Integrand size = 16, antiderivative size = 167 \[ \int \frac {x^6}{1-3 x^4+x^8} \, dx=\frac {\left (3+\sqrt {5}\right )^{3/4} \arctan \left (\sqrt [4]{\frac {2}{3+\sqrt {5}}} x\right )}{2\ 2^{3/4} \sqrt {5}}-\frac {\sqrt [4]{144-64 \sqrt {5}} \arctan \left (\sqrt [4]{\frac {1}{2} \left (3+\sqrt {5}\right )} x\right )}{4 \sqrt {5}}-\frac {\left (3+\sqrt {5}\right )^{3/4} \text {arctanh}\left (\sqrt [4]{\frac {2}{3+\sqrt {5}}} x\right )}{2\ 2^{3/4} \sqrt {5}}+\frac {\sqrt [4]{144-64 \sqrt {5}} \text {arctanh}\left (\sqrt [4]{\frac {1}{2} \left (3+\sqrt {5}\right )} x\right )}{4 \sqrt {5}} \]
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Time = 0.05 (sec) , antiderivative size = 167, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {1388, 304, 209, 212} \[ \int \frac {x^6}{1-3 x^4+x^8} \, dx=\frac {\left (3+\sqrt {5}\right )^{3/4} \arctan \left (\sqrt [4]{\frac {2}{3+\sqrt {5}}} x\right )}{2\ 2^{3/4} \sqrt {5}}-\frac {\sqrt [4]{144-64 \sqrt {5}} \arctan \left (\sqrt [4]{\frac {1}{2} \left (3+\sqrt {5}\right )} x\right )}{4 \sqrt {5}}-\frac {\left (3+\sqrt {5}\right )^{3/4} \text {arctanh}\left (\sqrt [4]{\frac {2}{3+\sqrt {5}}} x\right )}{2\ 2^{3/4} \sqrt {5}}+\frac {\sqrt [4]{144-64 \sqrt {5}} \text {arctanh}\left (\sqrt [4]{\frac {1}{2} \left (3+\sqrt {5}\right )} x\right )}{4 \sqrt {5}} \]
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Rule 209
Rule 212
Rule 304
Rule 1388
Rubi steps \begin{align*} \text {integral}& = \frac {1}{10} \left (5-3 \sqrt {5}\right ) \int \frac {x^2}{-\frac {3}{2}+\frac {\sqrt {5}}{2}+x^4} \, dx+\frac {1}{10} \left (5+3 \sqrt {5}\right ) \int \frac {x^2}{-\frac {3}{2}-\frac {\sqrt {5}}{2}+x^4} \, dx \\ & = \frac {\left (3-\sqrt {5}\right ) \int \frac {1}{\sqrt {3-\sqrt {5}}-\sqrt {2} x^2} \, dx}{2 \sqrt {10}}-\frac {\left (3-\sqrt {5}\right ) \int \frac {1}{\sqrt {3-\sqrt {5}}+\sqrt {2} x^2} \, dx}{2 \sqrt {10}}-\frac {\left (3+\sqrt {5}\right ) \int \frac {1}{\sqrt {3+\sqrt {5}}-\sqrt {2} x^2} \, dx}{2 \sqrt {10}}+\frac {\left (3+\sqrt {5}\right ) \int \frac {1}{\sqrt {3+\sqrt {5}}+\sqrt {2} x^2} \, dx}{2 \sqrt {10}} \\ & = \frac {\left (3+\sqrt {5}\right )^{3/4} \tan ^{-1}\left (\sqrt [4]{\frac {2}{3+\sqrt {5}}} x\right )}{2\ 2^{3/4} \sqrt {5}}-\frac {\left (3-\sqrt {5}\right )^{3/4} \tan ^{-1}\left (\sqrt [4]{\frac {1}{2} \left (3+\sqrt {5}\right )} x\right )}{2\ 2^{3/4} \sqrt {5}}-\frac {\left (3+\sqrt {5}\right )^{3/4} \tanh ^{-1}\left (\sqrt [4]{\frac {2}{3+\sqrt {5}}} x\right )}{2\ 2^{3/4} \sqrt {5}}+\frac {\left (3-\sqrt {5}\right )^{3/4} \tanh ^{-1}\left (\sqrt [4]{\frac {1}{2} \left (3+\sqrt {5}\right )} x\right )}{2\ 2^{3/4} \sqrt {5}} \\ \end{align*}
Time = 0.18 (sec) , antiderivative size = 160, normalized size of antiderivative = 0.96 \[ \int \frac {x^6}{1-3 x^4+x^8} \, dx=\frac {\frac {\left (-3+\sqrt {5}\right ) \arctan \left (\sqrt {\frac {2}{-1+\sqrt {5}}} x\right )}{\sqrt {-1+\sqrt {5}}}+\frac {\left (3+\sqrt {5}\right ) \arctan \left (\sqrt {\frac {2}{1+\sqrt {5}}} x\right )}{\sqrt {1+\sqrt {5}}}-\frac {\left (-3+\sqrt {5}\right ) \text {arctanh}\left (\sqrt {\frac {2}{-1+\sqrt {5}}} x\right )}{\sqrt {-1+\sqrt {5}}}-\frac {\left (3+\sqrt {5}\right ) \text {arctanh}\left (\sqrt {\frac {2}{1+\sqrt {5}}} x\right )}{\sqrt {1+\sqrt {5}}}}{2 \sqrt {10}} \]
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Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.08 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.41
method | result | size |
risch | \(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (25 \textit {\_Z}^{4}-20 \textit {\_Z}^{2}-1\right )}{\sum }\textit {\_R} \ln \left (5 \textit {\_R}^{3}-7 \textit {\_R} +2 x \right )\right )}{4}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (25 \textit {\_Z}^{4}+20 \textit {\_Z}^{2}-1\right )}{\sum }\textit {\_R} \ln \left (5 \textit {\_R}^{3}+7 \textit {\_R} +2 x \right )\right )}{4}\) | \(68\) |
default | \(-\frac {\left (3+\sqrt {5}\right ) \sqrt {5}\, \operatorname {arctanh}\left (\frac {2 x}{\sqrt {2 \sqrt {5}+2}}\right )}{10 \sqrt {2 \sqrt {5}+2}}+\frac {\left (\sqrt {5}-3\right ) \sqrt {5}\, \arctan \left (\frac {2 x}{\sqrt {2 \sqrt {5}-2}}\right )}{10 \sqrt {2 \sqrt {5}-2}}+\frac {\left (3+\sqrt {5}\right ) \sqrt {5}\, \arctan \left (\frac {2 x}{\sqrt {2 \sqrt {5}+2}}\right )}{10 \sqrt {2 \sqrt {5}+2}}-\frac {\left (\sqrt {5}-3\right ) \sqrt {5}\, \operatorname {arctanh}\left (\frac {2 x}{\sqrt {2 \sqrt {5}-2}}\right )}{10 \sqrt {2 \sqrt {5}-2}}\) | \(130\) |
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Leaf count of result is larger than twice the leaf count of optimal. 261 vs. \(2 (113) = 226\).
Time = 0.26 (sec) , antiderivative size = 261, normalized size of antiderivative = 1.56 \[ \int \frac {x^6}{1-3 x^4+x^8} \, dx=-\frac {1}{20} \, \sqrt {5} \sqrt {\sqrt {5} + 2} \log \left (\sqrt {\sqrt {5} + 2} {\left (\sqrt {5} - 1\right )} + 2 \, x\right ) + \frac {1}{20} \, \sqrt {5} \sqrt {\sqrt {5} + 2} \log \left (-\sqrt {\sqrt {5} + 2} {\left (\sqrt {5} - 1\right )} + 2 \, x\right ) + \frac {1}{20} \, \sqrt {5} \sqrt {\sqrt {5} - 2} \log \left ({\left (\sqrt {5} + 1\right )} \sqrt {\sqrt {5} - 2} + 2 \, x\right ) - \frac {1}{20} \, \sqrt {5} \sqrt {\sqrt {5} - 2} \log \left (-{\left (\sqrt {5} + 1\right )} \sqrt {\sqrt {5} - 2} + 2 \, x\right ) - \frac {1}{20} \, \sqrt {5} \sqrt {-\sqrt {5} + 2} \log \left ({\left (\sqrt {5} + 1\right )} \sqrt {-\sqrt {5} + 2} + 2 \, x\right ) + \frac {1}{20} \, \sqrt {5} \sqrt {-\sqrt {5} + 2} \log \left (-{\left (\sqrt {5} + 1\right )} \sqrt {-\sqrt {5} + 2} + 2 \, x\right ) + \frac {1}{20} \, \sqrt {5} \sqrt {-\sqrt {5} - 2} \log \left ({\left (\sqrt {5} - 1\right )} \sqrt {-\sqrt {5} - 2} + 2 \, x\right ) - \frac {1}{20} \, \sqrt {5} \sqrt {-\sqrt {5} - 2} \log \left (-{\left (\sqrt {5} - 1\right )} \sqrt {-\sqrt {5} - 2} + 2 \, x\right ) \]
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Time = 0.73 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.32 \[ \int \frac {x^6}{1-3 x^4+x^8} \, dx=\operatorname {RootSum} {\left (6400 t^{4} - 320 t^{2} - 1, \left ( t \mapsto t \log {\left (- 1792000 t^{7} + 4920 t^{3} + x \right )} \right )\right )} + \operatorname {RootSum} {\left (6400 t^{4} + 320 t^{2} - 1, \left ( t \mapsto t \log {\left (- 1792000 t^{7} + 4920 t^{3} + x \right )} \right )\right )} \]
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\[ \int \frac {x^6}{1-3 x^4+x^8} \, dx=\int { \frac {x^{6}}{x^{8} - 3 \, x^{4} + 1} \,d x } \]
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Time = 0.37 (sec) , antiderivative size = 147, normalized size of antiderivative = 0.88 \[ \int \frac {x^6}{1-3 x^4+x^8} \, dx=\frac {1}{10} \, \sqrt {5 \, \sqrt {5} + 10} \arctan \left (\frac {x}{\sqrt {\frac {1}{2} \, \sqrt {5} + \frac {1}{2}}}\right ) - \frac {1}{10} \, \sqrt {5 \, \sqrt {5} - 10} \arctan \left (\frac {x}{\sqrt {\frac {1}{2} \, \sqrt {5} - \frac {1}{2}}}\right ) - \frac {1}{20} \, \sqrt {5 \, \sqrt {5} + 10} \log \left ({\left | x + \sqrt {\frac {1}{2} \, \sqrt {5} + \frac {1}{2}} \right |}\right ) + \frac {1}{20} \, \sqrt {5 \, \sqrt {5} + 10} \log \left ({\left | x - \sqrt {\frac {1}{2} \, \sqrt {5} + \frac {1}{2}} \right |}\right ) + \frac {1}{20} \, \sqrt {5 \, \sqrt {5} - 10} \log \left ({\left | x + \sqrt {\frac {1}{2} \, \sqrt {5} - \frac {1}{2}} \right |}\right ) - \frac {1}{20} \, \sqrt {5 \, \sqrt {5} - 10} \log \left ({\left | x - \sqrt {\frac {1}{2} \, \sqrt {5} - \frac {1}{2}} \right |}\right ) \]
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Time = 0.12 (sec) , antiderivative size = 147, normalized size of antiderivative = 0.88 \[ \int \frac {x^6}{1-3 x^4+x^8} \, dx=\frac {\sqrt {5}\,\mathrm {atan}\left (\frac {16\,x\,\sqrt {2-\sqrt {5}}}{8\,\sqrt {5}-24}\right )\,\sqrt {\sqrt {5}-2}\,1{}\mathrm {i}}{10}+\frac {\sqrt {5}\,\mathrm {atan}\left (\frac {16\,x\,\sqrt {-\sqrt {5}-2}}{8\,\sqrt {5}+24}\right )\,\sqrt {\sqrt {5}+2}\,1{}\mathrm {i}}{10}+\frac {\sqrt {5}\,\mathrm {atan}\left (\frac {x\,\sqrt {2-\sqrt {5}}\,16{}\mathrm {i}}{8\,\sqrt {5}-24}\right )\,\sqrt {2-\sqrt {5}}\,1{}\mathrm {i}}{10}+\frac {\sqrt {5}\,\mathrm {atan}\left (\frac {x\,\sqrt {-\sqrt {5}-2}\,16{}\mathrm {i}}{8\,\sqrt {5}+24}\right )\,\sqrt {-\sqrt {5}-2}\,1{}\mathrm {i}}{10} \]
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