Integrand size = 16, antiderivative size = 113 \[ \int \frac {1}{x^8 \left (1+2 x^4+x^8\right )} \, dx=-\frac {11}{28 x^7}+\frac {11}{12 x^3}+\frac {1}{4 x^7 \left (1+x^4\right )}-\frac {11 \arctan \left (1-\sqrt {2} x\right )}{8 \sqrt {2}}+\frac {11 \arctan \left (1+\sqrt {2} x\right )}{8 \sqrt {2}}-\frac {11 \log \left (1-\sqrt {2} x+x^2\right )}{16 \sqrt {2}}+\frac {11 \log \left (1+\sqrt {2} x+x^2\right )}{16 \sqrt {2}} \]
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Time = 0.04 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.562, Rules used = {28, 296, 331, 217, 1179, 642, 1176, 631, 210} \[ \int \frac {1}{x^8 \left (1+2 x^4+x^8\right )} \, dx=-\frac {11 \arctan \left (1-\sqrt {2} x\right )}{8 \sqrt {2}}+\frac {11 \arctan \left (\sqrt {2} x+1\right )}{8 \sqrt {2}}-\frac {11}{28 x^7}+\frac {11}{12 x^3}-\frac {11 \log \left (x^2-\sqrt {2} x+1\right )}{16 \sqrt {2}}+\frac {11 \log \left (x^2+\sqrt {2} x+1\right )}{16 \sqrt {2}}+\frac {1}{4 x^7 \left (x^4+1\right )} \]
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Rule 28
Rule 210
Rule 217
Rule 296
Rule 331
Rule 631
Rule 642
Rule 1176
Rule 1179
Rubi steps \begin{align*} \text {integral}& = \int \frac {1}{x^8 \left (1+x^4\right )^2} \, dx \\ & = \frac {1}{4 x^7 \left (1+x^4\right )}+\frac {11}{4} \int \frac {1}{x^8 \left (1+x^4\right )} \, dx \\ & = -\frac {11}{28 x^7}+\frac {1}{4 x^7 \left (1+x^4\right )}-\frac {11}{4} \int \frac {1}{x^4 \left (1+x^4\right )} \, dx \\ & = -\frac {11}{28 x^7}+\frac {11}{12 x^3}+\frac {1}{4 x^7 \left (1+x^4\right )}+\frac {11}{4} \int \frac {1}{1+x^4} \, dx \\ & = -\frac {11}{28 x^7}+\frac {11}{12 x^3}+\frac {1}{4 x^7 \left (1+x^4\right )}+\frac {11}{8} \int \frac {1-x^2}{1+x^4} \, dx+\frac {11}{8} \int \frac {1+x^2}{1+x^4} \, dx \\ & = -\frac {11}{28 x^7}+\frac {11}{12 x^3}+\frac {1}{4 x^7 \left (1+x^4\right )}+\frac {11}{16} \int \frac {1}{1-\sqrt {2} x+x^2} \, dx+\frac {11}{16} \int \frac {1}{1+\sqrt {2} x+x^2} \, dx-\frac {11 \int \frac {\sqrt {2}+2 x}{-1-\sqrt {2} x-x^2} \, dx}{16 \sqrt {2}}-\frac {11 \int \frac {\sqrt {2}-2 x}{-1+\sqrt {2} x-x^2} \, dx}{16 \sqrt {2}} \\ & = -\frac {11}{28 x^7}+\frac {11}{12 x^3}+\frac {1}{4 x^7 \left (1+x^4\right )}-\frac {11 \log \left (1-\sqrt {2} x+x^2\right )}{16 \sqrt {2}}+\frac {11 \log \left (1+\sqrt {2} x+x^2\right )}{16 \sqrt {2}}+\frac {11 \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\sqrt {2} x\right )}{8 \sqrt {2}}-\frac {11 \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\sqrt {2} x\right )}{8 \sqrt {2}} \\ & = -\frac {11}{28 x^7}+\frac {11}{12 x^3}+\frac {1}{4 x^7 \left (1+x^4\right )}-\frac {11 \tan ^{-1}\left (1-\sqrt {2} x\right )}{8 \sqrt {2}}+\frac {11 \tan ^{-1}\left (1+\sqrt {2} x\right )}{8 \sqrt {2}}-\frac {11 \log \left (1-\sqrt {2} x+x^2\right )}{16 \sqrt {2}}+\frac {11 \log \left (1+\sqrt {2} x+x^2\right )}{16 \sqrt {2}} \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.89 \[ \int \frac {1}{x^8 \left (1+2 x^4+x^8\right )} \, dx=\frac {1}{672} \left (-\frac {96}{x^7}+\frac {448}{x^3}+\frac {168 x}{1+x^4}-462 \sqrt {2} \arctan \left (1-\sqrt {2} x\right )+462 \sqrt {2} \arctan \left (1+\sqrt {2} x\right )-231 \sqrt {2} \log \left (1-\sqrt {2} x+x^2\right )+231 \sqrt {2} \log \left (1+\sqrt {2} x+x^2\right )\right ) \]
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Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.09 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.37
method | result | size |
risch | \(\frac {\frac {11}{12} x^{8}+\frac {11}{21} x^{4}-\frac {1}{7}}{x^{7} \left (x^{4}+1\right )}+\frac {11 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )}{\sum }\textit {\_R} \ln \left (x +\textit {\_R} \right )\right )}{16}\) | \(42\) |
default | \(-\frac {1}{7 x^{7}}+\frac {2}{3 x^{3}}+\frac {x}{4 x^{4}+4}+\frac {11 \sqrt {2}\, \left (\ln \left (\frac {1+x^{2}+x \sqrt {2}}{1+x^{2}-x \sqrt {2}}\right )+2 \arctan \left (x \sqrt {2}+1\right )+2 \arctan \left (x \sqrt {2}-1\right )\right )}{32}\) | \(73\) |
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Result contains complex when optimal does not.
Time = 0.27 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.12 \[ \int \frac {1}{x^8 \left (1+2 x^4+x^8\right )} \, dx=\frac {616 \, x^{8} + 352 \, x^{4} - 231 \, \sqrt {2} {\left (-\left (i + 1\right ) \, x^{11} - \left (i + 1\right ) \, x^{7}\right )} \log \left (2 \, x + \left (i + 1\right ) \, \sqrt {2}\right ) - 231 \, \sqrt {2} {\left (\left (i - 1\right ) \, x^{11} + \left (i - 1\right ) \, x^{7}\right )} \log \left (2 \, x - \left (i - 1\right ) \, \sqrt {2}\right ) - 231 \, \sqrt {2} {\left (-\left (i - 1\right ) \, x^{11} - \left (i - 1\right ) \, x^{7}\right )} \log \left (2 \, x + \left (i - 1\right ) \, \sqrt {2}\right ) - 231 \, \sqrt {2} {\left (\left (i + 1\right ) \, x^{11} + \left (i + 1\right ) \, x^{7}\right )} \log \left (2 \, x - \left (i + 1\right ) \, \sqrt {2}\right ) - 96}{672 \, {\left (x^{11} + x^{7}\right )}} \]
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Time = 0.10 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.90 \[ \int \frac {1}{x^8 \left (1+2 x^4+x^8\right )} \, dx=- \frac {11 \sqrt {2} \log {\left (x^{2} - \sqrt {2} x + 1 \right )}}{32} + \frac {11 \sqrt {2} \log {\left (x^{2} + \sqrt {2} x + 1 \right )}}{32} + \frac {11 \sqrt {2} \operatorname {atan}{\left (\sqrt {2} x - 1 \right )}}{16} + \frac {11 \sqrt {2} \operatorname {atan}{\left (\sqrt {2} x + 1 \right )}}{16} + \frac {77 x^{8} + 44 x^{4} - 12}{84 x^{11} + 84 x^{7}} \]
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Time = 0.27 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.84 \[ \int \frac {1}{x^8 \left (1+2 x^4+x^8\right )} \, dx=\frac {11}{16} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (2 \, x + \sqrt {2}\right )}\right ) + \frac {11}{16} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (2 \, x - \sqrt {2}\right )}\right ) + \frac {11}{32} \, \sqrt {2} \log \left (x^{2} + \sqrt {2} x + 1\right ) - \frac {11}{32} \, \sqrt {2} \log \left (x^{2} - \sqrt {2} x + 1\right ) + \frac {77 \, x^{8} + 44 \, x^{4} - 12}{84 \, {\left (x^{11} + x^{7}\right )}} \]
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Time = 0.33 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.83 \[ \int \frac {1}{x^8 \left (1+2 x^4+x^8\right )} \, dx=\frac {11}{16} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (2 \, x + \sqrt {2}\right )}\right ) + \frac {11}{16} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (2 \, x - \sqrt {2}\right )}\right ) + \frac {11}{32} \, \sqrt {2} \log \left (x^{2} + \sqrt {2} x + 1\right ) - \frac {11}{32} \, \sqrt {2} \log \left (x^{2} - \sqrt {2} x + 1\right ) + \frac {x}{4 \, {\left (x^{4} + 1\right )}} + \frac {14 \, x^{4} - 3}{21 \, x^{7}} \]
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Time = 0.06 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.49 \[ \int \frac {1}{x^8 \left (1+2 x^4+x^8\right )} \, dx=\frac {\frac {11\,x^8}{12}+\frac {11\,x^4}{21}-\frac {1}{7}}{x^{11}+x^7}+\sqrt {2}\,\mathrm {atan}\left (\sqrt {2}\,x\,\left (\frac {1}{2}-\frac {1}{2}{}\mathrm {i}\right )\right )\,\left (\frac {11}{16}+\frac {11}{16}{}\mathrm {i}\right )+\sqrt {2}\,\mathrm {atan}\left (\sqrt {2}\,x\,\left (\frac {1}{2}+\frac {1}{2}{}\mathrm {i}\right )\right )\,\left (\frac {11}{16}-\frac {11}{16}{}\mathrm {i}\right ) \]
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