Integrand size = 16, antiderivative size = 275 \[ \int \frac {x^6}{1-x^4+x^8} \, dx=-\frac {\arctan \left (\frac {\sqrt {2-\sqrt {3}}-2 x}{\sqrt {2+\sqrt {3}}}\right )}{2 \sqrt {6}}-\frac {\arctan \left (\frac {\sqrt {2+\sqrt {3}}-2 x}{\sqrt {2-\sqrt {3}}}\right )}{2 \sqrt {6}}+\frac {\arctan \left (\frac {\sqrt {2-\sqrt {3}}+2 x}{\sqrt {2+\sqrt {3}}}\right )}{2 \sqrt {6}}+\frac {\arctan \left (\frac {\sqrt {2+\sqrt {3}}+2 x}{\sqrt {2-\sqrt {3}}}\right )}{2 \sqrt {6}}+\frac {\log \left (1-\sqrt {2-\sqrt {3}} x+x^2\right )}{4 \sqrt {6}}-\frac {\log \left (1+\sqrt {2-\sqrt {3}} x+x^2\right )}{4 \sqrt {6}}+\frac {\log \left (1-\sqrt {2+\sqrt {3}} x+x^2\right )}{4 \sqrt {6}}-\frac {\log \left (1+\sqrt {2+\sqrt {3}} x+x^2\right )}{4 \sqrt {6}} \]
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Time = 0.17 (sec) , antiderivative size = 275, normalized size of antiderivative = 1.00, number of steps used = 19, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {1386, 1183, 648, 632, 210, 642} \[ \int \frac {x^6}{1-x^4+x^8} \, dx=-\frac {\arctan \left (\frac {\sqrt {2-\sqrt {3}}-2 x}{\sqrt {2+\sqrt {3}}}\right )}{2 \sqrt {6}}-\frac {\arctan \left (\frac {\sqrt {2+\sqrt {3}}-2 x}{\sqrt {2-\sqrt {3}}}\right )}{2 \sqrt {6}}+\frac {\arctan \left (\frac {2 x+\sqrt {2-\sqrt {3}}}{\sqrt {2+\sqrt {3}}}\right )}{2 \sqrt {6}}+\frac {\arctan \left (\frac {2 x+\sqrt {2+\sqrt {3}}}{\sqrt {2-\sqrt {3}}}\right )}{2 \sqrt {6}}+\frac {\log \left (x^2-\sqrt {2-\sqrt {3}} x+1\right )}{4 \sqrt {6}}-\frac {\log \left (x^2+\sqrt {2-\sqrt {3}} x+1\right )}{4 \sqrt {6}}+\frac {\log \left (x^2-\sqrt {2+\sqrt {3}} x+1\right )}{4 \sqrt {6}}-\frac {\log \left (x^2+\sqrt {2+\sqrt {3}} x+1\right )}{4 \sqrt {6}} \]
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Rule 210
Rule 632
Rule 642
Rule 648
Rule 1183
Rule 1386
Rubi steps \begin{align*} \text {integral}& = -\frac {\int \frac {1-\sqrt {3} x^2}{1-\sqrt {3} x^2+x^4} \, dx}{2 \sqrt {3}}+\frac {\int \frac {1+\sqrt {3} x^2}{1+\sqrt {3} x^2+x^4} \, dx}{2 \sqrt {3}} \\ & = \frac {\int \frac {\sqrt {2-\sqrt {3}}-\left (1-\sqrt {3}\right ) x}{1-\sqrt {2-\sqrt {3}} x+x^2} \, dx}{4 \sqrt {3 \left (2-\sqrt {3}\right )}}+\frac {\int \frac {\sqrt {2-\sqrt {3}}+\left (1-\sqrt {3}\right ) x}{1+\sqrt {2-\sqrt {3}} x+x^2} \, dx}{4 \sqrt {3 \left (2-\sqrt {3}\right )}}-\frac {\int \frac {\sqrt {2+\sqrt {3}}-\left (1+\sqrt {3}\right ) x}{1-\sqrt {2+\sqrt {3}} x+x^2} \, dx}{4 \sqrt {3 \left (2+\sqrt {3}\right )}}-\frac {\int \frac {\sqrt {2+\sqrt {3}}+\left (1+\sqrt {3}\right ) x}{1+\sqrt {2+\sqrt {3}} x+x^2} \, dx}{4 \sqrt {3 \left (2+\sqrt {3}\right )}} \\ & = \frac {\int \frac {-\sqrt {2-\sqrt {3}}+2 x}{1-\sqrt {2-\sqrt {3}} x+x^2} \, dx}{4 \sqrt {6}}-\frac {\int \frac {\sqrt {2-\sqrt {3}}+2 x}{1+\sqrt {2-\sqrt {3}} x+x^2} \, dx}{4 \sqrt {6}}+\frac {\int \frac {-\sqrt {2+\sqrt {3}}+2 x}{1-\sqrt {2+\sqrt {3}} x+x^2} \, dx}{4 \sqrt {6}}-\frac {\int \frac {\sqrt {2+\sqrt {3}}+2 x}{1+\sqrt {2+\sqrt {3}} x+x^2} \, dx}{4 \sqrt {6}}+\frac {\int \frac {1}{1-\sqrt {2-\sqrt {3}} x+x^2} \, dx}{4 \sqrt {6 \left (2-\sqrt {3}\right )}}+\frac {\int \frac {1}{1+\sqrt {2-\sqrt {3}} x+x^2} \, dx}{4 \sqrt {6 \left (2-\sqrt {3}\right )}}+\frac {\int \frac {1}{1-\sqrt {2+\sqrt {3}} x+x^2} \, dx}{4 \sqrt {6 \left (2+\sqrt {3}\right )}}+\frac {\int \frac {1}{1+\sqrt {2+\sqrt {3}} x+x^2} \, dx}{4 \sqrt {6 \left (2+\sqrt {3}\right )}} \\ & = \frac {\log \left (1-\sqrt {2-\sqrt {3}} x+x^2\right )}{4 \sqrt {6}}-\frac {\log \left (1+\sqrt {2-\sqrt {3}} x+x^2\right )}{4 \sqrt {6}}+\frac {\log \left (1-\sqrt {2+\sqrt {3}} x+x^2\right )}{4 \sqrt {6}}-\frac {\log \left (1+\sqrt {2+\sqrt {3}} x+x^2\right )}{4 \sqrt {6}}-\frac {\text {Subst}\left (\int \frac {1}{-2-\sqrt {3}-x^2} \, dx,x,-\sqrt {2-\sqrt {3}}+2 x\right )}{2 \sqrt {6 \left (2-\sqrt {3}\right )}}-\frac {\text {Subst}\left (\int \frac {1}{-2-\sqrt {3}-x^2} \, dx,x,\sqrt {2-\sqrt {3}}+2 x\right )}{2 \sqrt {6 \left (2-\sqrt {3}\right )}}-\frac {\text {Subst}\left (\int \frac {1}{-2+\sqrt {3}-x^2} \, dx,x,-\sqrt {2+\sqrt {3}}+2 x\right )}{2 \sqrt {6 \left (2+\sqrt {3}\right )}}-\frac {\text {Subst}\left (\int \frac {1}{-2+\sqrt {3}-x^2} \, dx,x,\sqrt {2+\sqrt {3}}+2 x\right )}{2 \sqrt {6 \left (2+\sqrt {3}\right )}} \\ & = -\frac {\tan ^{-1}\left (\frac {\sqrt {2-\sqrt {3}}-2 x}{\sqrt {2+\sqrt {3}}}\right )}{2 \sqrt {6}}-\frac {\tan ^{-1}\left (\frac {\sqrt {2+\sqrt {3}}-2 x}{\sqrt {2-\sqrt {3}}}\right )}{2 \sqrt {6}}+\frac {\tan ^{-1}\left (\frac {\sqrt {2-\sqrt {3}}+2 x}{\sqrt {2+\sqrt {3}}}\right )}{2 \sqrt {6}}+\frac {\tan ^{-1}\left (\frac {\sqrt {2+\sqrt {3}}+2 x}{\sqrt {2-\sqrt {3}}}\right )}{2 \sqrt {6}}+\frac {\log \left (1-\sqrt {2-\sqrt {3}} x+x^2\right )}{4 \sqrt {6}}-\frac {\log \left (1+\sqrt {2-\sqrt {3}} x+x^2\right )}{4 \sqrt {6}}+\frac {\log \left (1-\sqrt {2+\sqrt {3}} x+x^2\right )}{4 \sqrt {6}}-\frac {\log \left (1+\sqrt {2+\sqrt {3}} x+x^2\right )}{4 \sqrt {6}} \\ \end{align*}
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 0.01 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.15 \[ \int \frac {x^6}{1-x^4+x^8} \, dx=\frac {1}{4} \text {RootSum}\left [1-\text {$\#$1}^4+\text {$\#$1}^8\&,\frac {\log (x-\text {$\#$1}) \text {$\#$1}^3}{-1+2 \text {$\#$1}^4}\&\right ] \]
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Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.06 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.12
method | result | size |
default | \(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (9 \textit {\_Z}^{4}+1\right )}{\sum }\textit {\_R} \ln \left (9 x \,\textit {\_R}^{3}-3 \textit {\_R}^{2}+x^{2}\right )\right )}{4}\) | \(32\) |
risch | \(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (9 \textit {\_Z}^{4}+1\right )}{\sum }\textit {\_R} \ln \left (9 x \,\textit {\_R}^{3}-3 \textit {\_R}^{2}+x^{2}\right )\right )}{4}\) | \(32\) |
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Result contains complex when optimal does not.
Time = 0.24 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.37 \[ \int \frac {x^6}{1-x^4+x^8} \, dx=\left (\frac {1}{24} i - \frac {1}{24}\right ) \, \sqrt {3} \sqrt {2} \log \left (\left (3 i + 3\right ) \, \sqrt {3} \sqrt {2} x + 6 \, x^{2} + 6 i\right ) - \left (\frac {1}{24} i + \frac {1}{24}\right ) \, \sqrt {3} \sqrt {2} \log \left (-\left (3 i - 3\right ) \, \sqrt {3} \sqrt {2} x + 6 \, x^{2} - 6 i\right ) + \left (\frac {1}{24} i + \frac {1}{24}\right ) \, \sqrt {3} \sqrt {2} \log \left (\left (3 i - 3\right ) \, \sqrt {3} \sqrt {2} x + 6 \, x^{2} - 6 i\right ) - \left (\frac {1}{24} i - \frac {1}{24}\right ) \, \sqrt {3} \sqrt {2} \log \left (-\left (3 i + 3\right ) \, \sqrt {3} \sqrt {2} x + 6 \, x^{2} + 6 i\right ) \]
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Time = 0.10 (sec) , antiderivative size = 165, normalized size of antiderivative = 0.60 \[ \int \frac {x^6}{1-x^4+x^8} \, dx=\frac {\sqrt {6} \cdot \left (2 \operatorname {atan}{\left (\frac {\sqrt {6} x}{3} - \frac {1}{3} \right )} + 2 \operatorname {atan}{\left (\sqrt {6} x^{3} - 4 x^{2} + 2 \sqrt {6} x - 3 \right )}\right )}{24} + \frac {\sqrt {6} \cdot \left (2 \operatorname {atan}{\left (\frac {\sqrt {6} x}{3} + \frac {1}{3} \right )} + 2 \operatorname {atan}{\left (\sqrt {6} x^{3} + 4 x^{2} + 2 \sqrt {6} x + 3 \right )}\right )}{24} + \frac {\sqrt {6} \log {\left (x^{4} - \sqrt {6} x^{3} + 3 x^{2} - \sqrt {6} x + 1 \right )}}{24} - \frac {\sqrt {6} \log {\left (x^{4} + \sqrt {6} x^{3} + 3 x^{2} + \sqrt {6} x + 1 \right )}}{24} \]
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\[ \int \frac {x^6}{1-x^4+x^8} \, dx=\int { \frac {x^{6}}{x^{8} - x^{4} + 1} \,d x } \]
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none
Time = 0.32 (sec) , antiderivative size = 205, normalized size of antiderivative = 0.75 \[ \int \frac {x^6}{1-x^4+x^8} \, dx=\frac {1}{12} \, \sqrt {6} \arctan \left (\frac {4 \, x + \sqrt {6} - \sqrt {2}}{\sqrt {6} + \sqrt {2}}\right ) + \frac {1}{12} \, \sqrt {6} \arctan \left (\frac {4 \, x - \sqrt {6} + \sqrt {2}}{\sqrt {6} + \sqrt {2}}\right ) + \frac {1}{12} \, \sqrt {6} \arctan \left (\frac {4 \, x + \sqrt {6} + \sqrt {2}}{\sqrt {6} - \sqrt {2}}\right ) + \frac {1}{12} \, \sqrt {6} \arctan \left (\frac {4 \, x - \sqrt {6} - \sqrt {2}}{\sqrt {6} - \sqrt {2}}\right ) - \frac {1}{24} \, \sqrt {6} \log \left (x^{2} + \frac {1}{2} \, x {\left (\sqrt {6} + \sqrt {2}\right )} + 1\right ) + \frac {1}{24} \, \sqrt {6} \log \left (x^{2} - \frac {1}{2} \, x {\left (\sqrt {6} + \sqrt {2}\right )} + 1\right ) - \frac {1}{24} \, \sqrt {6} \log \left (x^{2} + \frac {1}{2} \, x {\left (\sqrt {6} - \sqrt {2}\right )} + 1\right ) + \frac {1}{24} \, \sqrt {6} \log \left (x^{2} - \frac {1}{2} \, x {\left (\sqrt {6} - \sqrt {2}\right )} + 1\right ) \]
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Time = 0.06 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.19 \[ \int \frac {x^6}{1-x^4+x^8} \, dx=\sqrt {6}\,\mathrm {atan}\left (\frac {\sqrt {6}\,x\,\left (\frac {1}{3}+\frac {1}{3}{}\mathrm {i}\right )}{\frac {2\,x^2}{3}-\frac {2}{3}{}\mathrm {i}}\right )\,\left (-\frac {1}{12}+\frac {1}{12}{}\mathrm {i}\right )+\sqrt {6}\,\mathrm {atan}\left (\frac {\sqrt {6}\,x\,\left (\frac {1}{3}-\frac {1}{3}{}\mathrm {i}\right )}{\frac {2\,x^2}{3}+\frac {2}{3}{}\mathrm {i}}\right )\,\left (-\frac {1}{12}-\frac {1}{12}{}\mathrm {i}\right ) \]
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