{x^6} \, dx \\ & = -\frac {\left (-1+x^6\right )^{4/3}}{7 x^7}+\frac {1}{14} \int \frac {\sqrt [3]{-1+x^6} \left (14+2 x^3+14 x^6+14 x^9\right )}{x^5} \, dx \\ & = -\frac {\left (-1+x^6\right )^{4/3}}{7 x^7}+\frac {\left (-1+x^6\right )^{4/3}}{4 x^4}+\frac {1}{112} \int \frac {\sqrt [3]{-1+x^6} \left (16 x^2+112 x^8\right )}{x^4} \, dx \\ & = -\frac {\left (-1+x^6\right )^{4/3}}{7 x^7}+\frac {\left (-1+x^6\right )^{4/3}}{4 x^4}+\frac {1}{112} \int \frac {\sqrt [3]{-1+x^6} \left (16+112 x^6\right )}{x^2} \, dx \\ & = -\frac {\left (-1+x^6\right )^{4/3}}{7 x^7}+\frac {\left (-1+x^6\right )^{4/3}}{4 x^4}+\frac {\left (-1+x^6\right )^{4/3}}{7 x} \\ \end{align*}
Time = 0.71 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.74 \[ \int \frac {\sqrt [3]{-1+x^6} \left (1+x^6\right ) \left (-1+x^3+x^6\right )}{x^8} \, dx=\frac {\left (-1+x^6\right )^{4/3} \left (-4+7 x^3+4 x^6\right )}{28 x^7} \]
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Time = 0.96 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.66
method | result | size |
pseudoelliptic | \(\frac {\left (x^{6}-1\right )^{\frac {4}{3}} \left (4 x^{6}+7 x^{3}-4\right )}{28 x^{7}}\) | \(25\) |
trager | \(\frac {\left (x^{6}-1\right )^{\frac {1}{3}} \left (4 x^{12}+7 x^{9}-8 x^{6}-7 x^{3}+4\right )}{28 x^{7}}\) | \(35\) |
gosper | \(\frac {\left (x^{6}-1\right )^{\frac {1}{3}} \left (4 x^{6}+7 x^{3}-4\right ) \left (x -1\right ) \left (1+x \right ) \left (x^{2}+x +1\right ) \left (x^{2}-x +1\right )}{28 x^{7}}\) | \(45\) |
risch | \(\frac {4 x^{18}+7 x^{15}-12 x^{12}-14 x^{9}+12 x^{6}+7 x^{3}-4}{28 \left (x^{6}-1\right )^{\frac {2}{3}} x^{7}}\) | \(45\) |
meijerg | \(\frac {\operatorname {signum}\left (x^{6}-1\right )^{\frac {1}{3}} x^{5} \operatorname {hypergeom}\left (\left [-\frac {1}{3}, \frac {5}{6}\right ], \left [\frac {11}{6}\right ], x^{6}\right )}{5 {\left (-\operatorname {signum}\left (x^{6}-1\right )\right )}^{\frac {1}{3}}}+\frac {\operatorname {signum}\left (x^{6}-1\right )^{\frac {1}{3}} x^{2} \operatorname {hypergeom}\left (\left [-\frac {1}{3}, \frac {1}{3}\right ], \left [\frac {4}{3}\right ], x^{6}\right )}{2 {\left (-\operatorname {signum}\left (x^{6}-1\right )\right )}^{\frac {1}{3}}}-\frac {\operatorname {signum}\left (x^{6}-1\right )^{\frac {1}{3}} \operatorname {hypergeom}\left (\left [-\frac {2}{3}, -\frac {1}{3}\right ], \left [\frac {1}{3}\right ], x^{6}\right )}{4 {\left (-\operatorname {signum}\left (x^{6}-1\right )\right )}^{\frac {1}{3}} x^{4}}+\frac {\operatorname {signum}\left (x^{6}-1\right )^{\frac {1}{3}} \operatorname {hypergeom}\left (\left [-\frac {7}{6}, -\frac {1}{3}\right ], \left [-\frac {1}{6}\right ], x^{6}\right )}{7 {\left (-\operatorname {signum}\left (x^{6}-1\right )\right )}^{\frac {1}{3}} x^{7}}\) | \(130\) |
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Time = 0.24 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.89 \[ \int \frac {\sqrt [3]{-1+x^6} \left (1+x^6\right ) \left (-1+x^3+x^6\right )}{x^8} \, dx=\frac {{\left (4 \, x^{12} + 7 \, x^{9} - 8 \, x^{6} - 7 \, x^{3} + 4\right )} {\left (x^{6} - 1\right )}^{\frac {1}{3}}}{28 \, x^{7}} \]
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Result contains complex when optimal does not.
Time = 1.94 (sec) , antiderivative size = 148, normalized size of antiderivative = 3.89 \[ \int \frac {\sqrt [3]{-1+x^6} \left (1+x^6\right ) \left (-1+x^3+x^6\right )}{x^8} \, dx=\frac {x^{5} e^{\frac {i \pi }{3}} \Gamma \left (\frac {5}{6}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{3}, \frac {5}{6} \\ \frac {11}{6} \end {matrix}\middle | {x^{6}} \right )}}{6 \Gamma \left (\frac {11}{6}\right )} + \frac {x^{2} e^{\frac {i \pi }{3}} \Gamma \left (\frac {1}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{3}, \frac {1}{3} \\ \frac {4}{3} \end {matrix}\middle | {x^{6}} \right )}}{6 \Gamma \left (\frac {4}{3}\right )} - \frac {e^{- \frac {2 i \pi }{3}} \Gamma \left (- \frac {2}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {2}{3}, - \frac {1}{3} \\ \frac {1}{3} \end {matrix}\middle | {x^{6}} \right )}}{6 x^{4} \Gamma \left (\frac {1}{3}\right )} + \frac {e^{- \frac {2 i \pi }{3}} \Gamma \left (- \frac {7}{6}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {7}{6}, - \frac {1}{3} \\ - \frac {1}{6} \end {matrix}\middle | {x^{6}} \right )}}{6 x^{7} \Gamma \left (- \frac {1}{6}\right )} \]
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Time = 0.32 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.45 \[ \int \frac {\sqrt [3]{-1+x^6} \left (1+x^6\right ) \left (-1+x^3+x^6\right )}{x^8} \, dx=\frac {{\left (4 \, x^{12} + 7 \, x^{9} - 8 \, x^{6} - 7 \, x^{3} + 4\right )} {\left (x^{2} + x + 1\right )}^{\frac {1}{3}} {\left (x^{2} - x + 1\right )}^{\frac {1}{3}} {\left (x + 1\right )}^{\frac {1}{3}} {\left (x - 1\right )}^{\frac {1}{3}}}{28 \, x^{7}} \]
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\[ \int \frac {\sqrt [3]{-1+x^6} \left (1+x^6\right ) \left (-1+x^3+x^6\right )}{x^8} \, dx=\int { \frac {{\left (x^{6} + x^{3} - 1\right )} {\left (x^{6} + 1\right )} {\left (x^{6} - 1\right )}^{\frac {1}{3}}}{x^{8}} \,d x } \]
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Time = 5.21 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.47 \[ \int \frac {\sqrt [3]{-1+x^6} \left (1+x^6\right ) \left (-1+x^3+x^6\right )}{x^8} \, dx={\left (x^6-1\right )}^{1/3}\,\left (\frac {x^5}{7}+\frac {x^2}{4}\right )-\frac {2\,{\left (x^6-1\right )}^{1/3}}{7\,x}-\frac {{\left (x^6-1\right )}^{1/3}}{4\,x^4}+\frac {{\left (x^6-1\right )}^{1/3}}{7\,x^7} \]
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