Integrand size = 28, antiderivative size = 26 \[ \int \frac {\left (-2+x^6\right ) \left (1-x^4+x^6\right )}{x^6 \left (1+x^6\right )^{3/4}} \, dx=\frac {2 \sqrt [4]{1+x^6} \left (1-5 x^4+x^6\right )}{5 x^5} \]
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Time = 0.06 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.19, 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} \[ \int \frac {\left (-2+x^6\right ) \left (1-x^4+x^6\right )}{x^6 \left (1+x^6\right )^{3/4}} \, dx=\frac {2 \left (x^6+1\right )^{5/4}}{5 x^5}-\frac {2 \sqrt [4]{x^6+1}}{x} \]
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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^5 \left (1+x^6\right )^{3/4}}+\frac {-2-x^6+x^{12}}{x^6 \left (1+x^6\right )^{3/4}}\right ) \, dx \\ & = \int \frac {2 x^3-x^9}{x^5 \left (1+x^6\right )^{3/4}} \, dx+\int \frac {-2-x^6+x^{12}}{x^6 \left (1+x^6\right )^{3/4}} \, dx \\ & = \int \frac {2-x^6}{x^2 \left (1+x^6\right )^{3/4}} \, dx+\int \frac {\left (-2+x^6\right ) \sqrt [4]{1+x^6}}{x^6} \, dx \\ & = -\frac {2 \sqrt [4]{1+x^6}}{x}+\frac {2 \left (1+x^6\right )^{5/4}}{5 x^5} \\ \end{align*}
Time = 3.27 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00 \[ \int \frac {\left (-2+x^6\right ) \left (1-x^4+x^6\right )}{x^6 \left (1+x^6\right )^{3/4}} \, dx=\frac {2 \sqrt [4]{1+x^6} \left (1-5 x^4+x^6\right )}{5 x^5} \]
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Time = 0.90 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.88
method | result | size |
trager | \(\frac {2 \left (x^{6}+1\right )^{\frac {1}{4}} \left (x^{6}-5 x^{4}+1\right )}{5 x^{5}}\) | \(23\) |
pseudoelliptic | \(\frac {2 \left (x^{6}+1\right )^{\frac {1}{4}} \left (x^{6}-5 x^{4}+1\right )}{5 x^{5}}\) | \(23\) |
risch | \(\frac {\frac {2}{5} x^{12}+\frac {4}{5} x^{6}+\frac {2}{5}-2 x^{10}-2 x^{4}}{\left (x^{6}+1\right )^{\frac {3}{4}} x^{5}}\) | \(33\) |
gosper | \(\frac {2 \left (x^{4}-x^{2}+1\right ) \left (x^{2}+1\right ) \left (x^{6}-5 x^{4}+1\right )}{5 x^{5} \left (x^{6}+1\right )^{\frac {3}{4}}}\) | \(38\) |
meijerg | \(\frac {x^{7} \operatorname {hypergeom}\left (\left [\frac {3}{4}, \frac {7}{6}\right ], \left [\frac {13}{6}\right ], -x^{6}\right )}{7}-\frac {x^{5} \operatorname {hypergeom}\left (\left [\frac {3}{4}, \frac {5}{6}\right ], \left [\frac {11}{6}\right ], -x^{6}\right )}{5}-x \operatorname {hypergeom}\left (\left [\frac {1}{6}, \frac {3}{4}\right ], \left [\frac {7}{6}\right ], -x^{6}\right )+\frac {2 \operatorname {hypergeom}\left (\left [-\frac {5}{6}, \frac {3}{4}\right ], \left [\frac {1}{6}\right ], -x^{6}\right )}{5 x^{5}}-\frac {2 \operatorname {hypergeom}\left (\left [-\frac {1}{6}, \frac {3}{4}\right ], \left [\frac {5}{6}\right ], -x^{6}\right )}{x}\) | \(80\) |
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Time = 0.25 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.85 \[ \int \frac {\left (-2+x^6\right ) \left (1-x^4+x^6\right )}{x^6 \left (1+x^6\right )^{3/4}} \, dx=\frac {2 \, {\left (x^{6} - 5 \, x^{4} + 1\right )} {\left (x^{6} + 1\right )}^{\frac {1}{4}}}{5 \, x^{5}} \]
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Result contains complex when optimal does not.
Time = 2.09 (sec) , antiderivative size = 155, normalized size of antiderivative = 5.96 \[ \int \frac {\left (-2+x^6\right ) \left (1-x^4+x^6\right )}{x^6 \left (1+x^6\right )^{3/4}} \, dx=\frac {x^{7} \Gamma \left (\frac {7}{6}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {3}{4}, \frac {7}{6} \\ \frac {13}{6} \end {matrix}\middle | {x^{6} e^{i \pi }} \right )}}{6 \Gamma \left (\frac {13}{6}\right )} - \frac {x^{5} \Gamma \left (\frac {5}{6}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {3}{4}, \frac {5}{6} \\ \frac {11}{6} \end {matrix}\middle | {x^{6} e^{i \pi }} \right )}}{6 \Gamma \left (\frac {11}{6}\right )} - \frac {x \Gamma \left (\frac {1}{6}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{6}, \frac {3}{4} \\ \frac {7}{6} \end {matrix}\middle | {x^{6} e^{i \pi }} \right )}}{6 \Gamma \left (\frac {7}{6}\right )} + \frac {\Gamma \left (- \frac {1}{6}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{6}, \frac {3}{4} \\ \frac {5}{6} \end {matrix}\middle | {x^{6} e^{i \pi }} \right )}}{3 x \Gamma \left (\frac {5}{6}\right )} - \frac {\Gamma \left (- \frac {5}{6}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {5}{6}, \frac {3}{4} \\ \frac {1}{6} \end {matrix}\middle | {x^{6} e^{i \pi }} \right )}}{3 x^{5} \Gamma \left (\frac {1}{6}\right )} \]
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Time = 0.30 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.69 \[ \int \frac {\left (-2+x^6\right ) \left (1-x^4+x^6\right )}{x^6 \left (1+x^6\right )^{3/4}} \, dx=\frac {2 \, {\left (x^{12} - 5 \, x^{10} + 2 \, x^{6} - 5 \, x^{4} + 1\right )}}{5 \, {\left (x^{4} - x^{2} + 1\right )}^{\frac {3}{4}} {\left (x^{2} + 1\right )}^{\frac {3}{4}} x^{5}} \]
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\[ \int \frac {\left (-2+x^6\right ) \left (1-x^4+x^6\right )}{x^6 \left (1+x^6\right )^{3/4}} \, dx=\int { \frac {{\left (x^{6} - x^{4} + 1\right )} {\left (x^{6} - 2\right )}}{{\left (x^{6} + 1\right )}^{\frac {3}{4}} x^{6}} \,d x } \]
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Time = 5.16 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.04 \[ \int \frac {\left (-2+x^6\right ) \left (1-x^4+x^6\right )}{x^6 \left (1+x^6\right )^{3/4}} \, dx=\frac {2\,{\left (x^6+1\right )}^{5/4}-10\,x^4\,{\left (x^6+1\right )}^{1/4}}{5\,x^5} \]
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