Integrand size = 20, antiderivative size = 125 \[ \int \frac {1-x^4}{1-6 x^4+x^8} \, dx=\frac {\arctan \left (\frac {x}{\sqrt {-1+\sqrt {2}}}\right )}{4 \sqrt {2 \left (-1+\sqrt {2}\right )}}+\frac {\arctan \left (\frac {x}{\sqrt {1+\sqrt {2}}}\right )}{4 \sqrt {2 \left (1+\sqrt {2}\right )}}+\frac {\text {arctanh}\left (\frac {x}{\sqrt {-1+\sqrt {2}}}\right )}{4 \sqrt {2 \left (-1+\sqrt {2}\right )}}+\frac {\text {arctanh}\left (\frac {x}{\sqrt {1+\sqrt {2}}}\right )}{4 \sqrt {2 \left (1+\sqrt {2}\right )}} \]
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Time = 0.05 (sec) , antiderivative size = 125, 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} \[ \int \frac {1-x^4}{1-6 x^4+x^8} \, dx=\frac {\arctan \left (\frac {x}{\sqrt {\sqrt {2}-1}}\right )}{4 \sqrt {2 \left (\sqrt {2}-1\right )}}+\frac {\arctan \left (\frac {x}{\sqrt {1+\sqrt {2}}}\right )}{4 \sqrt {2 \left (1+\sqrt {2}\right )}}+\frac {\text {arctanh}\left (\frac {x}{\sqrt {\sqrt {2}-1}}\right )}{4 \sqrt {2 \left (\sqrt {2}-1\right )}}+\frac {\text {arctanh}\left (\frac {x}{\sqrt {1+\sqrt {2}}}\right )}{4 \sqrt {2 \left (1+\sqrt {2}\right )}} \]
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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}{-1-2 x^2+x^4} \, dx\right )-\frac {1}{2} \int \frac {1}{-1+2 x^2+x^4} \, dx \\ & = -\frac {\int \frac {1}{-1-\sqrt {2}+x^2} \, dx}{4 \sqrt {2}}-\frac {\int \frac {1}{1-\sqrt {2}+x^2} \, dx}{4 \sqrt {2}}+\frac {\int \frac {1}{-1+\sqrt {2}+x^2} \, dx}{4 \sqrt {2}}+\frac {\int \frac {1}{1+\sqrt {2}+x^2} \, dx}{4 \sqrt {2}} \\ & = \frac {\tan ^{-1}\left (\frac {x}{\sqrt {-1+\sqrt {2}}}\right )}{4 \sqrt {2 \left (-1+\sqrt {2}\right )}}+\frac {\tan ^{-1}\left (\frac {x}{\sqrt {1+\sqrt {2}}}\right )}{4 \sqrt {2 \left (1+\sqrt {2}\right )}}+\frac {\tanh ^{-1}\left (\frac {x}{\sqrt {-1+\sqrt {2}}}\right )}{4 \sqrt {2 \left (-1+\sqrt {2}\right )}}+\frac {\tanh ^{-1}\left (\frac {x}{\sqrt {1+\sqrt {2}}}\right )}{4 \sqrt {2 \left (1+\sqrt {2}\right )}} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.91 \[ \int \frac {1-x^4}{1-6 x^4+x^8} \, dx=\frac {\sqrt {1+\sqrt {2}} \arctan \left (\frac {x}{\sqrt {-1+\sqrt {2}}}\right )+\sqrt {-1+\sqrt {2}} \arctan \left (\frac {x}{\sqrt {1+\sqrt {2}}}\right )+\sqrt {1+\sqrt {2}} \text {arctanh}\left (\frac {x}{\sqrt {-1+\sqrt {2}}}\right )+\sqrt {-1+\sqrt {2}} \text {arctanh}\left (\frac {x}{\sqrt {1+\sqrt {2}}}\right )}{4 \sqrt {2}} \]
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Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.10 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.51
method | result | size |
risch | \(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (4 \textit {\_Z}^{4}-4 \textit {\_Z}^{2}-1\right )}{\sum }\textit {\_R} \ln \left (-2 \textit {\_R}^{3}+3 \textit {\_R} +x \right )\right )}{8}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (4 \textit {\_Z}^{4}+4 \textit {\_Z}^{2}-1\right )}{\sum }\textit {\_R} \ln \left (2 \textit {\_R}^{3}+3 \textit {\_R} +x \right )\right )}{8}\) | \(64\) |
default | \(\frac {\sqrt {2}\, \operatorname {arctanh}\left (\frac {x}{\sqrt {1+\sqrt {2}}}\right )}{8 \sqrt {1+\sqrt {2}}}+\frac {\sqrt {2}\, \arctan \left (\frac {x}{\sqrt {\sqrt {2}-1}}\right )}{8 \sqrt {\sqrt {2}-1}}+\frac {\sqrt {2}\, \operatorname {arctanh}\left (\frac {x}{\sqrt {\sqrt {2}-1}}\right )}{8 \sqrt {\sqrt {2}-1}}+\frac {\sqrt {2}\, \arctan \left (\frac {x}{\sqrt {1+\sqrt {2}}}\right )}{8 \sqrt {1+\sqrt {2}}}\) | \(90\) |
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Leaf count of result is larger than twice the leaf count of optimal. 245 vs. \(2 (85) = 170\).
Time = 0.28 (sec) , antiderivative size = 245, normalized size of antiderivative = 1.96 \[ \int \frac {1-x^4}{1-6 x^4+x^8} \, dx=\frac {1}{16} \, \sqrt {2} \sqrt {\sqrt {2} - 1} \log \left ({\left (\sqrt {2} + 1\right )} \sqrt {\sqrt {2} - 1} + x\right ) - \frac {1}{16} \, \sqrt {2} \sqrt {\sqrt {2} - 1} \log \left (-{\left (\sqrt {2} + 1\right )} \sqrt {\sqrt {2} - 1} + x\right ) + \frac {1}{16} \, \sqrt {2} \sqrt {\sqrt {2} + 1} \log \left (\sqrt {\sqrt {2} + 1} {\left (\sqrt {2} - 1\right )} + x\right ) - \frac {1}{16} \, \sqrt {2} \sqrt {\sqrt {2} + 1} \log \left (-\sqrt {\sqrt {2} + 1} {\left (\sqrt {2} - 1\right )} + x\right ) + \frac {1}{16} \, \sqrt {2} \sqrt {-\sqrt {2} + 1} \log \left ({\left (\sqrt {2} + 1\right )} \sqrt {-\sqrt {2} + 1} + x\right ) - \frac {1}{16} \, \sqrt {2} \sqrt {-\sqrt {2} + 1} \log \left (-{\left (\sqrt {2} + 1\right )} \sqrt {-\sqrt {2} + 1} + x\right ) + \frac {1}{16} \, \sqrt {2} \sqrt {-\sqrt {2} - 1} \log \left ({\left (\sqrt {2} - 1\right )} \sqrt {-\sqrt {2} - 1} + x\right ) - \frac {1}{16} \, \sqrt {2} \sqrt {-\sqrt {2} - 1} \log \left (-{\left (\sqrt {2} - 1\right )} \sqrt {-\sqrt {2} - 1} + x\right ) \]
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Time = 0.67 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.41 \[ \int \frac {1-x^4}{1-6 x^4+x^8} \, dx=- \operatorname {RootSum} {\left (16384 t^{4} - 256 t^{2} - 1, \left ( t \mapsto t \log {\left (65536 t^{5} - 28 t + x \right )} \right )\right )} - \operatorname {RootSum} {\left (16384 t^{4} + 256 t^{2} - 1, \left ( t \mapsto t \log {\left (65536 t^{5} - 28 t + x \right )} \right )\right )} \]
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\[ \int \frac {1-x^4}{1-6 x^4+x^8} \, dx=\int { -\frac {x^{4} - 1}{x^{8} - 6 \, x^{4} + 1} \,d x } \]
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none
Time = 0.35 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.08 \[ \int \frac {1-x^4}{1-6 x^4+x^8} \, dx=\frac {1}{8} \, \sqrt {2 \, \sqrt {2} - 2} \arctan \left (\frac {x}{\sqrt {\sqrt {2} + 1}}\right ) + \frac {1}{8} \, \sqrt {2 \, \sqrt {2} + 2} \arctan \left (\frac {x}{\sqrt {\sqrt {2} - 1}}\right ) + \frac {1}{16} \, \sqrt {2 \, \sqrt {2} - 2} \log \left ({\left | x + \sqrt {\sqrt {2} + 1} \right |}\right ) - \frac {1}{16} \, \sqrt {2 \, \sqrt {2} - 2} \log \left ({\left | x - \sqrt {\sqrt {2} + 1} \right |}\right ) + \frac {1}{16} \, \sqrt {2 \, \sqrt {2} + 2} \log \left ({\left | x + \sqrt {\sqrt {2} - 1} \right |}\right ) - \frac {1}{16} \, \sqrt {2 \, \sqrt {2} + 2} \log \left ({\left | x - \sqrt {\sqrt {2} - 1} \right |}\right ) \]
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Time = 8.51 (sec) , antiderivative size = 245, normalized size of antiderivative = 1.96 \[ \int \frac {1-x^4}{1-6 x^4+x^8} \, dx=-\frac {\sqrt {2}\,\mathrm {atan}\left (\frac {x\,\sqrt {1-\sqrt {2}}\,4352{}\mathrm {i}}{3072\,\sqrt {2}-4352}-\frac {\sqrt {2}\,x\,\sqrt {1-\sqrt {2}}\,3072{}\mathrm {i}}{3072\,\sqrt {2}-4352}\right )\,\sqrt {1-\sqrt {2}}\,1{}\mathrm {i}}{8}+\frac {\sqrt {2}\,\mathrm {atan}\left (\frac {x\,\sqrt {-\sqrt {2}-1}\,4352{}\mathrm {i}}{3072\,\sqrt {2}+4352}+\frac {\sqrt {2}\,x\,\sqrt {-\sqrt {2}-1}\,3072{}\mathrm {i}}{3072\,\sqrt {2}+4352}\right )\,\sqrt {-\sqrt {2}-1}\,1{}\mathrm {i}}{8}+\frac {\sqrt {2}\,\mathrm {atan}\left (\frac {x\,\sqrt {\sqrt {2}-1}\,4352{}\mathrm {i}}{3072\,\sqrt {2}-4352}-\frac {\sqrt {2}\,x\,\sqrt {\sqrt {2}-1}\,3072{}\mathrm {i}}{3072\,\sqrt {2}-4352}\right )\,\sqrt {\sqrt {2}-1}\,1{}\mathrm {i}}{8}-\frac {\sqrt {2}\,\mathrm {atan}\left (\frac {x\,\sqrt {\sqrt {2}+1}\,4352{}\mathrm {i}}{3072\,\sqrt {2}+4352}+\frac {\sqrt {2}\,x\,\sqrt {\sqrt {2}+1}\,3072{}\mathrm {i}}{3072\,\sqrt {2}+4352}\right )\,\sqrt {\sqrt {2}+1}\,1{}\mathrm {i}}{8} \]
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