Integrand size = 14, antiderivative size = 75 \[ \int \frac {x}{1-3 x^4+x^8} \, dx=-\frac {\text {arctanh}\left (\sqrt {\frac {2}{3+\sqrt {5}}} x^2\right )}{\sqrt {10 \left (3+\sqrt {5}\right )}}+\frac {1}{2} \sqrt {\frac {1}{10} \left (3+\sqrt {5}\right )} \text {arctanh}\left (\sqrt {\frac {1}{2} \left (3+\sqrt {5}\right )} x^2\right ) \]
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Time = 0.03 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {1373, 1107, 213} \[ \int \frac {x}{1-3 x^4+x^8} \, dx=\frac {1}{2} \sqrt {\frac {1}{10} \left (3+\sqrt {5}\right )} \text {arctanh}\left (\sqrt {\frac {1}{2} \left (3+\sqrt {5}\right )} x^2\right )-\frac {\text {arctanh}\left (\sqrt {\frac {2}{3+\sqrt {5}}} x^2\right )}{\sqrt {10 \left (3+\sqrt {5}\right )}} \]
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Rule 213
Rule 1107
Rule 1373
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \text {Subst}\left (\int \frac {1}{1-3 x^2+x^4} \, dx,x,x^2\right ) \\ & = \frac {\text {Subst}\left (\int \frac {1}{-\frac {3}{2}-\frac {\sqrt {5}}{2}+x^2} \, dx,x,x^2\right )}{2 \sqrt {5}}-\frac {\text {Subst}\left (\int \frac {1}{-\frac {3}{2}+\frac {\sqrt {5}}{2}+x^2} \, dx,x,x^2\right )}{2 \sqrt {5}} \\ & = -\frac {\tanh ^{-1}\left (\sqrt {\frac {2}{3+\sqrt {5}}} x^2\right )}{\sqrt {10 \left (3+\sqrt {5}\right )}}+\frac {1}{2} \sqrt {\frac {1}{10} \left (3+\sqrt {5}\right )} \tanh ^{-1}\left (\sqrt {\frac {1}{2} \left (3+\sqrt {5}\right )} x^2\right ) \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.21 \[ \int \frac {x}{1-3 x^4+x^8} \, dx=\frac {1}{40} \left (-\left (\left (5+\sqrt {5}\right ) \log \left (-1+\sqrt {5}-2 x^2\right )\right )-\left (-5+\sqrt {5}\right ) \log \left (1+\sqrt {5}-2 x^2\right )+\left (5+\sqrt {5}\right ) \log \left (-1+\sqrt {5}+2 x^2\right )+\left (-5+\sqrt {5}\right ) \log \left (1+\sqrt {5}+2 x^2\right )\right ) \]
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Time = 0.04 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.83
method | result | size |
default | \(-\frac {\ln \left (x^{4}+x^{2}-1\right )}{8}+\frac {\operatorname {arctanh}\left (\frac {\left (2 x^{2}+1\right ) \sqrt {5}}{5}\right ) \sqrt {5}}{20}+\frac {\ln \left (x^{4}-x^{2}-1\right )}{8}+\frac {\sqrt {5}\, \operatorname {arctanh}\left (\frac {\left (2 x^{2}-1\right ) \sqrt {5}}{5}\right )}{20}\) | \(62\) |
risch | \(\frac {\ln \left (2 x^{2}+\sqrt {5}-1\right )}{8}+\frac {\ln \left (2 x^{2}+\sqrt {5}-1\right ) \sqrt {5}}{40}+\frac {\ln \left (2 x^{2}-\sqrt {5}-1\right )}{8}-\frac {\ln \left (2 x^{2}-\sqrt {5}-1\right ) \sqrt {5}}{40}-\frac {\ln \left (2 x^{2}+\sqrt {5}+1\right )}{8}+\frac {\ln \left (2 x^{2}+\sqrt {5}+1\right ) \sqrt {5}}{40}-\frac {\ln \left (2 x^{2}-\sqrt {5}+1\right )}{8}-\frac {\ln \left (2 x^{2}-\sqrt {5}+1\right ) \sqrt {5}}{40}\) | \(126\) |
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Leaf count of result is larger than twice the leaf count of optimal. 107 vs. \(2 (43) = 86\).
Time = 0.25 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.43 \[ \int \frac {x}{1-3 x^4+x^8} \, dx=\frac {1}{40} \, \sqrt {5} \log \left (\frac {2 \, x^{4} + 2 \, x^{2} + \sqrt {5} {\left (2 \, x^{2} + 1\right )} + 3}{x^{4} + x^{2} - 1}\right ) + \frac {1}{40} \, \sqrt {5} \log \left (\frac {2 \, x^{4} - 2 \, x^{2} + \sqrt {5} {\left (2 \, x^{2} - 1\right )} + 3}{x^{4} - x^{2} - 1}\right ) - \frac {1}{8} \, \log \left (x^{4} + x^{2} - 1\right ) + \frac {1}{8} \, \log \left (x^{4} - x^{2} - 1\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. 165 vs. \(2 (53) = 106\).
Time = 0.19 (sec) , antiderivative size = 165, normalized size of antiderivative = 2.20 \[ \int \frac {x}{1-3 x^4+x^8} \, dx=\left (\frac {\sqrt {5}}{40} + \frac {1}{8}\right ) \log {\left (x^{2} - \frac {7}{2} - \frac {7 \sqrt {5}}{10} + 960 \left (\frac {\sqrt {5}}{40} + \frac {1}{8}\right )^{3} \right )} + \left (\frac {1}{8} - \frac {\sqrt {5}}{40}\right ) \log {\left (x^{2} - \frac {7}{2} + 960 \left (\frac {1}{8} - \frac {\sqrt {5}}{40}\right )^{3} + \frac {7 \sqrt {5}}{10} \right )} + \left (- \frac {1}{8} + \frac {\sqrt {5}}{40}\right ) \log {\left (x^{2} - \frac {7 \sqrt {5}}{10} + 960 \left (- \frac {1}{8} + \frac {\sqrt {5}}{40}\right )^{3} + \frac {7}{2} \right )} + \left (- \frac {1}{8} - \frac {\sqrt {5}}{40}\right ) \log {\left (x^{2} + 960 \left (- \frac {1}{8} - \frac {\sqrt {5}}{40}\right )^{3} + \frac {7 \sqrt {5}}{10} + \frac {7}{2} \right )} \]
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Leaf count of result is larger than twice the leaf count of optimal. 87 vs. \(2 (43) = 86\).
Time = 0.27 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.16 \[ \int \frac {x}{1-3 x^4+x^8} \, dx=-\frac {1}{40} \, \sqrt {5} \log \left (\frac {2 \, x^{2} - \sqrt {5} + 1}{2 \, x^{2} + \sqrt {5} + 1}\right ) - \frac {1}{40} \, \sqrt {5} \log \left (\frac {2 \, x^{2} - \sqrt {5} - 1}{2 \, x^{2} + \sqrt {5} - 1}\right ) - \frac {1}{8} \, \log \left (x^{4} + x^{2} - 1\right ) + \frac {1}{8} \, \log \left (x^{4} - x^{2} - 1\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. 92 vs. \(2 (43) = 86\).
Time = 0.32 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.23 \[ \int \frac {x}{1-3 x^4+x^8} \, dx=-\frac {1}{40} \, \sqrt {5} \log \left (\frac {{\left | 2 \, x^{2} - \sqrt {5} + 1 \right |}}{2 \, x^{2} + \sqrt {5} + 1}\right ) - \frac {1}{40} \, \sqrt {5} \log \left (\frac {{\left | 2 \, x^{2} - \sqrt {5} - 1 \right |}}{{\left | 2 \, x^{2} + \sqrt {5} - 1 \right |}}\right ) - \frac {1}{8} \, \log \left ({\left | x^{4} + x^{2} - 1 \right |}\right ) + \frac {1}{8} \, \log \left ({\left | x^{4} - x^{2} - 1 \right |}\right ) \]
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Time = 8.56 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.11 \[ \int \frac {x}{1-3 x^4+x^8} \, dx=\mathrm {atanh}\left (\frac {29\,x^2}{8\,\sqrt {5}-18}-\frac {13\,\sqrt {5}\,x^2}{8\,\sqrt {5}-18}\right )\,\left (\frac {\sqrt {5}}{20}-\frac {1}{4}\right )+\mathrm {atanh}\left (\frac {29\,x^2}{8\,\sqrt {5}+18}+\frac {13\,\sqrt {5}\,x^2}{8\,\sqrt {5}+18}\right )\,\left (\frac {\sqrt {5}}{20}+\frac {1}{4}\right ) \]
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