Integrand size = 16, antiderivative size = 89 \[ \int \frac {1}{x^3 \left (1-3 x^4+x^8\right )} \, dx=-\frac {1}{2 x^2}-\frac {1}{2} \sqrt {\frac {1}{5} \left (9-4 \sqrt {5}\right )} \text {arctanh}\left (\sqrt {\frac {2}{3+\sqrt {5}}} x^2\right )+\frac {\left (3+\sqrt {5}\right )^{3/2} \text {arctanh}\left (\sqrt {\frac {1}{2} \left (3+\sqrt {5}\right )} x^2\right )}{4 \sqrt {10}} \]
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Time = 0.04 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {1373, 1137, 1180, 213} \[ \int \frac {1}{x^3 \left (1-3 x^4+x^8\right )} \, dx=-\frac {1}{2} \sqrt {\frac {1}{5} \left (9-4 \sqrt {5}\right )} \text {arctanh}\left (\sqrt {\frac {2}{3+\sqrt {5}}} x^2\right )+\frac {\left (3+\sqrt {5}\right )^{3/2} \text {arctanh}\left (\sqrt {\frac {1}{2} \left (3+\sqrt {5}\right )} x^2\right )}{4 \sqrt {10}}-\frac {1}{2 x^2} \]
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Rule 213
Rule 1137
Rule 1180
Rule 1373
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \text {Subst}\left (\int \frac {1}{x^2 \left (1-3 x^2+x^4\right )} \, dx,x,x^2\right ) \\ & = -\frac {1}{2 x^2}+\frac {1}{2} \text {Subst}\left (\int \frac {3-x^2}{1-3 x^2+x^4} \, dx,x,x^2\right ) \\ & = -\frac {1}{2 x^2}+\frac {1}{20} \left (-5+3 \sqrt {5}\right ) \text {Subst}\left (\int \frac {1}{-\frac {3}{2}-\frac {\sqrt {5}}{2}+x^2} \, dx,x,x^2\right )-\frac {1}{20} \left (5+3 \sqrt {5}\right ) \text {Subst}\left (\int \frac {1}{-\frac {3}{2}+\frac {\sqrt {5}}{2}+x^2} \, dx,x,x^2\right ) \\ & = -\frac {1}{2 x^2}-\frac {1}{10} \sqrt {45-20 \sqrt {5}} \tanh ^{-1}\left (\sqrt {\frac {2}{3+\sqrt {5}}} x^2\right )+\frac {\left (3+\sqrt {5}\right )^{3/2} \tanh ^{-1}\left (\sqrt {\frac {1}{2} \left (3+\sqrt {5}\right )} x^2\right )}{4 \sqrt {10}} \\ \end{align*}
Time = 0.08 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.16 \[ \int \frac {1}{x^3 \left (1-3 x^4+x^8\right )} \, dx=\frac {1}{20} \left (-\frac {10}{x^2}-\left (5+2 \sqrt {5}\right ) \log \left (-1+\sqrt {5}-2 x^2\right )+\left (5-2 \sqrt {5}\right ) \log \left (1+\sqrt {5}-2 x^2\right )+\left (5+2 \sqrt {5}\right ) \log \left (-1+\sqrt {5}+2 x^2\right )+\left (-5+2 \sqrt {5}\right ) \log \left (1+\sqrt {5}+2 x^2\right )\right ) \]
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Time = 0.08 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.75
method | result | size |
default | \(-\frac {1}{2 x^{2}}+\frac {\ln \left (x^{4}-x^{2}-1\right )}{4}+\frac {\sqrt {5}\, \operatorname {arctanh}\left (\frac {\left (2 x^{2}-1\right ) \sqrt {5}}{5}\right )}{5}-\frac {\ln \left (x^{4}+x^{2}-1\right )}{4}+\frac {\operatorname {arctanh}\left (\frac {\left (2 x^{2}+1\right ) \sqrt {5}}{5}\right ) \sqrt {5}}{5}\) | \(67\) |
risch | \(-\frac {1}{2 x^{2}}+\frac {\ln \left (4 x^{2}-2+2 \sqrt {5}\right )}{4}+\frac {\ln \left (4 x^{2}-2+2 \sqrt {5}\right ) \sqrt {5}}{10}+\frac {\ln \left (4 x^{2}-2-2 \sqrt {5}\right )}{4}-\frac {\ln \left (4 x^{2}-2-2 \sqrt {5}\right ) \sqrt {5}}{10}-\frac {\ln \left (4 x^{2}+2+2 \sqrt {5}\right )}{4}+\frac {\ln \left (4 x^{2}+2+2 \sqrt {5}\right ) \sqrt {5}}{10}-\frac {\ln \left (4 x^{2}+2-2 \sqrt {5}\right )}{4}-\frac {\ln \left (4 x^{2}+2-2 \sqrt {5}\right ) \sqrt {5}}{10}\) | \(139\) |
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Leaf count of result is larger than twice the leaf count of optimal. 125 vs. \(2 (53) = 106\).
Time = 0.24 (sec) , antiderivative size = 125, normalized size of antiderivative = 1.40 \[ \int \frac {1}{x^3 \left (1-3 x^4+x^8\right )} \, dx=\frac {2 \, \sqrt {5} x^{2} \log \left (\frac {2 \, x^{4} + 2 \, x^{2} + \sqrt {5} {\left (2 \, x^{2} + 1\right )} + 3}{x^{4} + x^{2} - 1}\right ) + 2 \, \sqrt {5} x^{2} \log \left (\frac {2 \, x^{4} - 2 \, x^{2} + \sqrt {5} {\left (2 \, x^{2} - 1\right )} + 3}{x^{4} - x^{2} - 1}\right ) - 5 \, x^{2} \log \left (x^{4} + x^{2} - 1\right ) + 5 \, x^{2} \log \left (x^{4} - x^{2} - 1\right ) - 10}{20 \, x^{2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 172 vs. \(2 (70) = 140\).
Time = 0.22 (sec) , antiderivative size = 172, normalized size of antiderivative = 1.93 \[ \int \frac {1}{x^3 \left (1-3 x^4+x^8\right )} \, dx=\left (\frac {\sqrt {5}}{10} + \frac {1}{4}\right ) \log {\left (x^{2} - \frac {123}{8} - \frac {123 \sqrt {5}}{20} + 280 \left (\frac {\sqrt {5}}{10} + \frac {1}{4}\right )^{3} \right )} + \left (\frac {1}{4} - \frac {\sqrt {5}}{10}\right ) \log {\left (x^{2} - \frac {123}{8} + 280 \left (\frac {1}{4} - \frac {\sqrt {5}}{10}\right )^{3} + \frac {123 \sqrt {5}}{20} \right )} + \left (- \frac {1}{4} + \frac {\sqrt {5}}{10}\right ) \log {\left (x^{2} - \frac {123 \sqrt {5}}{20} + 280 \left (- \frac {1}{4} + \frac {\sqrt {5}}{10}\right )^{3} + \frac {123}{8} \right )} + \left (- \frac {1}{4} - \frac {\sqrt {5}}{10}\right ) \log {\left (x^{2} + 280 \left (- \frac {1}{4} - \frac {\sqrt {5}}{10}\right )^{3} + \frac {123 \sqrt {5}}{20} + \frac {123}{8} \right )} - \frac {1}{2 x^{2}} \]
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Time = 0.27 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.03 \[ \int \frac {1}{x^3 \left (1-3 x^4+x^8\right )} \, dx=-\frac {1}{10} \, \sqrt {5} \log \left (\frac {2 \, x^{2} - \sqrt {5} + 1}{2 \, x^{2} + \sqrt {5} + 1}\right ) - \frac {1}{10} \, \sqrt {5} \log \left (\frac {2 \, x^{2} - \sqrt {5} - 1}{2 \, x^{2} + \sqrt {5} - 1}\right ) - \frac {1}{2 \, x^{2}} - \frac {1}{4} \, \log \left (x^{4} + x^{2} - 1\right ) + \frac {1}{4} \, \log \left (x^{4} - x^{2} - 1\right ) \]
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Time = 0.32 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.09 \[ \int \frac {1}{x^3 \left (1-3 x^4+x^8\right )} \, dx=-\frac {1}{10} \, \sqrt {5} \log \left (\frac {{\left | 2 \, x^{2} - \sqrt {5} + 1 \right |}}{2 \, x^{2} + \sqrt {5} + 1}\right ) - \frac {1}{10} \, \sqrt {5} \log \left (\frac {{\left | 2 \, x^{2} - \sqrt {5} - 1 \right |}}{{\left | 2 \, x^{2} + \sqrt {5} - 1 \right |}}\right ) - \frac {1}{2 \, x^{2}} - \frac {1}{4} \, \log \left ({\left | x^{4} + x^{2} - 1 \right |}\right ) + \frac {1}{4} \, \log \left ({\left | x^{4} - x^{2} - 1 \right |}\right ) \]
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Time = 0.04 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.99 \[ \int \frac {1}{x^3 \left (1-3 x^4+x^8\right )} \, dx=\mathrm {atanh}\left (\frac {12736\,x^2}{3520\,\sqrt {5}-7872}-\frac {5696\,\sqrt {5}\,x^2}{3520\,\sqrt {5}-7872}\right )\,\left (\frac {\sqrt {5}}{5}-\frac {1}{2}\right )+\mathrm {atanh}\left (\frac {12736\,x^2}{3520\,\sqrt {5}+7872}+\frac {5696\,\sqrt {5}\,x^2}{3520\,\sqrt {5}+7872}\right )\,\left (\frac {\sqrt {5}}{5}+\frac {1}{2}\right )-\frac {1}{2\,x^2} \]
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