Integrand size = 20, antiderivative size = 355 \[ \int \frac {1-x^4}{1-x^4+x^8} \, dx=-\frac {\arctan \left (\frac {\sqrt {2-\sqrt {3}}-2 x}{\sqrt {2+\sqrt {3}}}\right )}{4 \sqrt {3 \left (2-\sqrt {3}\right )}}+\frac {\arctan \left (\frac {\sqrt {2+\sqrt {3}}-2 x}{\sqrt {2-\sqrt {3}}}\right )}{4 \sqrt {3 \left (2+\sqrt {3}\right )}}+\frac {\arctan \left (\frac {\sqrt {2-\sqrt {3}}+2 x}{\sqrt {2+\sqrt {3}}}\right )}{4 \sqrt {3 \left (2-\sqrt {3}\right )}}-\frac {\arctan \left (\frac {\sqrt {2+\sqrt {3}}+2 x}{\sqrt {2-\sqrt {3}}}\right )}{4 \sqrt {3 \left (2+\sqrt {3}\right )}}+\frac {1}{8} \sqrt {\frac {1}{3} \left (2-\sqrt {3}\right )} \log \left (1-\sqrt {2-\sqrt {3}} x+x^2\right )-\frac {1}{8} \sqrt {\frac {1}{3} \left (2-\sqrt {3}\right )} \log \left (1+\sqrt {2-\sqrt {3}} x+x^2\right )-\frac {1}{8} \sqrt {\frac {1}{3} \left (2+\sqrt {3}\right )} \log \left (1-\sqrt {2+\sqrt {3}} x+x^2\right )+\frac {1}{8} \sqrt {\frac {1}{3} \left (2+\sqrt {3}\right )} \log \left (1+\sqrt {2+\sqrt {3}} x+x^2\right ) \]
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Time = 0.18 (sec) , antiderivative size = 355, normalized size of antiderivative = 1.00, number of steps used = 19, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {1435, 1183, 648, 632, 210, 642} \[ \int \frac {1-x^4}{1-x^4+x^8} \, dx=-\frac {\arctan \left (\frac {\sqrt {2-\sqrt {3}}-2 x}{\sqrt {2+\sqrt {3}}}\right )}{4 \sqrt {3 \left (2-\sqrt {3}\right )}}+\frac {\arctan \left (\frac {\sqrt {2+\sqrt {3}}-2 x}{\sqrt {2-\sqrt {3}}}\right )}{4 \sqrt {3 \left (2+\sqrt {3}\right )}}+\frac {\arctan \left (\frac {2 x+\sqrt {2-\sqrt {3}}}{\sqrt {2+\sqrt {3}}}\right )}{4 \sqrt {3 \left (2-\sqrt {3}\right )}}-\frac {\arctan \left (\frac {2 x+\sqrt {2+\sqrt {3}}}{\sqrt {2-\sqrt {3}}}\right )}{4 \sqrt {3 \left (2+\sqrt {3}\right )}}+\frac {1}{8} \sqrt {\frac {1}{3} \left (2-\sqrt {3}\right )} \log \left (x^2-\sqrt {2-\sqrt {3}} x+1\right )-\frac {1}{8} \sqrt {\frac {1}{3} \left (2-\sqrt {3}\right )} \log \left (x^2+\sqrt {2-\sqrt {3}} x+1\right )-\frac {1}{8} \sqrt {\frac {1}{3} \left (2+\sqrt {3}\right )} \log \left (x^2-\sqrt {2+\sqrt {3}} x+1\right )+\frac {1}{8} \sqrt {\frac {1}{3} \left (2+\sqrt {3}\right )} \log \left (x^2+\sqrt {2+\sqrt {3}} x+1\right ) \]
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Rule 210
Rule 632
Rule 642
Rule 648
Rule 1183
Rule 1435
Rubi steps \begin{align*} \text {integral}& = -\frac {\int \frac {\sqrt {3}+2 x^2}{-1-\sqrt {3} x^2-x^4} \, dx}{2 \sqrt {3}}-\frac {\int \frac {\sqrt {3}-2 x^2}{-1+\sqrt {3} x^2-x^4} \, dx}{2 \sqrt {3}} \\ & = \frac {\int \frac {\sqrt {3 \left (2-\sqrt {3}\right )}-\left (-2+\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 {3 \left (2-\sqrt {3}\right )}+\left (-2+\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 {3 \left (2+\sqrt {3}\right )}-\left (2+\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 {3 \left (2+\sqrt {3}\right )}+\left (2+\sqrt {3}\right ) x}{1+\sqrt {2+\sqrt {3}} x+x^2} \, dx}{4 \sqrt {3 \left (2+\sqrt {3}\right )}} \\ & = -\left (\frac {1}{8} \sqrt {\frac {1}{3} \left (7-4 \sqrt {3}\right )} \int \frac {1}{1-\sqrt {2+\sqrt {3}} x+x^2} \, dx\right )-\frac {1}{8} \sqrt {\frac {1}{3} \left (7-4 \sqrt {3}\right )} \int \frac {1}{1+\sqrt {2+\sqrt {3}} x+x^2} \, dx+\frac {1}{8} \sqrt {\frac {1}{3} \left (2-\sqrt {3}\right )} \int \frac {-\sqrt {2-\sqrt {3}}+2 x}{1-\sqrt {2-\sqrt {3}} x+x^2} \, dx+\frac {\left (-2+\sqrt {3}\right ) \int \frac {\sqrt {2-\sqrt {3}}+2 x}{1+\sqrt {2-\sqrt {3}} x+x^2} \, dx}{8 \sqrt {3 \left (2-\sqrt {3}\right )}}-\frac {1}{8} \sqrt {\frac {1}{3} \left (2+\sqrt {3}\right )} \int \frac {-\sqrt {2+\sqrt {3}}+2 x}{1-\sqrt {2+\sqrt {3}} x+x^2} \, dx+\frac {1}{8} \sqrt {\frac {1}{3} \left (2+\sqrt {3}\right )} \int \frac {\sqrt {2+\sqrt {3}}+2 x}{1+\sqrt {2+\sqrt {3}} x+x^2} \, dx+\frac {1}{8} \sqrt {\frac {1}{3} \left (7+4 \sqrt {3}\right )} \int \frac {1}{1-\sqrt {2-\sqrt {3}} x+x^2} \, dx+\frac {1}{8} \sqrt {\frac {1}{3} \left (7+4 \sqrt {3}\right )} \int \frac {1}{1+\sqrt {2-\sqrt {3}} x+x^2} \, dx \\ & = \frac {1}{8} \sqrt {\frac {1}{3} \left (2-\sqrt {3}\right )} \log \left (1-\sqrt {2-\sqrt {3}} x+x^2\right )-\frac {1}{8} \sqrt {\frac {2}{3}-\frac {1}{\sqrt {3}}} \log \left (1+\sqrt {2-\sqrt {3}} x+x^2\right )-\frac {1}{8} \sqrt {\frac {1}{3} \left (2+\sqrt {3}\right )} \log \left (1-\sqrt {2+\sqrt {3}} x+x^2\right )+\frac {1}{8} \sqrt {\frac {1}{3} \left (2+\sqrt {3}\right )} \log \left (1+\sqrt {2+\sqrt {3}} x+x^2\right )+\frac {1}{4} \sqrt {\frac {1}{3} \left (7-4 \sqrt {3}\right )} \text {Subst}\left (\int \frac {1}{-2+\sqrt {3}-x^2} \, dx,x,-\sqrt {2+\sqrt {3}}+2 x\right )+\frac {1}{4} \sqrt {\frac {1}{3} \left (7-4 \sqrt {3}\right )} \text {Subst}\left (\int \frac {1}{-2+\sqrt {3}-x^2} \, dx,x,\sqrt {2+\sqrt {3}}+2 x\right )-\frac {1}{4} \sqrt {\frac {1}{3} \left (7+4 \sqrt {3}\right )} \text {Subst}\left (\int \frac {1}{-2-\sqrt {3}-x^2} \, dx,x,-\sqrt {2-\sqrt {3}}+2 x\right )-\frac {1}{4} \sqrt {\frac {1}{3} \left (7+4 \sqrt {3}\right )} \text {Subst}\left (\int \frac {1}{-2-\sqrt {3}-x^2} \, dx,x,\sqrt {2-\sqrt {3}}+2 x\right ) \\ & = -\frac {1}{4} \sqrt {\frac {1}{3} \left (2+\sqrt {3}\right )} \tan ^{-1}\left (\frac {\sqrt {2-\sqrt {3}}-2 x}{\sqrt {2+\sqrt {3}}}\right )+\frac {1}{4} \sqrt {\frac {1}{3} \left (2-\sqrt {3}\right )} \tan ^{-1}\left (\frac {\sqrt {2+\sqrt {3}}-2 x}{\sqrt {2-\sqrt {3}}}\right )+\frac {1}{4} \sqrt {\frac {1}{3} \left (2+\sqrt {3}\right )} \tan ^{-1}\left (\frac {\sqrt {2-\sqrt {3}}+2 x}{\sqrt {2+\sqrt {3}}}\right )-\frac {1}{4} \sqrt {\frac {1}{3} \left (2-\sqrt {3}\right )} \tan ^{-1}\left (\frac {\sqrt {2+\sqrt {3}}+2 x}{\sqrt {2-\sqrt {3}}}\right )+\frac {1}{8} \sqrt {\frac {1}{3} \left (2-\sqrt {3}\right )} \log \left (1-\sqrt {2-\sqrt {3}} x+x^2\right )-\frac {1}{8} \sqrt {\frac {2}{3}-\frac {1}{\sqrt {3}}} \log \left (1+\sqrt {2-\sqrt {3}} x+x^2\right )-\frac {1}{8} \sqrt {\frac {1}{3} \left (2+\sqrt {3}\right )} \log \left (1-\sqrt {2+\sqrt {3}} x+x^2\right )+\frac {1}{8} \sqrt {\frac {1}{3} \left (2+\sqrt {3}\right )} \log \left (1+\sqrt {2+\sqrt {3}} x+x^2\right ) \\ \end{align*}
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 0.02 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.16 \[ \int \frac {1-x^4}{1-x^4+x^8} \, dx=-\frac {1}{4} \text {RootSum}\left [1-\text {$\#$1}^4+\text {$\#$1}^8\&,\frac {-\log (x-\text {$\#$1})+\log (x-\text {$\#$1}) \text {$\#$1}^4}{-\text {$\#$1}^3+2 \text {$\#$1}^7}\&\right ] \]
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Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.07 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.12
method | result | size |
default | \(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{8}-\textit {\_Z}^{4}+1\right )}{\sum }\frac {\left (-\textit {\_R}^{4}+1\right ) \ln \left (x -\textit {\_R} \right )}{2 \textit {\_R}^{7}-\textit {\_R}^{3}}\right )}{4}\) | \(44\) |
risch | \(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{8}-\textit {\_Z}^{4}+1\right )}{\sum }\frac {\left (-\textit {\_R}^{4}+1\right ) \ln \left (x -\textit {\_R} \right )}{2 \textit {\_R}^{7}-\textit {\_R}^{3}}\right )}{4}\) | \(44\) |
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Result contains complex when optimal does not.
Time = 0.30 (sec) , antiderivative size = 417, normalized size of antiderivative = 1.17 \[ \int \frac {1-x^4}{1-x^4+x^8} \, dx=\frac {1}{24} \, \sqrt {6} \sqrt {\sqrt {2} \sqrt {-i \, \sqrt {3} + 1}} \log \left (\sqrt {6} \sqrt {\sqrt {2} \sqrt {-i \, \sqrt {3} + 1}} {\left (i \, \sqrt {3} + 3\right )} + 12 \, x\right ) + \frac {1}{24} \, \sqrt {6} \sqrt {-\sqrt {2} \sqrt {-i \, \sqrt {3} + 1}} \log \left (\sqrt {6} \sqrt {-\sqrt {2} \sqrt {-i \, \sqrt {3} + 1}} {\left (i \, \sqrt {3} + 3\right )} + 12 \, x\right ) - \frac {1}{24} \, \sqrt {6} \sqrt {\sqrt {2} \sqrt {i \, \sqrt {3} + 1}} \log \left (\sqrt {6} \sqrt {\sqrt {2} \sqrt {i \, \sqrt {3} + 1}} {\left (i \, \sqrt {3} - 3\right )} + 12 \, x\right ) - \frac {1}{24} \, \sqrt {6} \sqrt {-\sqrt {2} \sqrt {i \, \sqrt {3} + 1}} \log \left (\sqrt {6} \sqrt {-\sqrt {2} \sqrt {i \, \sqrt {3} + 1}} {\left (i \, \sqrt {3} - 3\right )} + 12 \, x\right ) + \frac {1}{24} \, \sqrt {6} \sqrt {\sqrt {2} \sqrt {i \, \sqrt {3} + 1}} \log \left (\sqrt {6} \sqrt {\sqrt {2} \sqrt {i \, \sqrt {3} + 1}} {\left (-i \, \sqrt {3} + 3\right )} + 12 \, x\right ) + \frac {1}{24} \, \sqrt {6} \sqrt {-\sqrt {2} \sqrt {i \, \sqrt {3} + 1}} \log \left (\sqrt {6} \sqrt {-\sqrt {2} \sqrt {i \, \sqrt {3} + 1}} {\left (-i \, \sqrt {3} + 3\right )} + 12 \, x\right ) - \frac {1}{24} \, \sqrt {6} \sqrt {\sqrt {2} \sqrt {-i \, \sqrt {3} + 1}} \log \left (\sqrt {6} \sqrt {\sqrt {2} \sqrt {-i \, \sqrt {3} + 1}} {\left (-i \, \sqrt {3} - 3\right )} + 12 \, x\right ) - \frac {1}{24} \, \sqrt {6} \sqrt {-\sqrt {2} \sqrt {-i \, \sqrt {3} + 1}} \log \left (\sqrt {6} \sqrt {-\sqrt {2} \sqrt {-i \, \sqrt {3} + 1}} {\left (-i \, \sqrt {3} - 3\right )} + 12 \, x\right ) \]
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Time = 1.30 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.07 \[ \int \frac {1-x^4}{1-x^4+x^8} \, dx=- \operatorname {RootSum} {\left (5308416 t^{8} - 2304 t^{4} + 1, \left ( t \mapsto t \log {\left (9216 t^{5} - 8 t + x \right )} \right )\right )} \]
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\[ \int \frac {1-x^4}{1-x^4+x^8} \, dx=\int { -\frac {x^{4} - 1}{x^{8} - x^{4} + 1} \,d x } \]
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none
Time = 0.30 (sec) , antiderivative size = 253, normalized size of antiderivative = 0.71 \[ \int \frac {1-x^4}{1-x^4+x^8} \, dx=\frac {1}{24} \, {\left (\sqrt {6} + 3 \, \sqrt {2}\right )} \arctan \left (\frac {4 \, x + \sqrt {6} - \sqrt {2}}{\sqrt {6} + \sqrt {2}}\right ) + \frac {1}{24} \, {\left (\sqrt {6} + 3 \, \sqrt {2}\right )} \arctan \left (\frac {4 \, x - \sqrt {6} + \sqrt {2}}{\sqrt {6} + \sqrt {2}}\right ) + \frac {1}{24} \, {\left (\sqrt {6} - 3 \, \sqrt {2}\right )} \arctan \left (\frac {4 \, x + \sqrt {6} + \sqrt {2}}{\sqrt {6} - \sqrt {2}}\right ) + \frac {1}{24} \, {\left (\sqrt {6} - 3 \, \sqrt {2}\right )} \arctan \left (\frac {4 \, x - \sqrt {6} - \sqrt {2}}{\sqrt {6} - \sqrt {2}}\right ) + \frac {1}{48} \, {\left (\sqrt {6} + 3 \, \sqrt {2}\right )} \log \left (x^{2} + \frac {1}{2} \, x {\left (\sqrt {6} + \sqrt {2}\right )} + 1\right ) - \frac {1}{48} \, {\left (\sqrt {6} + 3 \, \sqrt {2}\right )} \log \left (x^{2} - \frac {1}{2} \, x {\left (\sqrt {6} + \sqrt {2}\right )} + 1\right ) + \frac {1}{48} \, {\left (\sqrt {6} - 3 \, \sqrt {2}\right )} \log \left (x^{2} + \frac {1}{2} \, x {\left (\sqrt {6} - \sqrt {2}\right )} + 1\right ) - \frac {1}{48} \, {\left (\sqrt {6} - 3 \, \sqrt {2}\right )} \log \left (x^{2} - \frac {1}{2} \, x {\left (\sqrt {6} - \sqrt {2}\right )} + 1\right ) \]
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Time = 8.44 (sec) , antiderivative size = 208, normalized size of antiderivative = 0.59 \[ \int \frac {1-x^4}{1-x^4+x^8} \, dx=-\frac {\sqrt {3}\,\mathrm {atan}\left (\frac {x}{{\left (8-\sqrt {3}\,8{}\mathrm {i}\right )}^{1/4}}+\frac {\sqrt {3}\,x\,1{}\mathrm {i}}{{\left (8-\sqrt {3}\,8{}\mathrm {i}\right )}^{1/4}}\right )\,{\left (8-\sqrt {3}\,8{}\mathrm {i}\right )}^{1/4}\,1{}\mathrm {i}}{12}-\frac {\sqrt {3}\,\mathrm {atan}\left (\frac {x\,1{}\mathrm {i}}{{\left (8-\sqrt {3}\,8{}\mathrm {i}\right )}^{1/4}}-\frac {\sqrt {3}\,x}{{\left (8-\sqrt {3}\,8{}\mathrm {i}\right )}^{1/4}}\right )\,{\left (8-\sqrt {3}\,8{}\mathrm {i}\right )}^{1/4}}{12}+\frac {2^{3/4}\,\sqrt {3}\,\mathrm {atan}\left (\frac {2^{1/4}\,x}{2\,{\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}^{1/4}}-\frac {2^{1/4}\,\sqrt {3}\,x\,1{}\mathrm {i}}{2\,{\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}^{1/4}}\right )\,{\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}^{1/4}\,1{}\mathrm {i}}{12}+\frac {2^{3/4}\,\sqrt {3}\,\mathrm {atan}\left (\frac {2^{1/4}\,x\,1{}\mathrm {i}}{2\,{\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}^{1/4}}+\frac {2^{1/4}\,\sqrt {3}\,x}{2\,{\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}^{1/4}}\right )\,{\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}^{1/4}}{12} \]
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