Integrand size = 16, antiderivative size = 170 \[ \int \frac {x^8}{1-3 x^4+x^8} \, dx=x-\frac {\sqrt [4]{\frac {1}{2} \left (123+55 \sqrt {5}\right )} \arctan \left (\sqrt [4]{\frac {2}{3+\sqrt {5}}} x\right )}{2 \sqrt {5}}+\frac {\sqrt [4]{984-440 \sqrt {5}} \arctan \left (\sqrt [4]{\frac {1}{2} \left (3+\sqrt {5}\right )} x\right )}{4 \sqrt {5}}-\frac {\sqrt [4]{\frac {1}{2} \left (123+55 \sqrt {5}\right )} \text {arctanh}\left (\sqrt [4]{\frac {2}{3+\sqrt {5}}} x\right )}{2 \sqrt {5}}+\frac {\sqrt [4]{984-440 \sqrt {5}} \text {arctanh}\left (\sqrt [4]{\frac {1}{2} \left (3+\sqrt {5}\right )} x\right )}{4 \sqrt {5}} \]
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Time = 0.09 (sec) , antiderivative size = 170, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.312, Rules used = {1381, 1436, 218, 212, 209} \[ \int \frac {x^8}{1-3 x^4+x^8} \, dx=-\frac {\sqrt [4]{\frac {1}{2} \left (123+55 \sqrt {5}\right )} \arctan \left (\sqrt [4]{\frac {2}{3+\sqrt {5}}} x\right )}{2 \sqrt {5}}+\frac {\sqrt [4]{984-440 \sqrt {5}} \arctan \left (\sqrt [4]{\frac {1}{2} \left (3+\sqrt {5}\right )} x\right )}{4 \sqrt {5}}-\frac {\sqrt [4]{\frac {1}{2} \left (123+55 \sqrt {5}\right )} \text {arctanh}\left (\sqrt [4]{\frac {2}{3+\sqrt {5}}} x\right )}{2 \sqrt {5}}+\frac {\sqrt [4]{984-440 \sqrt {5}} \text {arctanh}\left (\sqrt [4]{\frac {1}{2} \left (3+\sqrt {5}\right )} x\right )}{4 \sqrt {5}}+x \]
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Rule 209
Rule 212
Rule 218
Rule 1381
Rule 1436
Rubi steps \begin{align*} \text {integral}& = x-\int \frac {1-3 x^4}{1-3 x^4+x^8} \, dx \\ & = x-\frac {1}{10} \left (-15+7 \sqrt {5}\right ) \int \frac {1}{-\frac {3}{2}+\frac {\sqrt {5}}{2}+x^4} \, dx+\frac {1}{10} \left (15+7 \sqrt {5}\right ) \int \frac {1}{-\frac {3}{2}-\frac {\sqrt {5}}{2}+x^4} \, dx \\ & = x+\sqrt {\frac {1}{10} \left (9-4 \sqrt {5}\right )} \int \frac {1}{\sqrt {3-\sqrt {5}}-\sqrt {2} x^2} \, dx+\sqrt {\frac {1}{10} \left (9-4 \sqrt {5}\right )} \int \frac {1}{\sqrt {3-\sqrt {5}}+\sqrt {2} x^2} \, dx--\frac {\left (-15-7 \sqrt {5}\right ) \int \frac {1}{\sqrt {3+\sqrt {5}}-\sqrt {2} x^2} \, dx}{10 \sqrt {3+\sqrt {5}}}--\frac {\left (-15-7 \sqrt {5}\right ) \int \frac {1}{\sqrt {3+\sqrt {5}}+\sqrt {2} x^2} \, dx}{10 \sqrt {3+\sqrt {5}}} \\ & = x-\frac {\sqrt [4]{\frac {1}{2} \left (123+55 \sqrt {5}\right )} \tan ^{-1}\left (\sqrt [4]{\frac {2}{3+\sqrt {5}}} x\right )}{2 \sqrt {5}}+\frac {\sqrt [4]{\frac {1}{2} \left (123-55 \sqrt {5}\right )} \tan ^{-1}\left (\sqrt [4]{\frac {1}{2} \left (3+\sqrt {5}\right )} x\right )}{2 \sqrt {5}}-\frac {\sqrt [4]{\frac {1}{2} \left (123+55 \sqrt {5}\right )} \tanh ^{-1}\left (\sqrt [4]{\frac {2}{3+\sqrt {5}}} x\right )}{2 \sqrt {5}}+\frac {\sqrt [4]{\frac {1}{2} \left (123-55 \sqrt {5}\right )} \tanh ^{-1}\left (\sqrt [4]{\frac {1}{2} \left (3+\sqrt {5}\right )} x\right )}{2 \sqrt {5}} \\ \end{align*}
Time = 0.34 (sec) , antiderivative size = 160, normalized size of antiderivative = 0.94 \[ \int \frac {x^8}{1-3 x^4+x^8} \, dx=x+\frac {\left (-2+\sqrt {5}\right ) \arctan \left (\sqrt {\frac {2}{-1+\sqrt {5}}} x\right )}{\sqrt {10 \left (-1+\sqrt {5}\right )}}-\frac {\left (2+\sqrt {5}\right ) \arctan \left (\sqrt {\frac {2}{1+\sqrt {5}}} x\right )}{\sqrt {10 \left (1+\sqrt {5}\right )}}+\frac {\left (-2+\sqrt {5}\right ) \text {arctanh}\left (\sqrt {\frac {2}{-1+\sqrt {5}}} x\right )}{\sqrt {10 \left (-1+\sqrt {5}\right )}}-\frac {\left (2+\sqrt {5}\right ) \text {arctanh}\left (\sqrt {\frac {2}{1+\sqrt {5}}} x\right )}{\sqrt {10 \left (1+\sqrt {5}\right )}} \]
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Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.12 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.41
method | result | size |
risch | \(x +\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (25 \textit {\_Z}^{4}+55 \textit {\_Z}^{2}-1\right )}{\sum }\textit {\_R} \ln \left (15 \textit {\_R}^{3}+29 \textit {\_R} +5 x \right )\right )}{4}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (25 \textit {\_Z}^{4}-55 \textit {\_Z}^{2}-1\right )}{\sum }\textit {\_R} \ln \left (-15 \textit {\_R}^{3}+29 \textit {\_R} +5 x \right )\right )}{4}\) | \(69\) |
default | \(x -\frac {\left (2+\sqrt {5}\right ) \sqrt {5}\, \operatorname {arctanh}\left (\frac {2 x}{\sqrt {2 \sqrt {5}+2}}\right )}{5 \sqrt {2 \sqrt {5}+2}}+\frac {\left (\sqrt {5}-2\right ) \sqrt {5}\, \arctan \left (\frac {2 x}{\sqrt {2 \sqrt {5}-2}}\right )}{5 \sqrt {2 \sqrt {5}-2}}-\frac {\left (2+\sqrt {5}\right ) \sqrt {5}\, \arctan \left (\frac {2 x}{\sqrt {2 \sqrt {5}+2}}\right )}{5 \sqrt {2 \sqrt {5}+2}}+\frac {\left (\sqrt {5}-2\right ) \sqrt {5}\, \operatorname {arctanh}\left (\frac {2 x}{\sqrt {2 \sqrt {5}-2}}\right )}{5 \sqrt {2 \sqrt {5}-2}}\) | \(131\) |
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Leaf count of result is larger than twice the leaf count of optimal. 318 vs. \(2 (118) = 236\).
Time = 0.25 (sec) , antiderivative size = 318, normalized size of antiderivative = 1.87 \[ \int \frac {x^8}{1-3 x^4+x^8} \, dx=\frac {1}{40} \, \sqrt {10} \sqrt {5 \, \sqrt {5} - 11} \log \left (\sqrt {10} \sqrt {5 \, \sqrt {5} - 11} {\left (3 \, \sqrt {5} + 5\right )} + 20 \, x\right ) - \frac {1}{40} \, \sqrt {10} \sqrt {5 \, \sqrt {5} - 11} \log \left (-\sqrt {10} \sqrt {5 \, \sqrt {5} - 11} {\left (3 \, \sqrt {5} + 5\right )} + 20 \, x\right ) - \frac {1}{40} \, \sqrt {10} \sqrt {5 \, \sqrt {5} + 11} \log \left (\sqrt {10} \sqrt {5 \, \sqrt {5} + 11} {\left (3 \, \sqrt {5} - 5\right )} + 20 \, x\right ) + \frac {1}{40} \, \sqrt {10} \sqrt {5 \, \sqrt {5} + 11} \log \left (-\sqrt {10} \sqrt {5 \, \sqrt {5} + 11} {\left (3 \, \sqrt {5} - 5\right )} + 20 \, x\right ) + \frac {1}{40} \, \sqrt {10} \sqrt {-5 \, \sqrt {5} + 11} \log \left (\sqrt {10} {\left (3 \, \sqrt {5} + 5\right )} \sqrt {-5 \, \sqrt {5} + 11} + 20 \, x\right ) - \frac {1}{40} \, \sqrt {10} \sqrt {-5 \, \sqrt {5} + 11} \log \left (-\sqrt {10} {\left (3 \, \sqrt {5} + 5\right )} \sqrt {-5 \, \sqrt {5} + 11} + 20 \, x\right ) - \frac {1}{40} \, \sqrt {10} \sqrt {-5 \, \sqrt {5} - 11} \log \left (\sqrt {10} {\left (3 \, \sqrt {5} - 5\right )} \sqrt {-5 \, \sqrt {5} - 11} + 20 \, x\right ) + \frac {1}{40} \, \sqrt {10} \sqrt {-5 \, \sqrt {5} - 11} \log \left (-\sqrt {10} {\left (3 \, \sqrt {5} - 5\right )} \sqrt {-5 \, \sqrt {5} - 11} + 20 \, x\right ) + x \]
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Time = 0.73 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.34 \[ \int \frac {x^8}{1-3 x^4+x^8} \, dx=x + \operatorname {RootSum} {\left (6400 t^{4} - 880 t^{2} - 1, \left ( t \mapsto t \log {\left (- \frac {15360 t^{5}}{11} + \frac {1288 t}{55} + x \right )} \right )\right )} + \operatorname {RootSum} {\left (6400 t^{4} + 880 t^{2} - 1, \left ( t \mapsto t \log {\left (- \frac {15360 t^{5}}{11} + \frac {1288 t}{55} + x \right )} \right )\right )} \]
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\[ \int \frac {x^8}{1-3 x^4+x^8} \, dx=\int { \frac {x^{8}}{x^{8} - 3 \, x^{4} + 1} \,d x } \]
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Time = 0.37 (sec) , antiderivative size = 148, normalized size of antiderivative = 0.87 \[ \int \frac {x^8}{1-3 x^4+x^8} \, dx=-\frac {1}{20} \, \sqrt {50 \, \sqrt {5} + 110} \arctan \left (\frac {x}{\sqrt {\frac {1}{2} \, \sqrt {5} + \frac {1}{2}}}\right ) + \frac {1}{20} \, \sqrt {50 \, \sqrt {5} - 110} \arctan \left (\frac {x}{\sqrt {\frac {1}{2} \, \sqrt {5} - \frac {1}{2}}}\right ) - \frac {1}{40} \, \sqrt {50 \, \sqrt {5} + 110} \log \left ({\left | x + \sqrt {\frac {1}{2} \, \sqrt {5} + \frac {1}{2}} \right |}\right ) + \frac {1}{40} \, \sqrt {50 \, \sqrt {5} + 110} \log \left ({\left | x - \sqrt {\frac {1}{2} \, \sqrt {5} + \frac {1}{2}} \right |}\right ) + \frac {1}{40} \, \sqrt {50 \, \sqrt {5} - 110} \log \left ({\left | x + \sqrt {\frac {1}{2} \, \sqrt {5} - \frac {1}{2}} \right |}\right ) - \frac {1}{40} \, \sqrt {50 \, \sqrt {5} - 110} \log \left ({\left | x - \sqrt {\frac {1}{2} \, \sqrt {5} - \frac {1}{2}} \right |}\right ) + x \]
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Time = 8.57 (sec) , antiderivative size = 246, normalized size of antiderivative = 1.45 \[ \int \frac {x^8}{1-3 x^4+x^8} \, dx=x-\frac {\mathrm {atan}\left (\frac {x\,\sqrt {-50\,\sqrt {5}-110}\,55{}\mathrm {i}}{2\,\left (275\,\sqrt {5}+605\right )}+\frac {\sqrt {5}\,x\,\sqrt {-50\,\sqrt {5}-110}\,33{}\mathrm {i}}{2\,\left (275\,\sqrt {5}+605\right )}\right )\,\sqrt {-50\,\sqrt {5}-110}\,1{}\mathrm {i}}{20}-\frac {\mathrm {atan}\left (\frac {x\,\sqrt {110-50\,\sqrt {5}}\,55{}\mathrm {i}}{2\,\left (275\,\sqrt {5}-605\right )}-\frac {\sqrt {5}\,x\,\sqrt {110-50\,\sqrt {5}}\,33{}\mathrm {i}}{2\,\left (275\,\sqrt {5}-605\right )}\right )\,\sqrt {110-50\,\sqrt {5}}\,1{}\mathrm {i}}{20}+\frac {\mathrm {atan}\left (\frac {x\,\sqrt {50\,\sqrt {5}-110}\,55{}\mathrm {i}}{2\,\left (275\,\sqrt {5}-605\right )}-\frac {\sqrt {5}\,x\,\sqrt {50\,\sqrt {5}-110}\,33{}\mathrm {i}}{2\,\left (275\,\sqrt {5}-605\right )}\right )\,\sqrt {50\,\sqrt {5}-110}\,1{}\mathrm {i}}{20}+\frac {\mathrm {atan}\left (\frac {x\,\sqrt {50\,\sqrt {5}+110}\,55{}\mathrm {i}}{2\,\left (275\,\sqrt {5}+605\right )}+\frac {\sqrt {5}\,x\,\sqrt {50\,\sqrt {5}+110}\,33{}\mathrm {i}}{2\,\left (275\,\sqrt {5}+605\right )}\right )\,\sqrt {50\,\sqrt {5}+110}\,1{}\mathrm {i}}{20} \]
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