Integrand size = 12, antiderivative size = 169 \[ \int \frac {1}{1-3 x^4+x^8} \, dx=-\frac {\arctan \left (\sqrt [4]{\frac {2}{3+\sqrt {5}}} x\right )}{\sqrt [4]{2} \sqrt {5} \left (3+\sqrt {5}\right )^{3/4}}+\frac {\left (3+\sqrt {5}\right )^{3/4} \arctan \left (\sqrt [4]{\frac {1}{2} \left (3+\sqrt {5}\right )} x\right )}{2\ 2^{3/4} \sqrt {5}}-\frac {\text {arctanh}\left (\sqrt [4]{\frac {2}{3+\sqrt {5}}} x\right )}{\sqrt [4]{2} \sqrt {5} \left (3+\sqrt {5}\right )^{3/4}}+\frac {\left (3+\sqrt {5}\right )^{3/4} \text {arctanh}\left (\sqrt [4]{\frac {1}{2} \left (3+\sqrt {5}\right )} x\right )}{2\ 2^{3/4} \sqrt {5}} \]
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Time = 0.04 (sec) , antiderivative size = 169, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {1361, 218, 212, 209} \[ \int \frac {1}{1-3 x^4+x^8} \, dx=-\frac {\arctan \left (\sqrt [4]{\frac {2}{3+\sqrt {5}}} x\right )}{\sqrt [4]{2} \sqrt {5} \left (3+\sqrt {5}\right )^{3/4}}+\frac {\left (3+\sqrt {5}\right )^{3/4} \arctan \left (\sqrt [4]{\frac {1}{2} \left (3+\sqrt {5}\right )} x\right )}{2\ 2^{3/4} \sqrt {5}}-\frac {\text {arctanh}\left (\sqrt [4]{\frac {2}{3+\sqrt {5}}} x\right )}{\sqrt [4]{2} \sqrt {5} \left (3+\sqrt {5}\right )^{3/4}}+\frac {\left (3+\sqrt {5}\right )^{3/4} \text {arctanh}\left (\sqrt [4]{\frac {1}{2} \left (3+\sqrt {5}\right )} x\right )}{2\ 2^{3/4} \sqrt {5}} \]
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Rule 209
Rule 212
Rule 218
Rule 1361
Rubi steps \begin{align*} \text {integral}& = \frac {\int \frac {1}{-\frac {3}{2}-\frac {\sqrt {5}}{2}+x^4} \, dx}{\sqrt {5}}-\frac {\int \frac {1}{-\frac {3}{2}+\frac {\sqrt {5}}{2}+x^4} \, dx}{\sqrt {5}} \\ & = \frac {\int \frac {1}{\sqrt {3-\sqrt {5}}-\sqrt {2} x^2} \, dx}{\sqrt {5 \left (3-\sqrt {5}\right )}}+\frac {\int \frac {1}{\sqrt {3-\sqrt {5}}+\sqrt {2} x^2} \, dx}{\sqrt {5 \left (3-\sqrt {5}\right )}}-\frac {\int \frac {1}{\sqrt {3+\sqrt {5}}-\sqrt {2} x^2} \, dx}{\sqrt {5 \left (3+\sqrt {5}\right )}}-\frac {\int \frac {1}{\sqrt {3+\sqrt {5}}+\sqrt {2} x^2} \, dx}{\sqrt {5 \left (3+\sqrt {5}\right )}} \\ & = -\frac {\tan ^{-1}\left (\sqrt [4]{\frac {2}{3+\sqrt {5}}} x\right )}{\sqrt [4]{2} \sqrt {5} \left (3+\sqrt {5}\right )^{3/4}}+\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 {\tanh ^{-1}\left (\sqrt [4]{\frac {2}{3+\sqrt {5}}} x\right )}{\sqrt [4]{2} \sqrt {5} \left (3+\sqrt {5}\right )^{3/4}}+\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 = 0.13 (sec) , antiderivative size = 160, normalized size of antiderivative = 0.95 \[ \int \frac {1}{1-3 x^4+x^8} \, dx=\frac {\frac {\left (1+\sqrt {5}\right ) \arctan \left (\sqrt {\frac {2}{-1+\sqrt {5}}} x\right )}{\sqrt {-1+\sqrt {5}}}-\frac {\left (-1+\sqrt {5}\right ) \arctan \left (\sqrt {\frac {2}{1+\sqrt {5}}} x\right )}{\sqrt {1+\sqrt {5}}}+\frac {\left (1+\sqrt {5}\right ) \text {arctanh}\left (\sqrt {\frac {2}{-1+\sqrt {5}}} x\right )}{\sqrt {-1+\sqrt {5}}}-\frac {\left (-1+\sqrt {5}\right ) \text {arctanh}\left (\sqrt {\frac {2}{1+\sqrt {5}}} x\right )}{\sqrt {1+\sqrt {5}}}}{2 \sqrt {10}} \]
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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 = 0.40
method | result | size |
risch | \(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (25 \textit {\_Z}^{4}+20 \textit {\_Z}^{2}-1\right )}{\sum }\textit {\_R} \ln \left (-15 \textit {\_R}^{3}-11 \textit {\_R} +2 x \right )\right )}{4}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (25 \textit {\_Z}^{4}-20 \textit {\_Z}^{2}-1\right )}{\sum }\textit {\_R} \ln \left (15 \textit {\_R}^{3}-11 \textit {\_R} +2 x \right )\right )}{4}\) | \(68\) |
default | \(-\frac {\sqrt {5}\, \left (\sqrt {5}-1\right ) \operatorname {arctanh}\left (\frac {2 x}{\sqrt {2 \sqrt {5}+2}}\right )}{10 \sqrt {2 \sqrt {5}+2}}+\frac {\left (\sqrt {5}+1\right ) \sqrt {5}\, \arctan \left (\frac {2 x}{\sqrt {2 \sqrt {5}-2}}\right )}{10 \sqrt {2 \sqrt {5}-2}}-\frac {\sqrt {5}\, \left (\sqrt {5}-1\right ) \arctan \left (\frac {2 x}{\sqrt {2 \sqrt {5}+2}}\right )}{10 \sqrt {2 \sqrt {5}+2}}+\frac {\left (\sqrt {5}+1\right ) \sqrt {5}\, \operatorname {arctanh}\left (\frac {2 x}{\sqrt {2 \sqrt {5}-2}}\right )}{10 \sqrt {2 \sqrt {5}-2}}\) | \(130\) |
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Leaf count of result is larger than twice the leaf count of optimal. 261 vs. \(2 (113) = 226\).
Time = 0.27 (sec) , antiderivative size = 261, normalized size of antiderivative = 1.54 \[ \int \frac {1}{1-3 x^4+x^8} \, dx=-\frac {1}{20} \, \sqrt {5} \sqrt {\sqrt {5} - 2} \log \left ({\left (\sqrt {5} + 3\right )} \sqrt {\sqrt {5} - 2} + 2 \, x\right ) + \frac {1}{20} \, \sqrt {5} \sqrt {\sqrt {5} - 2} \log \left (-{\left (\sqrt {5} + 3\right )} \sqrt {\sqrt {5} - 2} + 2 \, x\right ) - \frac {1}{20} \, \sqrt {5} \sqrt {\sqrt {5} + 2} \log \left (\sqrt {\sqrt {5} + 2} {\left (\sqrt {5} - 3\right )} + 2 \, x\right ) + \frac {1}{20} \, \sqrt {5} \sqrt {\sqrt {5} + 2} \log \left (-\sqrt {\sqrt {5} + 2} {\left (\sqrt {5} - 3\right )} + 2 \, x\right ) - \frac {1}{20} \, \sqrt {5} \sqrt {-\sqrt {5} + 2} \log \left ({\left (\sqrt {5} + 3\right )} \sqrt {-\sqrt {5} + 2} + 2 \, x\right ) + \frac {1}{20} \, \sqrt {5} \sqrt {-\sqrt {5} + 2} \log \left (-{\left (\sqrt {5} + 3\right )} \sqrt {-\sqrt {5} + 2} + 2 \, x\right ) - \frac {1}{20} \, \sqrt {5} \sqrt {-\sqrt {5} - 2} \log \left ({\left (\sqrt {5} - 3\right )} \sqrt {-\sqrt {5} - 2} + 2 \, x\right ) + \frac {1}{20} \, \sqrt {5} \sqrt {-\sqrt {5} - 2} \log \left (-{\left (\sqrt {5} - 3\right )} \sqrt {-\sqrt {5} - 2} + 2 \, x\right ) \]
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Time = 0.73 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.31 \[ \int \frac {1}{1-3 x^4+x^8} \, dx=\operatorname {RootSum} {\left (6400 t^{4} - 320 t^{2} - 1, \left ( t \mapsto t \log {\left (9600 t^{5} - \frac {47 t}{2} + x \right )} \right )\right )} + \operatorname {RootSum} {\left (6400 t^{4} + 320 t^{2} - 1, \left ( t \mapsto t \log {\left (9600 t^{5} - \frac {47 t}{2} + x \right )} \right )\right )} \]
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\[ \int \frac {1}{1-3 x^4+x^8} \, dx=\int { \frac {1}{x^{8} - 3 \, x^{4} + 1} \,d x } \]
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none
Time = 0.33 (sec) , antiderivative size = 147, normalized size of antiderivative = 0.87 \[ \int \frac {1}{1-3 x^4+x^8} \, dx=-\frac {1}{10} \, \sqrt {5 \, \sqrt {5} - 10} \arctan \left (\frac {x}{\sqrt {\frac {1}{2} \, \sqrt {5} + \frac {1}{2}}}\right ) + \frac {1}{10} \, \sqrt {5 \, \sqrt {5} + 10} \arctan \left (\frac {x}{\sqrt {\frac {1}{2} \, \sqrt {5} - \frac {1}{2}}}\right ) - \frac {1}{20} \, \sqrt {5 \, \sqrt {5} - 10} \log \left ({\left | x + \sqrt {\frac {1}{2} \, \sqrt {5} + \frac {1}{2}} \right |}\right ) + \frac {1}{20} \, \sqrt {5 \, \sqrt {5} - 10} \log \left ({\left | x - \sqrt {\frac {1}{2} \, \sqrt {5} + \frac {1}{2}} \right |}\right ) + \frac {1}{20} \, \sqrt {5 \, \sqrt {5} + 10} \log \left ({\left | x + \sqrt {\frac {1}{2} \, \sqrt {5} - \frac {1}{2}} \right |}\right ) - \frac {1}{20} \, \sqrt {5 \, \sqrt {5} + 10} \log \left ({\left | x - \sqrt {\frac {1}{2} \, \sqrt {5} - \frac {1}{2}} \right |}\right ) \]
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Time = 8.38 (sec) , antiderivative size = 245, normalized size of antiderivative = 1.45 \[ \int \frac {1}{1-3 x^4+x^8} \, dx=-\frac {\sqrt {5}\,\mathrm {atan}\left (\frac {x\,\sqrt {2-\sqrt {5}}\,144{}\mathrm {i}}{104\,\sqrt {5}-232}-\frac {\sqrt {5}\,x\,\sqrt {2-\sqrt {5}}\,64{}\mathrm {i}}{104\,\sqrt {5}-232}\right )\,\sqrt {2-\sqrt {5}}\,1{}\mathrm {i}}{10}+\frac {\sqrt {5}\,\mathrm {atan}\left (\frac {x\,\sqrt {-\sqrt {5}-2}\,144{}\mathrm {i}}{104\,\sqrt {5}+232}+\frac {\sqrt {5}\,x\,\sqrt {-\sqrt {5}-2}\,64{}\mathrm {i}}{104\,\sqrt {5}+232}\right )\,\sqrt {-\sqrt {5}-2}\,1{}\mathrm {i}}{10}+\frac {\sqrt {5}\,\mathrm {atan}\left (\frac {x\,\sqrt {\sqrt {5}-2}\,144{}\mathrm {i}}{104\,\sqrt {5}-232}-\frac {\sqrt {5}\,x\,\sqrt {\sqrt {5}-2}\,64{}\mathrm {i}}{104\,\sqrt {5}-232}\right )\,\sqrt {\sqrt {5}-2}\,1{}\mathrm {i}}{10}-\frac {\sqrt {5}\,\mathrm {atan}\left (\frac {x\,\sqrt {\sqrt {5}+2}\,144{}\mathrm {i}}{104\,\sqrt {5}+232}+\frac {\sqrt {5}\,x\,\sqrt {\sqrt {5}+2}\,64{}\mathrm {i}}{104\,\sqrt {5}+232}\right )\,\sqrt {\sqrt {5}+2}\,1{}\mathrm {i}}{10} \]
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