Integrand size = 20, antiderivative size = 129 \[ \int \frac {1-x^4}{1-3 x^4+x^8} \, dx=\frac {\arctan \left (\sqrt {\frac {2}{-1+\sqrt {5}}} x\right )}{\sqrt {10 \left (-1+\sqrt {5}\right )}}+\frac {\arctan \left (\sqrt {\frac {2}{1+\sqrt {5}}} x\right )}{\sqrt {10 \left (1+\sqrt {5}\right )}}+\frac {\text {arctanh}\left (\sqrt {\frac {2}{-1+\sqrt {5}}} x\right )}{\sqrt {10 \left (-1+\sqrt {5}\right )}}+\frac {\text {arctanh}\left (\sqrt {\frac {2}{1+\sqrt {5}}} x\right )}{\sqrt {10 \left (1+\sqrt {5}\right )}} \]
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Time = 0.08 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {1433, 1107, 213, 209} \[ \int \frac {1-x^4}{1-3 x^4+x^8} \, dx=\frac {\arctan \left (\sqrt {\frac {2}{\sqrt {5}-1}} x\right )}{\sqrt {10 \left (\sqrt {5}-1\right )}}+\frac {\arctan \left (\sqrt {\frac {2}{1+\sqrt {5}}} x\right )}{\sqrt {10 \left (1+\sqrt {5}\right )}}+\frac {\text {arctanh}\left (\sqrt {\frac {2}{\sqrt {5}-1}} x\right )}{\sqrt {10 \left (\sqrt {5}-1\right )}}+\frac {\text {arctanh}\left (\sqrt {\frac {2}{1+\sqrt {5}}} x\right )}{\sqrt {10 \left (1+\sqrt {5}\right )}} \]
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Rule 209
Rule 213
Rule 1107
Rule 1433
Rubi steps \begin{align*} \text {integral}& = -\left (\frac {1}{2} \int \frac {1}{-1-x^2+x^4} \, dx\right )-\frac {1}{2} \int \frac {1}{-1+x^2+x^4} \, dx \\ & = -\frac {\int \frac {1}{-\frac {1}{2}-\frac {\sqrt {5}}{2}+x^2} \, dx}{2 \sqrt {5}}-\frac {\int \frac {1}{\frac {1}{2}-\frac {\sqrt {5}}{2}+x^2} \, dx}{2 \sqrt {5}}+\frac {\int \frac {1}{-\frac {1}{2}+\frac {\sqrt {5}}{2}+x^2} \, dx}{2 \sqrt {5}}+\frac {\int \frac {1}{\frac {1}{2}+\frac {\sqrt {5}}{2}+x^2} \, dx}{2 \sqrt {5}} \\ & = \frac {\tan ^{-1}\left (\sqrt {\frac {2}{-1+\sqrt {5}}} x\right )}{\sqrt {10 \left (-1+\sqrt {5}\right )}}+\frac {\tan ^{-1}\left (\sqrt {\frac {2}{1+\sqrt {5}}} x\right )}{\sqrt {10 \left (1+\sqrt {5}\right )}}+\frac {\tanh ^{-1}\left (\sqrt {\frac {2}{-1+\sqrt {5}}} x\right )}{\sqrt {10 \left (-1+\sqrt {5}\right )}}+\frac {\tanh ^{-1}\left (\sqrt {\frac {2}{1+\sqrt {5}}} x\right )}{\sqrt {10 \left (1+\sqrt {5}\right )}} \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.00 \[ \int \frac {1-x^4}{1-3 x^4+x^8} \, dx=\frac {\arctan \left (\sqrt {\frac {2}{-1+\sqrt {5}}} x\right )}{\sqrt {10 \left (-1+\sqrt {5}\right )}}+\frac {\arctan \left (\sqrt {\frac {2}{1+\sqrt {5}}} x\right )}{\sqrt {10 \left (1+\sqrt {5}\right )}}+\frac {\text {arctanh}\left (\sqrt {\frac {2}{-1+\sqrt {5}}} x\right )}{\sqrt {10 \left (-1+\sqrt {5}\right )}}+\frac {\text {arctanh}\left (\sqrt {\frac {2}{1+\sqrt {5}}} x\right )}{\sqrt {10 \left (1+\sqrt {5}\right )}} \]
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Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.09 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.50
method | result | size |
risch | \(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (25 \textit {\_Z}^{4}-5 \textit {\_Z}^{2}-1\right )}{\sum }\textit {\_R} \ln \left (-5 \textit {\_R}^{3}+3 \textit {\_R} +x \right )\right )}{4}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (25 \textit {\_Z}^{4}+5 \textit {\_Z}^{2}-1\right )}{\sum }\textit {\_R} \ln \left (5 \textit {\_R}^{3}+3 \textit {\_R} +x \right )\right )}{4}\) | \(64\) |
default | \(\frac {\sqrt {5}\, \operatorname {arctanh}\left (\frac {2 x}{\sqrt {2 \sqrt {5}+2}}\right )}{5 \sqrt {2 \sqrt {5}+2}}+\frac {\sqrt {5}\, \arctan \left (\frac {2 x}{\sqrt {2 \sqrt {5}-2}}\right )}{5 \sqrt {2 \sqrt {5}-2}}+\frac {\sqrt {5}\, \arctan \left (\frac {2 x}{\sqrt {2 \sqrt {5}+2}}\right )}{5 \sqrt {2 \sqrt {5}+2}}+\frac {\sqrt {5}\, \operatorname {arctanh}\left (\frac {2 x}{\sqrt {2 \sqrt {5}-2}}\right )}{5 \sqrt {2 \sqrt {5}-2}}\) | \(110\) |
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Leaf count of result is larger than twice the leaf count of optimal. 285 vs. \(2 (93) = 186\).
Time = 0.29 (sec) , antiderivative size = 285, normalized size of antiderivative = 2.21 \[ \int \frac {1-x^4}{1-3 x^4+x^8} \, dx=\frac {1}{40} \, \sqrt {10} \sqrt {\sqrt {5} - 1} \log \left (\sqrt {10} {\left (\sqrt {5} + 5\right )} \sqrt {\sqrt {5} - 1} + 20 \, x\right ) - \frac {1}{40} \, \sqrt {10} \sqrt {\sqrt {5} - 1} \log \left (-\sqrt {10} {\left (\sqrt {5} + 5\right )} \sqrt {\sqrt {5} - 1} + 20 \, x\right ) - \frac {1}{40} \, \sqrt {10} \sqrt {\sqrt {5} + 1} \log \left (\sqrt {10} \sqrt {\sqrt {5} + 1} {\left (\sqrt {5} - 5\right )} + 20 \, x\right ) + \frac {1}{40} \, \sqrt {10} \sqrt {\sqrt {5} + 1} \log \left (-\sqrt {10} \sqrt {\sqrt {5} + 1} {\left (\sqrt {5} - 5\right )} + 20 \, x\right ) + \frac {1}{40} \, \sqrt {10} \sqrt {-\sqrt {5} + 1} \log \left (\sqrt {10} {\left (\sqrt {5} + 5\right )} \sqrt {-\sqrt {5} + 1} + 20 \, x\right ) - \frac {1}{40} \, \sqrt {10} \sqrt {-\sqrt {5} + 1} \log \left (-\sqrt {10} {\left (\sqrt {5} + 5\right )} \sqrt {-\sqrt {5} + 1} + 20 \, x\right ) - \frac {1}{40} \, \sqrt {10} \sqrt {-\sqrt {5} - 1} \log \left (\sqrt {10} {\left (\sqrt {5} - 5\right )} \sqrt {-\sqrt {5} - 1} + 20 \, x\right ) + \frac {1}{40} \, \sqrt {10} \sqrt {-\sqrt {5} - 1} \log \left (-\sqrt {10} {\left (\sqrt {5} - 5\right )} \sqrt {-\sqrt {5} - 1} + 20 \, x\right ) \]
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Time = 0.68 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.40 \[ \int \frac {1-x^4}{1-3 x^4+x^8} \, dx=- \operatorname {RootSum} {\left (6400 t^{4} - 80 t^{2} - 1, \left ( t \mapsto t \log {\left (25600 t^{5} - 16 t + x \right )} \right )\right )} - \operatorname {RootSum} {\left (6400 t^{4} + 80 t^{2} - 1, \left ( t \mapsto t \log {\left (25600 t^{5} - 16 t + x \right )} \right )\right )} \]
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\[ \int \frac {1-x^4}{1-3 x^4+x^8} \, dx=\int { -\frac {x^{4} - 1}{x^{8} - 3 \, x^{4} + 1} \,d x } \]
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Time = 0.48 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.14 \[ \int \frac {1-x^4}{1-3 x^4+x^8} \, dx=\frac {1}{20} \, \sqrt {10 \, \sqrt {5} - 10} \arctan \left (\frac {x}{\sqrt {\frac {1}{2} \, \sqrt {5} + \frac {1}{2}}}\right ) + \frac {1}{20} \, \sqrt {10 \, \sqrt {5} + 10} \arctan \left (\frac {x}{\sqrt {\frac {1}{2} \, \sqrt {5} - \frac {1}{2}}}\right ) + \frac {1}{40} \, \sqrt {10 \, \sqrt {5} - 10} \log \left ({\left | x + \sqrt {\frac {1}{2} \, \sqrt {5} + \frac {1}{2}} \right |}\right ) - \frac {1}{40} \, \sqrt {10 \, \sqrt {5} - 10} \log \left ({\left | x - \sqrt {\frac {1}{2} \, \sqrt {5} + \frac {1}{2}} \right |}\right ) + \frac {1}{40} \, \sqrt {10 \, \sqrt {5} + 10} \log \left ({\left | x + \sqrt {\frac {1}{2} \, \sqrt {5} - \frac {1}{2}} \right |}\right ) - \frac {1}{40} \, \sqrt {10 \, \sqrt {5} + 10} \log \left ({\left | x - \sqrt {\frac {1}{2} \, \sqrt {5} - \frac {1}{2}} \right |}\right ) \]
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Time = 0.10 (sec) , antiderivative size = 269, normalized size of antiderivative = 2.09 \[ \int \frac {1-x^4}{1-3 x^4+x^8} \, dx=-\frac {\sqrt {10}\,\mathrm {atan}\left (\frac {\sqrt {10}\,x\,\sqrt {\sqrt {5}-1}\,3{}\mathrm {i}}{2\,\left (3\,\sqrt {5}-7\right )}-\frac {\sqrt {5}\,\sqrt {10}\,x\,\sqrt {\sqrt {5}-1}\,7{}\mathrm {i}}{10\,\left (3\,\sqrt {5}-7\right )}\right )\,\sqrt {\sqrt {5}-1}\,1{}\mathrm {i}}{20}-\frac {\sqrt {10}\,\mathrm {atan}\left (\frac {\sqrt {10}\,x\,\sqrt {\sqrt {5}+1}\,3{}\mathrm {i}}{2\,\left (3\,\sqrt {5}+7\right )}+\frac {\sqrt {5}\,\sqrt {10}\,x\,\sqrt {\sqrt {5}+1}\,7{}\mathrm {i}}{10\,\left (3\,\sqrt {5}+7\right )}\right )\,\sqrt {\sqrt {5}+1}\,1{}\mathrm {i}}{20}+\frac {\sqrt {10}\,\mathrm {atan}\left (\frac {\sqrt {10}\,x\,\sqrt {1-\sqrt {5}}\,3{}\mathrm {i}}{2\,\left (3\,\sqrt {5}-7\right )}-\frac {\sqrt {5}\,\sqrt {10}\,x\,\sqrt {1-\sqrt {5}}\,7{}\mathrm {i}}{10\,\left (3\,\sqrt {5}-7\right )}\right )\,\sqrt {1-\sqrt {5}}\,1{}\mathrm {i}}{20}+\frac {\sqrt {10}\,\mathrm {atan}\left (\frac {\sqrt {10}\,x\,\sqrt {-\sqrt {5}-1}\,3{}\mathrm {i}}{2\,\left (3\,\sqrt {5}+7\right )}+\frac {\sqrt {5}\,\sqrt {10}\,x\,\sqrt {-\sqrt {5}-1}\,7{}\mathrm {i}}{10\,\left (3\,\sqrt {5}+7\right )}\right )\,\sqrt {-\sqrt {5}-1}\,1{}\mathrm {i}}{20} \]
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