Integrand size = 18, antiderivative size = 131 \[ \int \frac {1+x^4}{1-3 x^4+x^8} \, dx=\frac {\arctan \left (\sqrt {\frac {2}{-1+\sqrt {5}}} x\right )}{\sqrt {2 \left (-1+\sqrt {5}\right )}}-\frac {\arctan \left (\sqrt {\frac {2}{1+\sqrt {5}}} x\right )}{\sqrt {2 \left (1+\sqrt {5}\right )}}+\frac {\text {arctanh}\left (\sqrt {\frac {2}{-1+\sqrt {5}}} x\right )}{\sqrt {2 \left (-1+\sqrt {5}\right )}}-\frac {\text {arctanh}\left (\sqrt {\frac {2}{1+\sqrt {5}}} x\right )}{\sqrt {2 \left (1+\sqrt {5}\right )}} \]
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Time = 0.06 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {1433, 1107, 209, 213} \[ \int \frac {1+x^4}{1-3 x^4+x^8} \, dx=\frac {\arctan \left (\sqrt {\frac {2}{\sqrt {5}-1}} x\right )}{\sqrt {2 \left (\sqrt {5}-1\right )}}-\frac {\arctan \left (\sqrt {\frac {2}{1+\sqrt {5}}} x\right )}{\sqrt {2 \left (1+\sqrt {5}\right )}}+\frac {\text {arctanh}\left (\sqrt {\frac {2}{\sqrt {5}-1}} x\right )}{\sqrt {2 \left (\sqrt {5}-1\right )}}-\frac {\text {arctanh}\left (\sqrt {\frac {2}{1+\sqrt {5}}} x\right )}{\sqrt {2 \left (1+\sqrt {5}\right )}} \]
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Rule 209
Rule 213
Rule 1107
Rule 1433
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \int \frac {1}{1-\sqrt {5} x^2+x^4} \, dx+\frac {1}{2} \int \frac {1}{1+\sqrt {5} x^2+x^4} \, dx \\ & = \frac {1}{2} \int \frac {1}{-\frac {1}{2}-\frac {\sqrt {5}}{2}+x^2} \, dx-\frac {1}{2} \int \frac {1}{\frac {1}{2}-\frac {\sqrt {5}}{2}+x^2} \, dx+\frac {1}{2} \int \frac {1}{-\frac {1}{2}+\frac {\sqrt {5}}{2}+x^2} \, dx-\frac {1}{2} \int \frac {1}{\frac {1}{2}+\frac {\sqrt {5}}{2}+x^2} \, dx \\ & = \frac {\tan ^{-1}\left (\sqrt {\frac {2}{-1+\sqrt {5}}} x\right )}{\sqrt {2 \left (-1+\sqrt {5}\right )}}-\frac {\tan ^{-1}\left (\sqrt {\frac {2}{1+\sqrt {5}}} x\right )}{\sqrt {2 \left (1+\sqrt {5}\right )}}+\frac {\tanh ^{-1}\left (\sqrt {\frac {2}{-1+\sqrt {5}}} x\right )}{\sqrt {2 \left (-1+\sqrt {5}\right )}}-\frac {\tanh ^{-1}\left (\sqrt {\frac {2}{1+\sqrt {5}}} x\right )}{\sqrt {2 \left (1+\sqrt {5}\right )}} \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.00 \[ \int \frac {1+x^4}{1-3 x^4+x^8} \, dx=\frac {\arctan \left (\sqrt {\frac {2}{-1+\sqrt {5}}} x\right )}{\sqrt {2 \left (-1+\sqrt {5}\right )}}-\frac {\arctan \left (\sqrt {\frac {2}{1+\sqrt {5}}} x\right )}{\sqrt {2 \left (1+\sqrt {5}\right )}}+\frac {\text {arctanh}\left (\sqrt {\frac {2}{-1+\sqrt {5}}} x\right )}{\sqrt {2 \left (-1+\sqrt {5}\right )}}-\frac {\text {arctanh}\left (\sqrt {\frac {2}{1+\sqrt {5}}} x\right )}{\sqrt {2 \left (1+\sqrt {5}\right )}} \]
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Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.11 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.43
method | result | size |
risch | \(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{4}-\textit {\_Z}^{2}-1\right )}{\sum }\textit {\_R} \ln \left (\textit {\_R}^{3}-\textit {\_R} +x \right )\right )}{4}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{4}+\textit {\_Z}^{2}-1\right )}{\sum }\textit {\_R} \ln \left (-\textit {\_R}^{3}-\textit {\_R} +x \right )\right )}{4}\) | \(56\) |
default | \(-\frac {\operatorname {arctanh}\left (\frac {2 x}{\sqrt {2 \sqrt {5}+2}}\right )}{\sqrt {2 \sqrt {5}+2}}+\frac {\arctan \left (\frac {2 x}{\sqrt {2 \sqrt {5}-2}}\right )}{\sqrt {2 \sqrt {5}-2}}-\frac {\arctan \left (\frac {2 x}{\sqrt {2 \sqrt {5}+2}}\right )}{\sqrt {2 \sqrt {5}+2}}+\frac {\operatorname {arctanh}\left (\frac {2 x}{\sqrt {2 \sqrt {5}-2}}\right )}{\sqrt {2 \sqrt {5}-2}}\) | \(96\) |
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Leaf count of result is larger than twice the leaf count of optimal. 317 vs. \(2 (95) = 190\).
Time = 0.28 (sec) , antiderivative size = 317, normalized size of antiderivative = 2.42 \[ \int \frac {1+x^4}{1-3 x^4+x^8} \, dx=\frac {1}{8} \, \sqrt {2} \sqrt {\sqrt {5} + 1} \log \left ({\left (\sqrt {5} \sqrt {2} - \sqrt {2}\right )} \sqrt {\sqrt {5} + 1} + 4 \, x\right ) - \frac {1}{8} \, \sqrt {2} \sqrt {\sqrt {5} + 1} \log \left (-{\left (\sqrt {5} \sqrt {2} - \sqrt {2}\right )} \sqrt {\sqrt {5} + 1} + 4 \, x\right ) - \frac {1}{8} \, \sqrt {2} \sqrt {\sqrt {5} - 1} \log \left ({\left (\sqrt {5} \sqrt {2} + \sqrt {2}\right )} \sqrt {\sqrt {5} - 1} + 4 \, x\right ) + \frac {1}{8} \, \sqrt {2} \sqrt {\sqrt {5} - 1} \log \left (-{\left (\sqrt {5} \sqrt {2} + \sqrt {2}\right )} \sqrt {\sqrt {5} - 1} + 4 \, x\right ) - \frac {1}{8} \, \sqrt {2} \sqrt {-\sqrt {5} + 1} \log \left ({\left (\sqrt {5} \sqrt {2} + \sqrt {2}\right )} \sqrt {-\sqrt {5} + 1} + 4 \, x\right ) + \frac {1}{8} \, \sqrt {2} \sqrt {-\sqrt {5} + 1} \log \left (-{\left (\sqrt {5} \sqrt {2} + \sqrt {2}\right )} \sqrt {-\sqrt {5} + 1} + 4 \, x\right ) + \frac {1}{8} \, \sqrt {2} \sqrt {-\sqrt {5} - 1} \log \left ({\left (\sqrt {5} \sqrt {2} - \sqrt {2}\right )} \sqrt {-\sqrt {5} - 1} + 4 \, x\right ) - \frac {1}{8} \, \sqrt {2} \sqrt {-\sqrt {5} - 1} \log \left (-{\left (\sqrt {5} \sqrt {2} - \sqrt {2}\right )} \sqrt {-\sqrt {5} - 1} + 4 \, x\right ) \]
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Time = 0.66 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.37 \[ \int \frac {1+x^4}{1-3 x^4+x^8} \, dx=\operatorname {RootSum} {\left (256 t^{4} - 16 t^{2} - 1, \left ( t \mapsto t \log {\left (1024 t^{5} - 8 t + x \right )} \right )\right )} + \operatorname {RootSum} {\left (256 t^{4} + 16 t^{2} - 1, \left ( t \mapsto t \log {\left (1024 t^{5} - 8 t + x \right )} \right )\right )} \]
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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 = 0.37 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.12 \[ \int \frac {1+x^4}{1-3 x^4+x^8} \, dx=-\frac {1}{4} \, \sqrt {2 \, \sqrt {5} - 2} \arctan \left (\frac {x}{\sqrt {\frac {1}{2} \, \sqrt {5} + \frac {1}{2}}}\right ) + \frac {1}{4} \, \sqrt {2 \, \sqrt {5} + 2} \arctan \left (\frac {x}{\sqrt {\frac {1}{2} \, \sqrt {5} - \frac {1}{2}}}\right ) - \frac {1}{8} \, \sqrt {2 \, \sqrt {5} - 2} \log \left ({\left | x + \sqrt {\frac {1}{2} \, \sqrt {5} + \frac {1}{2}} \right |}\right ) + \frac {1}{8} \, \sqrt {2 \, \sqrt {5} - 2} \log \left ({\left | x - \sqrt {\frac {1}{2} \, \sqrt {5} + \frac {1}{2}} \right |}\right ) + \frac {1}{8} \, \sqrt {2 \, \sqrt {5} + 2} \log \left ({\left | x + \sqrt {\frac {1}{2} \, \sqrt {5} - \frac {1}{2}} \right |}\right ) - \frac {1}{8} \, \sqrt {2 \, \sqrt {5} + 2} \log \left ({\left | x - \sqrt {\frac {1}{2} \, \sqrt {5} - \frac {1}{2}} \right |}\right ) \]
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Time = 0.12 (sec) , antiderivative size = 269, normalized size of antiderivative = 2.05 \[ \int \frac {1+x^4}{1-3 x^4+x^8} \, dx=-\frac {\sqrt {2}\,\mathrm {atan}\left (\frac {\sqrt {2}\,x\,\sqrt {\sqrt {5}-1}\,1875{}\mathrm {i}}{2\,\left (875\,\sqrt {5}-1875\right )}-\frac {\sqrt {2}\,\sqrt {5}\,x\,\sqrt {\sqrt {5}-1}\,875{}\mathrm {i}}{2\,\left (875\,\sqrt {5}-1875\right )}\right )\,\sqrt {\sqrt {5}-1}\,1{}\mathrm {i}}{4}-\frac {\sqrt {2}\,\mathrm {atan}\left (\frac {\sqrt {2}\,x\,\sqrt {\sqrt {5}+1}\,1875{}\mathrm {i}}{2\,\left (875\,\sqrt {5}+1875\right )}+\frac {\sqrt {2}\,\sqrt {5}\,x\,\sqrt {\sqrt {5}+1}\,875{}\mathrm {i}}{2\,\left (875\,\sqrt {5}+1875\right )}\right )\,\sqrt {\sqrt {5}+1}\,1{}\mathrm {i}}{4}+\frac {\sqrt {2}\,\mathrm {atan}\left (\frac {\sqrt {2}\,x\,\sqrt {1-\sqrt {5}}\,1875{}\mathrm {i}}{2\,\left (875\,\sqrt {5}-1875\right )}-\frac {\sqrt {2}\,\sqrt {5}\,x\,\sqrt {1-\sqrt {5}}\,875{}\mathrm {i}}{2\,\left (875\,\sqrt {5}-1875\right )}\right )\,\sqrt {1-\sqrt {5}}\,1{}\mathrm {i}}{4}+\frac {\sqrt {2}\,\mathrm {atan}\left (\frac {\sqrt {2}\,x\,\sqrt {-\sqrt {5}-1}\,1875{}\mathrm {i}}{2\,\left (875\,\sqrt {5}+1875\right )}+\frac {\sqrt {2}\,\sqrt {5}\,x\,\sqrt {-\sqrt {5}-1}\,875{}\mathrm {i}}{2\,\left (875\,\sqrt {5}+1875\right )}\right )\,\sqrt {-\sqrt {5}-1}\,1{}\mathrm {i}}{4} \]
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