Integrand size = 16, antiderivative size = 188 \[ \int \frac {x^2}{2-2 x^2+x^4} \, dx=-\frac {1}{2} \sqrt {\frac {1}{2} \left (1+\sqrt {2}\right )} \arctan \left (\frac {\sqrt {2 \left (1+\sqrt {2}\right )}-2 x}{\sqrt {2 \left (-1+\sqrt {2}\right )}}\right )+\frac {1}{2} \sqrt {\frac {1}{2} \left (1+\sqrt {2}\right )} \arctan \left (\frac {\sqrt {2 \left (1+\sqrt {2}\right )}+2 x}{\sqrt {2 \left (-1+\sqrt {2}\right )}}\right )+\frac {\log \left (\sqrt {2}-\sqrt {2 \left (1+\sqrt {2}\right )} x+x^2\right )}{4 \sqrt {2 \left (1+\sqrt {2}\right )}}-\frac {\log \left (\sqrt {2}+\sqrt {2 \left (1+\sqrt {2}\right )} x+x^2\right )}{4 \sqrt {2 \left (1+\sqrt {2}\right )}} \]
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Time = 0.12 (sec) , antiderivative size = 188, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {1141, 1175, 632, 210, 1178, 642} \[ \int \frac {x^2}{2-2 x^2+x^4} \, dx=-\frac {1}{2} \sqrt {\frac {1}{2} \left (1+\sqrt {2}\right )} \arctan \left (\frac {\sqrt {2 \left (1+\sqrt {2}\right )}-2 x}{\sqrt {2 \left (\sqrt {2}-1\right )}}\right )+\frac {1}{2} \sqrt {\frac {1}{2} \left (1+\sqrt {2}\right )} \arctan \left (\frac {2 x+\sqrt {2 \left (1+\sqrt {2}\right )}}{\sqrt {2 \left (\sqrt {2}-1\right )}}\right )+\frac {\log \left (x^2-\sqrt {2 \left (1+\sqrt {2}\right )} x+\sqrt {2}\right )}{4 \sqrt {2 \left (1+\sqrt {2}\right )}}-\frac {\log \left (x^2+\sqrt {2 \left (1+\sqrt {2}\right )} x+\sqrt {2}\right )}{4 \sqrt {2 \left (1+\sqrt {2}\right )}} \]
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Rule 210
Rule 632
Rule 642
Rule 1141
Rule 1175
Rule 1178
Rubi steps \begin{align*} \text {integral}& = -\left (\frac {1}{2} \int \frac {\sqrt {2}-x^2}{2-2 x^2+x^4} \, dx\right )+\frac {1}{2} \int \frac {\sqrt {2}+x^2}{2-2 x^2+x^4} \, dx \\ & = \frac {1}{4} \int \frac {1}{\sqrt {2}-\sqrt {2 \left (1+\sqrt {2}\right )} x+x^2} \, dx+\frac {1}{4} \int \frac {1}{\sqrt {2}+\sqrt {2 \left (1+\sqrt {2}\right )} x+x^2} \, dx+\frac {\int \frac {\sqrt {2 \left (1+\sqrt {2}\right )}+2 x}{-\sqrt {2}-\sqrt {2 \left (1+\sqrt {2}\right )} x-x^2} \, dx}{4 \sqrt {2 \left (1+\sqrt {2}\right )}}+\frac {\int \frac {\sqrt {2 \left (1+\sqrt {2}\right )}-2 x}{-\sqrt {2}+\sqrt {2 \left (1+\sqrt {2}\right )} x-x^2} \, dx}{4 \sqrt {2 \left (1+\sqrt {2}\right )}} \\ & = \frac {\log \left (\sqrt {2}-\sqrt {2 \left (1+\sqrt {2}\right )} x+x^2\right )}{4 \sqrt {2 \left (1+\sqrt {2}\right )}}-\frac {\log \left (\sqrt {2}+\sqrt {2 \left (1+\sqrt {2}\right )} x+x^2\right )}{4 \sqrt {2 \left (1+\sqrt {2}\right )}}-\frac {1}{2} \text {Subst}\left (\int \frac {1}{2 \left (1-\sqrt {2}\right )-x^2} \, dx,x,-\sqrt {2 \left (1+\sqrt {2}\right )}+2 x\right )-\frac {1}{2} \text {Subst}\left (\int \frac {1}{2 \left (1-\sqrt {2}\right )-x^2} \, dx,x,\sqrt {2 \left (1+\sqrt {2}\right )}+2 x\right ) \\ & = -\frac {\tan ^{-1}\left (\frac {\sqrt {2 \left (1+\sqrt {2}\right )}-2 x}{\sqrt {2 \left (-1+\sqrt {2}\right )}}\right )}{2 \sqrt {2 \left (-1+\sqrt {2}\right )}}+\frac {\tan ^{-1}\left (\frac {\sqrt {2 \left (1+\sqrt {2}\right )}+2 x}{\sqrt {2 \left (-1+\sqrt {2}\right )}}\right )}{2 \sqrt {2 \left (-1+\sqrt {2}\right )}}+\frac {\log \left (\sqrt {2}-\sqrt {2 \left (1+\sqrt {2}\right )} x+x^2\right )}{4 \sqrt {2 \left (1+\sqrt {2}\right )}}-\frac {\log \left (\sqrt {2}+\sqrt {2 \left (1+\sqrt {2}\right )} x+x^2\right )}{4 \sqrt {2 \left (1+\sqrt {2}\right )}} \\ \end{align*}
Result contains complex when optimal does not.
Time = 0.04 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.21 \[ \int \frac {x^2}{2-2 x^2+x^4} \, dx=-\frac {\arctan \left (\frac {x}{\sqrt {-1-i}}\right )}{(-1-i)^{3/2}}-\frac {\arctan \left (\frac {x}{\sqrt {-1+i}}\right )}{(-1+i)^{3/2}} \]
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Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.16 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.19
method | result | size |
risch | \(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{4}-2 \textit {\_Z}^{2}+2\right )}{\sum }\frac {\textit {\_R}^{2} \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{3}-\textit {\_R}}\right )}{4}\) | \(36\) |
default | \(-\frac {\sqrt {2+2 \sqrt {2}}\, \left (\sqrt {2}-1\right ) \left (-\frac {\ln \left (x^{2}+\sqrt {2}-x \sqrt {2+2 \sqrt {2}}\right )}{2}-\frac {\sqrt {2+2 \sqrt {2}}\, \arctan \left (\frac {2 x -\sqrt {2+2 \sqrt {2}}}{\sqrt {-2+2 \sqrt {2}}}\right )}{\sqrt {-2+2 \sqrt {2}}}\right )}{4}-\frac {\sqrt {2+2 \sqrt {2}}\, \left (\sqrt {2}-1\right ) \left (\frac {\ln \left (x^{2}+\sqrt {2}+x \sqrt {2+2 \sqrt {2}}\right )}{2}-\frac {\sqrt {2+2 \sqrt {2}}\, \arctan \left (\frac {2 x +\sqrt {2+2 \sqrt {2}}}{\sqrt {-2+2 \sqrt {2}}}\right )}{\sqrt {-2+2 \sqrt {2}}}\right )}{4}\) | \(169\) |
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Result contains complex when optimal does not.
Time = 0.26 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.28 \[ \int \frac {x^2}{2-2 x^2+x^4} \, dx=\frac {1}{4} \, \sqrt {i - 1} \log \left (x + i \, \sqrt {i - 1}\right ) - \frac {1}{4} \, \sqrt {i - 1} \log \left (x - i \, \sqrt {i - 1}\right ) - \frac {1}{4} \, \sqrt {-i - 1} \log \left (x + i \, \sqrt {-i - 1}\right ) + \frac {1}{4} \, \sqrt {-i - 1} \log \left (x - i \, \sqrt {-i - 1}\right ) \]
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Time = 0.28 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.13 \[ \int \frac {x^2}{2-2 x^2+x^4} \, dx=\operatorname {RootSum} {\left (128 t^{4} + 16 t^{2} + 1, \left ( t \mapsto t \log {\left (64 t^{3} + 4 t + x \right )} \right )\right )} \]
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\[ \int \frac {x^2}{2-2 x^2+x^4} \, dx=\int { \frac {x^{2}}{x^{4} - 2 \, x^{2} + 2} \,d x } \]
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none
Time = 0.68 (sec) , antiderivative size = 147, normalized size of antiderivative = 0.78 \[ \int \frac {x^2}{2-2 x^2+x^4} \, dx=\frac {1}{4} \, \sqrt {2 \, \sqrt {2} + 2} \arctan \left (\frac {2^{\frac {3}{4}} {\left (2 \, x + 2^{\frac {1}{4}} \sqrt {\sqrt {2} + 2}\right )}}{2 \, \sqrt {-\sqrt {2} + 2}}\right ) + \frac {1}{4} \, \sqrt {2 \, \sqrt {2} + 2} \arctan \left (\frac {2^{\frac {3}{4}} {\left (2 \, x - 2^{\frac {1}{4}} \sqrt {\sqrt {2} + 2}\right )}}{2 \, \sqrt {-\sqrt {2} + 2}}\right ) - \frac {1}{8} \, \sqrt {2 \, \sqrt {2} - 2} \log \left (x^{2} + 2^{\frac {1}{4}} x \sqrt {\sqrt {2} + 2} + \sqrt {2}\right ) + \frac {1}{8} \, \sqrt {2 \, \sqrt {2} - 2} \log \left (x^{2} - 2^{\frac {1}{4}} x \sqrt {\sqrt {2} + 2} + \sqrt {2}\right ) \]
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Time = 13.10 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.54 \[ \int \frac {x^2}{2-2 x^2+x^4} \, dx=\mathrm {atanh}\left (32\,x\,{\left (\sqrt {-\frac {\sqrt {2}}{32}-\frac {1}{32}}+\sqrt {\frac {\sqrt {2}}{32}-\frac {1}{32}}\right )}^3\right )\,\left (2\,\sqrt {-\frac {\sqrt {2}}{32}-\frac {1}{32}}+2\,\sqrt {\frac {\sqrt {2}}{32}-\frac {1}{32}}\right )+\mathrm {atanh}\left (32\,x\,{\left (\sqrt {-\frac {\sqrt {2}}{32}-\frac {1}{32}}-\sqrt {\frac {\sqrt {2}}{32}-\frac {1}{32}}\right )}^3\right )\,\left (2\,\sqrt {-\frac {\sqrt {2}}{32}-\frac {1}{32}}-2\,\sqrt {\frac {\sqrt {2}}{32}-\frac {1}{32}}\right ) \]
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