Integrand size = 16, antiderivative size = 104 \[ \int \frac {x^8}{1+2 x^4+x^8} \, dx=\frac {5 x}{4}-\frac {x^5}{4 \left (1+x^4\right )}+\frac {5 \arctan \left (1-\sqrt {2} x\right )}{8 \sqrt {2}}-\frac {5 \arctan \left (1+\sqrt {2} x\right )}{8 \sqrt {2}}+\frac {5 \log \left (1-\sqrt {2} x+x^2\right )}{16 \sqrt {2}}-\frac {5 \log \left (1+\sqrt {2} x+x^2\right )}{16 \sqrt {2}} \]
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Time = 0.04 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.562, Rules used = {28, 294, 327, 217, 1179, 642, 1176, 631, 210} \[ \int \frac {x^8}{1+2 x^4+x^8} \, dx=\frac {5 \arctan \left (1-\sqrt {2} x\right )}{8 \sqrt {2}}-\frac {5 \arctan \left (\sqrt {2} x+1\right )}{8 \sqrt {2}}+\frac {5 \log \left (x^2-\sqrt {2} x+1\right )}{16 \sqrt {2}}-\frac {5 \log \left (x^2+\sqrt {2} x+1\right )}{16 \sqrt {2}}-\frac {x^5}{4 \left (x^4+1\right )}+\frac {5 x}{4} \]
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Rule 28
Rule 210
Rule 217
Rule 294
Rule 327
Rule 631
Rule 642
Rule 1176
Rule 1179
Rubi steps \begin{align*} \text {integral}& = \int \frac {x^8}{\left (1+x^4\right )^2} \, dx \\ & = -\frac {x^5}{4 \left (1+x^4\right )}+\frac {5}{4} \int \frac {x^4}{1+x^4} \, dx \\ & = \frac {5 x}{4}-\frac {x^5}{4 \left (1+x^4\right )}-\frac {5}{4} \int \frac {1}{1+x^4} \, dx \\ & = \frac {5 x}{4}-\frac {x^5}{4 \left (1+x^4\right )}-\frac {5}{8} \int \frac {1-x^2}{1+x^4} \, dx-\frac {5}{8} \int \frac {1+x^2}{1+x^4} \, dx \\ & = \frac {5 x}{4}-\frac {x^5}{4 \left (1+x^4\right )}-\frac {5}{16} \int \frac {1}{1-\sqrt {2} x+x^2} \, dx-\frac {5}{16} \int \frac {1}{1+\sqrt {2} x+x^2} \, dx+\frac {5 \int \frac {\sqrt {2}+2 x}{-1-\sqrt {2} x-x^2} \, dx}{16 \sqrt {2}}+\frac {5 \int \frac {\sqrt {2}-2 x}{-1+\sqrt {2} x-x^2} \, dx}{16 \sqrt {2}} \\ & = \frac {5 x}{4}-\frac {x^5}{4 \left (1+x^4\right )}+\frac {5 \log \left (1-\sqrt {2} x+x^2\right )}{16 \sqrt {2}}-\frac {5 \log \left (1+\sqrt {2} x+x^2\right )}{16 \sqrt {2}}-\frac {5 \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\sqrt {2} x\right )}{8 \sqrt {2}}+\frac {5 \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\sqrt {2} x\right )}{8 \sqrt {2}} \\ & = \frac {5 x}{4}-\frac {x^5}{4 \left (1+x^4\right )}+\frac {5 \tan ^{-1}\left (1-\sqrt {2} x\right )}{8 \sqrt {2}}-\frac {5 \tan ^{-1}\left (1+\sqrt {2} x\right )}{8 \sqrt {2}}+\frac {5 \log \left (1-\sqrt {2} x+x^2\right )}{16 \sqrt {2}}-\frac {5 \log \left (1+\sqrt {2} x+x^2\right )}{16 \sqrt {2}} \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.90 \[ \int \frac {x^8}{1+2 x^4+x^8} \, dx=\frac {1}{32} \left (32 x+\frac {8 x}{1+x^4}+10 \sqrt {2} \arctan \left (1-\sqrt {2} x\right )-10 \sqrt {2} \arctan \left (1+\sqrt {2} x\right )+5 \sqrt {2} \log \left (1-\sqrt {2} x+x^2\right )-5 \sqrt {2} \log \left (1+\sqrt {2} x+x^2\right )\right ) \]
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Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.06 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.33
method | result | size |
risch | \(x +\frac {x}{4 x^{4}+4}-\frac {5 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )}{\sum }\frac {\ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{3}}\right )}{16}\) | \(34\) |
default | \(x +\frac {x}{4 x^{4}+4}-\frac {5 \sqrt {2}\, \left (\ln \left (\frac {1+x^{2}+x \sqrt {2}}{1+x^{2}-x \sqrt {2}}\right )+2 \arctan \left (x \sqrt {2}+1\right )+2 \arctan \left (x \sqrt {2}-1\right )\right )}{32}\) | \(64\) |
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Result contains complex when optimal does not.
Time = 0.25 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.02 \[ \int \frac {x^8}{1+2 x^4+x^8} \, dx=\frac {32 \, x^{5} - 5 \, \sqrt {2} {\left (\left (i + 1\right ) \, x^{4} + i + 1\right )} \log \left (2 \, x + \left (i + 1\right ) \, \sqrt {2}\right ) - 5 \, \sqrt {2} {\left (-\left (i - 1\right ) \, x^{4} - i + 1\right )} \log \left (2 \, x - \left (i - 1\right ) \, \sqrt {2}\right ) - 5 \, \sqrt {2} {\left (\left (i - 1\right ) \, x^{4} + i - 1\right )} \log \left (2 \, x + \left (i - 1\right ) \, \sqrt {2}\right ) - 5 \, \sqrt {2} {\left (-\left (i + 1\right ) \, x^{4} - i - 1\right )} \log \left (2 \, x - \left (i + 1\right ) \, \sqrt {2}\right ) + 40 \, x}{32 \, {\left (x^{4} + 1\right )}} \]
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Time = 0.07 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.87 \[ \int \frac {x^8}{1+2 x^4+x^8} \, dx=x + \frac {x}{4 x^{4} + 4} + \frac {5 \sqrt {2} \log {\left (x^{2} - \sqrt {2} x + 1 \right )}}{32} - \frac {5 \sqrt {2} \log {\left (x^{2} + \sqrt {2} x + 1 \right )}}{32} - \frac {5 \sqrt {2} \operatorname {atan}{\left (\sqrt {2} x - 1 \right )}}{16} - \frac {5 \sqrt {2} \operatorname {atan}{\left (\sqrt {2} x + 1 \right )}}{16} \]
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Time = 0.27 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.80 \[ \int \frac {x^8}{1+2 x^4+x^8} \, dx=-\frac {5}{16} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (2 \, x + \sqrt {2}\right )}\right ) - \frac {5}{16} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (2 \, x - \sqrt {2}\right )}\right ) - \frac {5}{32} \, \sqrt {2} \log \left (x^{2} + \sqrt {2} x + 1\right ) + \frac {5}{32} \, \sqrt {2} \log \left (x^{2} - \sqrt {2} x + 1\right ) + x + \frac {x}{4 \, {\left (x^{4} + 1\right )}} \]
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Time = 0.31 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.80 \[ \int \frac {x^8}{1+2 x^4+x^8} \, dx=-\frac {5}{16} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (2 \, x + \sqrt {2}\right )}\right ) - \frac {5}{16} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (2 \, x - \sqrt {2}\right )}\right ) - \frac {5}{32} \, \sqrt {2} \log \left (x^{2} + \sqrt {2} x + 1\right ) + \frac {5}{32} \, \sqrt {2} \log \left (x^{2} - \sqrt {2} x + 1\right ) + x + \frac {x}{4 \, {\left (x^{4} + 1\right )}} \]
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Time = 8.29 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.43 \[ \int \frac {x^8}{1+2 x^4+x^8} \, dx=x+\frac {x}{4\,\left (x^4+1\right )}+\sqrt {2}\,\mathrm {atan}\left (\sqrt {2}\,x\,\left (\frac {1}{2}-\frac {1}{2}{}\mathrm {i}\right )\right )\,\left (-\frac {5}{16}-\frac {5}{16}{}\mathrm {i}\right )+\sqrt {2}\,\mathrm {atan}\left (\sqrt {2}\,x\,\left (\frac {1}{2}+\frac {1}{2}{}\mathrm {i}\right )\right )\,\left (-\frac {5}{16}+\frac {5}{16}{}\mathrm {i}\right ) \]
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