Integrand size = 28, antiderivative size = 28 \[ \int \frac {\left (-2+x^6\right ) \left (1-x^4+x^6\right )}{x^8 \sqrt [4]{1+x^6}} \, dx=\frac {2 \left (1+x^6\right )^{3/4} \left (3-7 x^4+3 x^6\right )}{21 x^7} \]
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Time = 0.06 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.18, 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} \[ \int \frac {\left (-2+x^6\right ) \left (1-x^4+x^6\right )}{x^8 \sqrt [4]{1+x^6}} \, dx=\frac {2 \left (x^6+1\right )^{7/4}}{7 x^7}-\frac {2 \left (x^6+1\right )^{3/4}}{3 x^3} \]
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Rule 460
Rule 1492
Rule 1598
Rule 1847
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {2 x^3-x^9}{x^7 \sqrt [4]{1+x^6}}+\frac {-2-x^6+x^{12}}{x^8 \sqrt [4]{1+x^6}}\right ) \, dx \\ & = \int \frac {2 x^3-x^9}{x^7 \sqrt [4]{1+x^6}} \, dx+\int \frac {-2-x^6+x^{12}}{x^8 \sqrt [4]{1+x^6}} \, dx \\ & = \int \frac {2-x^6}{x^4 \sqrt [4]{1+x^6}} \, dx+\int \frac {\left (-2+x^6\right ) \left (1+x^6\right )^{3/4}}{x^8} \, dx \\ & = -\frac {2 \left (1+x^6\right )^{3/4}}{3 x^3}+\frac {2 \left (1+x^6\right )^{7/4}}{7 x^7} \\ \end{align*}
Time = 1.91 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00 \[ \int \frac {\left (-2+x^6\right ) \left (1-x^4+x^6\right )}{x^8 \sqrt [4]{1+x^6}} \, dx=\frac {2 \left (1+x^6\right )^{3/4} \left (3-7 x^4+3 x^6\right )}{21 x^7} \]
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Time = 0.92 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.89
method | result | size |
trager | \(\frac {2 \left (x^{6}+1\right )^{\frac {3}{4}} \left (3 x^{6}-7 x^{4}+3\right )}{21 x^{7}}\) | \(25\) |
pseudoelliptic | \(\frac {2 \left (x^{6}+1\right )^{\frac {3}{4}} \left (3 x^{6}-7 x^{4}+3\right )}{21 x^{7}}\) | \(25\) |
risch | \(\frac {\frac {2}{7} x^{12}+\frac {4}{7} x^{6}+\frac {2}{7}-\frac {2}{3} x^{10}-\frac {2}{3} x^{4}}{x^{7} \left (x^{6}+1\right )^{\frac {1}{4}}}\) | \(35\) |
gosper | \(\frac {2 \left (x^{2}+1\right ) \left (x^{4}-x^{2}+1\right ) \left (3 x^{6}-7 x^{4}+3\right )}{21 x^{7} \left (x^{6}+1\right )^{\frac {1}{4}}}\) | \(40\) |
meijerg | \(\frac {2 \operatorname {hypergeom}\left (\left [-\frac {7}{6}, \frac {1}{4}\right ], \left [-\frac {1}{6}\right ], -x^{6}\right )}{7 x^{7}}+\frac {\operatorname {hypergeom}\left (\left [-\frac {1}{6}, \frac {1}{4}\right ], \left [\frac {5}{6}\right ], -x^{6}\right )}{x}-\frac {2 \operatorname {hypergeom}\left (\left [-\frac {1}{2}, \frac {1}{4}\right ], \left [\frac {1}{2}\right ], -x^{6}\right )}{3 x^{3}}+\frac {x^{5} \operatorname {hypergeom}\left (\left [\frac {1}{4}, \frac {5}{6}\right ], \left [\frac {11}{6}\right ], -x^{6}\right )}{5}-\frac {x^{3} \operatorname {hypergeom}\left (\left [\frac {1}{4}, \frac {1}{2}\right ], \left [\frac {3}{2}\right ], -x^{6}\right )}{3}\) | \(81\) |
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Time = 0.26 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.86 \[ \int \frac {\left (-2+x^6\right ) \left (1-x^4+x^6\right )}{x^8 \sqrt [4]{1+x^6}} \, dx=\frac {2 \, {\left (3 \, x^{6} - 7 \, x^{4} + 3\right )} {\left (x^{6} + 1\right )}^{\frac {3}{4}}}{21 \, x^{7}} \]
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Result contains complex when optimal does not.
Time = 2.15 (sec) , antiderivative size = 143, normalized size of antiderivative = 5.11 \[ \int \frac {\left (-2+x^6\right ) \left (1-x^4+x^6\right )}{x^8 \sqrt [4]{1+x^6}} \, dx=\frac {x^{5} \Gamma \left (\frac {5}{6}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{4}, \frac {5}{6} \\ \frac {11}{6} \end {matrix}\middle | {x^{6} e^{i \pi }} \right )}}{6 \Gamma \left (\frac {11}{6}\right )} - \frac {x^{3} {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{4}, \frac {1}{2} \\ \frac {3}{2} \end {matrix}\middle | {x^{6} e^{i \pi }} \right )}}{3} - \frac {\Gamma \left (- \frac {1}{6}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{6}, \frac {1}{4} \\ \frac {5}{6} \end {matrix}\middle | {x^{6} e^{i \pi }} \right )}}{6 x \Gamma \left (\frac {5}{6}\right )} - \frac {2 {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {1}{4} \\ \frac {1}{2} \end {matrix}\middle | {x^{6} e^{i \pi }} \right )}}{3 x^{3}} - \frac {\Gamma \left (- \frac {7}{6}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {7}{6}, \frac {1}{4} \\ - \frac {1}{6} \end {matrix}\middle | {x^{6} e^{i \pi }} \right )}}{3 x^{7} \Gamma \left (- \frac {1}{6}\right )} \]
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Time = 0.32 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.64 \[ \int \frac {\left (-2+x^6\right ) \left (1-x^4+x^6\right )}{x^8 \sqrt [4]{1+x^6}} \, dx=\frac {2 \, {\left (3 \, x^{12} - 7 \, x^{10} + 6 \, x^{6} - 7 \, x^{4} + 3\right )}}{21 \, {\left (x^{4} - x^{2} + 1\right )}^{\frac {1}{4}} {\left (x^{2} + 1\right )}^{\frac {1}{4}} x^{7}} \]
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\[ \int \frac {\left (-2+x^6\right ) \left (1-x^4+x^6\right )}{x^8 \sqrt [4]{1+x^6}} \, dx=\int { \frac {{\left (x^{6} - x^{4} + 1\right )} {\left (x^{6} - 2\right )}}{{\left (x^{6} + 1\right )}^{\frac {1}{4}} x^{8}} \,d x } \]
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Time = 0.20 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.39 \[ \int \frac {\left (-2+x^6\right ) \left (1-x^4+x^6\right )}{x^8 \sqrt [4]{1+x^6}} \, dx=\frac {6\,{\left (x^6+1\right )}^{3/4}-14\,x^4\,{\left (x^6+1\right )}^{3/4}+6\,x^6\,{\left (x^6+1\right )}^{3/4}}{21\,x^7} \]
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