Integrand size = 11, antiderivative size = 354 \[ \int \frac {1}{x^4 \left (1+x^8\right )} \, dx=-\frac {1}{3 x^3}-\frac {\arctan \left (\frac {\sqrt {2-\sqrt {2}}-2 x}{\sqrt {2+\sqrt {2}}}\right )}{4 \sqrt {2 \left (2+\sqrt {2}\right )}}+\frac {\arctan \left (\frac {\sqrt {2+\sqrt {2}}-2 x}{\sqrt {2-\sqrt {2}}}\right )}{4 \sqrt {2 \left (2-\sqrt {2}\right )}}+\frac {\arctan \left (\frac {\sqrt {2-\sqrt {2}}+2 x}{\sqrt {2+\sqrt {2}}}\right )}{4 \sqrt {2 \left (2+\sqrt {2}\right )}}-\frac {\arctan \left (\frac {\sqrt {2+\sqrt {2}}+2 x}{\sqrt {2-\sqrt {2}}}\right )}{4 \sqrt {2 \left (2-\sqrt {2}\right )}}+\frac {\log \left (1-\sqrt {2-\sqrt {2}} x+x^2\right )}{8 \sqrt {2 \left (2-\sqrt {2}\right )}}-\frac {\log \left (1+\sqrt {2-\sqrt {2}} x+x^2\right )}{8 \sqrt {2 \left (2-\sqrt {2}\right )}}-\frac {\log \left (1-\sqrt {2+\sqrt {2}} x+x^2\right )}{8 \sqrt {2 \left (2+\sqrt {2}\right )}}+\frac {\log \left (1+\sqrt {2+\sqrt {2}} x+x^2\right )}{8 \sqrt {2 \left (2+\sqrt {2}\right )}} \]
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Time = 0.14 (sec) , antiderivative size = 354, normalized size of antiderivative = 1.00, number of steps used = 20, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.727, Rules used = {331, 305, 1141, 1175, 632, 210, 1178, 642} \[ \int \frac {1}{x^4 \left (1+x^8\right )} \, dx=-\frac {\arctan \left (\frac {\sqrt {2-\sqrt {2}}-2 x}{\sqrt {2+\sqrt {2}}}\right )}{4 \sqrt {2 \left (2+\sqrt {2}\right )}}+\frac {\arctan \left (\frac {\sqrt {2+\sqrt {2}}-2 x}{\sqrt {2-\sqrt {2}}}\right )}{4 \sqrt {2 \left (2-\sqrt {2}\right )}}+\frac {\arctan \left (\frac {2 x+\sqrt {2-\sqrt {2}}}{\sqrt {2+\sqrt {2}}}\right )}{4 \sqrt {2 \left (2+\sqrt {2}\right )}}-\frac {\arctan \left (\frac {2 x+\sqrt {2+\sqrt {2}}}{\sqrt {2-\sqrt {2}}}\right )}{4 \sqrt {2 \left (2-\sqrt {2}\right )}}-\frac {1}{3 x^3}+\frac {\log \left (x^2-\sqrt {2-\sqrt {2}} x+1\right )}{8 \sqrt {2 \left (2-\sqrt {2}\right )}}-\frac {\log \left (x^2+\sqrt {2-\sqrt {2}} x+1\right )}{8 \sqrt {2 \left (2-\sqrt {2}\right )}}-\frac {\log \left (x^2-\sqrt {2+\sqrt {2}} x+1\right )}{8 \sqrt {2 \left (2+\sqrt {2}\right )}}+\frac {\log \left (x^2+\sqrt {2+\sqrt {2}} x+1\right )}{8 \sqrt {2 \left (2+\sqrt {2}\right )}} \]
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Rule 210
Rule 305
Rule 331
Rule 632
Rule 642
Rule 1141
Rule 1175
Rule 1178
Rubi steps \begin{align*} \text {integral}& = -\frac {1}{3 x^3}-\int \frac {x^4}{1+x^8} \, dx \\ & = -\frac {1}{3 x^3}-\frac {\int \frac {x^2}{1-\sqrt {2} x^2+x^4} \, dx}{2 \sqrt {2}}+\frac {\int \frac {x^2}{1+\sqrt {2} x^2+x^4} \, dx}{2 \sqrt {2}} \\ & = -\frac {1}{3 x^3}+\frac {\int \frac {1-x^2}{1-\sqrt {2} x^2+x^4} \, dx}{4 \sqrt {2}}-\frac {\int \frac {1+x^2}{1-\sqrt {2} x^2+x^4} \, dx}{4 \sqrt {2}}-\frac {\int \frac {1-x^2}{1+\sqrt {2} x^2+x^4} \, dx}{4 \sqrt {2}}+\frac {\int \frac {1+x^2}{1+\sqrt {2} x^2+x^4} \, dx}{4 \sqrt {2}} \\ & = -\frac {1}{3 x^3}+\frac {\int \frac {1}{1-\sqrt {2-\sqrt {2}} x+x^2} \, dx}{8 \sqrt {2}}+\frac {\int \frac {1}{1+\sqrt {2-\sqrt {2}} x+x^2} \, dx}{8 \sqrt {2}}-\frac {\int \frac {1}{1-\sqrt {2+\sqrt {2}} x+x^2} \, dx}{8 \sqrt {2}}-\frac {\int \frac {1}{1+\sqrt {2+\sqrt {2}} x+x^2} \, dx}{8 \sqrt {2}}+\frac {\int \frac {\sqrt {2-\sqrt {2}}+2 x}{-1-\sqrt {2-\sqrt {2}} x-x^2} \, dx}{8 \sqrt {2 \left (2-\sqrt {2}\right )}}+\frac {\int \frac {\sqrt {2-\sqrt {2}}-2 x}{-1+\sqrt {2-\sqrt {2}} x-x^2} \, dx}{8 \sqrt {2 \left (2-\sqrt {2}\right )}}-\frac {\int \frac {\sqrt {2+\sqrt {2}}+2 x}{-1-\sqrt {2+\sqrt {2}} x-x^2} \, dx}{8 \sqrt {2 \left (2+\sqrt {2}\right )}}-\frac {\int \frac {\sqrt {2+\sqrt {2}}-2 x}{-1+\sqrt {2+\sqrt {2}} x-x^2} \, dx}{8 \sqrt {2 \left (2+\sqrt {2}\right )}} \\ & = -\frac {1}{3 x^3}+\frac {\log \left (1-\sqrt {2-\sqrt {2}} x+x^2\right )}{8 \sqrt {2 \left (2-\sqrt {2}\right )}}-\frac {\log \left (1+\sqrt {2-\sqrt {2}} x+x^2\right )}{8 \sqrt {2 \left (2-\sqrt {2}\right )}}-\frac {\log \left (1-\sqrt {2+\sqrt {2}} x+x^2\right )}{8 \sqrt {2 \left (2+\sqrt {2}\right )}}+\frac {\log \left (1+\sqrt {2+\sqrt {2}} x+x^2\right )}{8 \sqrt {2 \left (2+\sqrt {2}\right )}}-\frac {\text {Subst}\left (\int \frac {1}{-2-\sqrt {2}-x^2} \, dx,x,-\sqrt {2-\sqrt {2}}+2 x\right )}{4 \sqrt {2}}-\frac {\text {Subst}\left (\int \frac {1}{-2-\sqrt {2}-x^2} \, dx,x,\sqrt {2-\sqrt {2}}+2 x\right )}{4 \sqrt {2}}+\frac {\text {Subst}\left (\int \frac {1}{-2+\sqrt {2}-x^2} \, dx,x,-\sqrt {2+\sqrt {2}}+2 x\right )}{4 \sqrt {2}}+\frac {\text {Subst}\left (\int \frac {1}{-2+\sqrt {2}-x^2} \, dx,x,\sqrt {2+\sqrt {2}}+2 x\right )}{4 \sqrt {2}} \\ & = -\frac {1}{3 x^3}-\frac {\tan ^{-1}\left (\frac {\sqrt {2-\sqrt {2}}-2 x}{\sqrt {2+\sqrt {2}}}\right )}{4 \sqrt {2 \left (2+\sqrt {2}\right )}}+\frac {\tan ^{-1}\left (\frac {\sqrt {2+\sqrt {2}}-2 x}{\sqrt {2-\sqrt {2}}}\right )}{4 \sqrt {2 \left (2-\sqrt {2}\right )}}+\frac {\tan ^{-1}\left (\frac {\sqrt {2-\sqrt {2}}+2 x}{\sqrt {2+\sqrt {2}}}\right )}{4 \sqrt {2 \left (2+\sqrt {2}\right )}}-\frac {\tan ^{-1}\left (\frac {\sqrt {2+\sqrt {2}}+2 x}{\sqrt {2-\sqrt {2}}}\right )}{4 \sqrt {2 \left (2-\sqrt {2}\right )}}+\frac {\log \left (1-\sqrt {2-\sqrt {2}} x+x^2\right )}{8 \sqrt {2 \left (2-\sqrt {2}\right )}}-\frac {\log \left (1+\sqrt {2-\sqrt {2}} x+x^2\right )}{8 \sqrt {2 \left (2-\sqrt {2}\right )}}-\frac {\log \left (1-\sqrt {2+\sqrt {2}} x+x^2\right )}{8 \sqrt {2 \left (2+\sqrt {2}\right )}}+\frac {\log \left (1+\sqrt {2+\sqrt {2}} x+x^2\right )}{8 \sqrt {2 \left (2+\sqrt {2}\right )}} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 216, normalized size of antiderivative = 0.61 \[ \int \frac {1}{x^4 \left (1+x^8\right )} \, dx=-\frac {1}{3 x^3}-\frac {1}{4} \arctan \left (\left (x-\cos \left (\frac {\pi }{8}\right )\right ) \csc \left (\frac {\pi }{8}\right )\right ) \cos \left (\frac {\pi }{8}\right )-\frac {1}{4} \arctan \left (\left (x+\cos \left (\frac {\pi }{8}\right )\right ) \csc \left (\frac {\pi }{8}\right )\right ) \cos \left (\frac {\pi }{8}\right )+\frac {1}{8} \cos \left (\frac {\pi }{8}\right ) \log \left (1+x^2-2 x \sin \left (\frac {\pi }{8}\right )\right )-\frac {1}{8} \cos \left (\frac {\pi }{8}\right ) \log \left (1+x^2+2 x \sin \left (\frac {\pi }{8}\right )\right )+\frac {1}{4} \arctan \left (\sec \left (\frac {\pi }{8}\right ) \left (x-\sin \left (\frac {\pi }{8}\right )\right )\right ) \sin \left (\frac {\pi }{8}\right )+\frac {1}{4} \arctan \left (\sec \left (\frac {\pi }{8}\right ) \left (x+\sin \left (\frac {\pi }{8}\right )\right )\right ) \sin \left (\frac {\pi }{8}\right )-\frac {1}{8} \log \left (1+x^2-2 x \cos \left (\frac {\pi }{8}\right )\right ) \sin \left (\frac {\pi }{8}\right )+\frac {1}{8} \log \left (1+x^2+2 x \cos \left (\frac {\pi }{8}\right )\right ) \sin \left (\frac {\pi }{8}\right ) \]
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Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 3.41 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.07
method | result | size |
risch | \(-\frac {1}{3 x^{3}}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{8}+1\right )}{\sum }\textit {\_R} \ln \left (\textit {\_R}^{5}+x \right )\right )}{8}\) | \(26\) |
default | \(-\frac {1}{3 x^{3}}-\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{8}+1\right )}{\sum }\frac {\ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{3}}\right )}{8}\) | \(28\) |
meijerg | \(-\frac {1}{3 x^{3}}-\frac {x^{5} \left (\frac {\cos \left (\frac {3 \pi }{8}\right ) \ln \left (1-2 \cos \left (\frac {\pi }{8}\right ) \left (x^{8}\right )^{\frac {1}{8}}+\left (x^{8}\right )^{\frac {1}{4}}\right )}{\left (x^{8}\right )^{\frac {5}{8}}}+\frac {2 \sin \left (\frac {3 \pi }{8}\right ) \arctan \left (\frac {\sin \left (\frac {\pi }{8}\right ) \left (x^{8}\right )^{\frac {1}{8}}}{1-\cos \left (\frac {\pi }{8}\right ) \left (x^{8}\right )^{\frac {1}{8}}}\right )}{\left (x^{8}\right )^{\frac {5}{8}}}-\frac {\cos \left (\frac {\pi }{8}\right ) \ln \left (1-2 \cos \left (\frac {3 \pi }{8}\right ) \left (x^{8}\right )^{\frac {1}{8}}+\left (x^{8}\right )^{\frac {1}{4}}\right )}{\left (x^{8}\right )^{\frac {5}{8}}}-\frac {2 \sin \left (\frac {\pi }{8}\right ) \arctan \left (\frac {\sin \left (\frac {3 \pi }{8}\right ) \left (x^{8}\right )^{\frac {1}{8}}}{1-\cos \left (\frac {3 \pi }{8}\right ) \left (x^{8}\right )^{\frac {1}{8}}}\right )}{\left (x^{8}\right )^{\frac {5}{8}}}+\frac {\cos \left (\frac {\pi }{8}\right ) \ln \left (1+2 \cos \left (\frac {3 \pi }{8}\right ) \left (x^{8}\right )^{\frac {1}{8}}+\left (x^{8}\right )^{\frac {1}{4}}\right )}{\left (x^{8}\right )^{\frac {5}{8}}}-\frac {2 \sin \left (\frac {\pi }{8}\right ) \arctan \left (\frac {\sin \left (\frac {3 \pi }{8}\right ) \left (x^{8}\right )^{\frac {1}{8}}}{1+\cos \left (\frac {3 \pi }{8}\right ) \left (x^{8}\right )^{\frac {1}{8}}}\right )}{\left (x^{8}\right )^{\frac {5}{8}}}-\frac {\cos \left (\frac {3 \pi }{8}\right ) \ln \left (1+2 \cos \left (\frac {\pi }{8}\right ) \left (x^{8}\right )^{\frac {1}{8}}+\left (x^{8}\right )^{\frac {1}{4}}\right )}{\left (x^{8}\right )^{\frac {5}{8}}}+\frac {2 \sin \left (\frac {3 \pi }{8}\right ) \arctan \left (\frac {\sin \left (\frac {\pi }{8}\right ) \left (x^{8}\right )^{\frac {1}{8}}}{1+\cos \left (\frac {\pi }{8}\right ) \left (x^{8}\right )^{\frac {1}{8}}}\right )}{\left (x^{8}\right )^{\frac {5}{8}}}\right )}{8}\) | \(277\) |
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Result contains complex when optimal does not.
Time = 0.31 (sec) , antiderivative size = 165, normalized size of antiderivative = 0.47 \[ \int \frac {1}{x^4 \left (1+x^8\right )} \, dx=\frac {-\left (3 i + 3\right ) \, \sqrt {2} \left (-1\right )^{\frac {1}{8}} x^{3} \log \left (\left (i + 1\right ) \, \sqrt {2} \left (-1\right )^{\frac {5}{8}} + 2 \, x\right ) + \left (3 i - 3\right ) \, \sqrt {2} \left (-1\right )^{\frac {1}{8}} x^{3} \log \left (-\left (i - 1\right ) \, \sqrt {2} \left (-1\right )^{\frac {5}{8}} + 2 \, x\right ) - \left (3 i - 3\right ) \, \sqrt {2} \left (-1\right )^{\frac {1}{8}} x^{3} \log \left (\left (i - 1\right ) \, \sqrt {2} \left (-1\right )^{\frac {5}{8}} + 2 \, x\right ) + \left (3 i + 3\right ) \, \sqrt {2} \left (-1\right )^{\frac {1}{8}} x^{3} \log \left (-\left (i + 1\right ) \, \sqrt {2} \left (-1\right )^{\frac {5}{8}} + 2 \, x\right ) + 6 \, \left (-1\right )^{\frac {1}{8}} x^{3} \log \left (x + \left (-1\right )^{\frac {5}{8}}\right ) + 6 i \, \left (-1\right )^{\frac {1}{8}} x^{3} \log \left (x + i \, \left (-1\right )^{\frac {5}{8}}\right ) - 6 i \, \left (-1\right )^{\frac {1}{8}} x^{3} \log \left (x - i \, \left (-1\right )^{\frac {5}{8}}\right ) - 6 \, \left (-1\right )^{\frac {1}{8}} x^{3} \log \left (x - \left (-1\right )^{\frac {5}{8}}\right ) - 16}{48 \, x^{3}} \]
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Time = 1.21 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.06 \[ \int \frac {1}{x^4 \left (1+x^8\right )} \, dx=\operatorname {RootSum} {\left (16777216 t^{8} + 1, \left ( t \mapsto t \log {\left (32768 t^{5} + x \right )} \right )\right )} - \frac {1}{3 x^{3}} \]
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\[ \int \frac {1}{x^4 \left (1+x^8\right )} \, dx=\int { \frac {1}{{\left (x^{8} + 1\right )} x^{4}} \,d x } \]
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none
Time = 0.39 (sec) , antiderivative size = 244, normalized size of antiderivative = 0.69 \[ \int \frac {1}{x^4 \left (1+x^8\right )} \, dx=\frac {1}{8} \, \sqrt {-\sqrt {2} + 2} \arctan \left (\frac {2 \, x + \sqrt {-\sqrt {2} + 2}}{\sqrt {\sqrt {2} + 2}}\right ) + \frac {1}{8} \, \sqrt {-\sqrt {2} + 2} \arctan \left (\frac {2 \, x - \sqrt {-\sqrt {2} + 2}}{\sqrt {\sqrt {2} + 2}}\right ) - \frac {1}{8} \, \sqrt {\sqrt {2} + 2} \arctan \left (\frac {2 \, x + \sqrt {\sqrt {2} + 2}}{\sqrt {-\sqrt {2} + 2}}\right ) - \frac {1}{8} \, \sqrt {\sqrt {2} + 2} \arctan \left (\frac {2 \, x - \sqrt {\sqrt {2} + 2}}{\sqrt {-\sqrt {2} + 2}}\right ) + \frac {1}{16} \, \sqrt {-\sqrt {2} + 2} \log \left (x^{2} + x \sqrt {\sqrt {2} + 2} + 1\right ) - \frac {1}{16} \, \sqrt {-\sqrt {2} + 2} \log \left (x^{2} - x \sqrt {\sqrt {2} + 2} + 1\right ) - \frac {1}{16} \, \sqrt {\sqrt {2} + 2} \log \left (x^{2} + x \sqrt {-\sqrt {2} + 2} + 1\right ) + \frac {1}{16} \, \sqrt {\sqrt {2} + 2} \log \left (x^{2} - x \sqrt {-\sqrt {2} + 2} + 1\right ) - \frac {1}{3 \, x^{3}} \]
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Time = 5.92 (sec) , antiderivative size = 298, normalized size of antiderivative = 0.84 \[ \int \frac {1}{x^4 \left (1+x^8\right )} \, dx=\mathrm {atan}\left (\frac {x\,\sqrt {-\sqrt {2}-2}\,1{}\mathrm {i}}{\sqrt {2-\sqrt {2}}\,\sqrt {-\sqrt {2}-2}-\sqrt {2}}-\frac {x\,\sqrt {2-\sqrt {2}}\,1{}\mathrm {i}}{\sqrt {2-\sqrt {2}}\,\sqrt {-\sqrt {2}-2}-\sqrt {2}}\right )\,\left (\frac {\sqrt {-\sqrt {2}-2}\,1{}\mathrm {i}}{8}-\frac {\sqrt {2-\sqrt {2}}\,1{}\mathrm {i}}{8}\right )-\frac {1}{3\,x^3}+\mathrm {atan}\left (\frac {x\,\sqrt {\sqrt {2}-2}\,1{}\mathrm {i}}{\sqrt {2}-\sqrt {\sqrt {2}-2}\,\sqrt {\sqrt {2}+2}}+\frac {x\,\sqrt {\sqrt {2}+2}\,1{}\mathrm {i}}{\sqrt {2}-\sqrt {\sqrt {2}-2}\,\sqrt {\sqrt {2}+2}}\right )\,\left (\frac {\sqrt {\sqrt {2}-2}\,1{}\mathrm {i}}{8}+\frac {\sqrt {\sqrt {2}+2}\,1{}\mathrm {i}}{8}\right )-\mathrm {atan}\left (-\frac {\sqrt {2}\,x\,\sqrt {\sqrt {2}+2}}{2}+x\,\sqrt {\sqrt {2}+2}\,\left (\frac {1}{2}+\frac {1}{2}{}\mathrm {i}\right )\right )\,\left (\frac {\sqrt {2}}{16}-\frac {1}{16}+\frac {1}{16}{}\mathrm {i}\right )\,\sqrt {\sqrt {2}+2}\,2{}\mathrm {i}-\mathrm {atan}\left (x\,\sqrt {\sqrt {2}+2}\,\left (\frac {1}{2}-\frac {1}{2}{}\mathrm {i}\right )+\frac {\sqrt {2}\,x\,\sqrt {\sqrt {2}+2}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {\sqrt {2}\,1{}\mathrm {i}}{16}-\frac {1}{16}-\frac {1}{16}{}\mathrm {i}\right )\,\sqrt {\sqrt {2}+2}\,2{}\mathrm {i} \]
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