Integrand size = 40, antiderivative size = 93 \[ \int \frac {\left (1+x^4\right ) \sqrt {-1-x^2+x^4}}{1-x^2-3 x^4+x^6+x^8} \, dx=\frac {1}{10} \left (-5-\sqrt {5}\right ) \arctan \left (\frac {\sqrt {\frac {3}{2}+\frac {\sqrt {5}}{2}} x}{\sqrt {-1-x^2+x^4}}\right )+\frac {1}{10} \left (5-\sqrt {5}\right ) \arctan \left (\frac {\sqrt {\frac {2}{3+\sqrt {5}}} x}{\sqrt {-1-x^2+x^4}}\right ) \]
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\[ \int \frac {\left (1+x^4\right ) \sqrt {-1-x^2+x^4}}{1-x^2-3 x^4+x^6+x^8} \, dx=\int \frac {\left (1+x^4\right ) \sqrt {-1-x^2+x^4}}{1-x^2-3 x^4+x^6+x^8} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {\sqrt {-1-x^2+x^4}}{1-x^2-3 x^4+x^6+x^8}+\frac {x^4 \sqrt {-1-x^2+x^4}}{1-x^2-3 x^4+x^6+x^8}\right ) \, dx \\ & = \int \frac {\sqrt {-1-x^2+x^4}}{1-x^2-3 x^4+x^6+x^8} \, dx+\int \frac {x^4 \sqrt {-1-x^2+x^4}}{1-x^2-3 x^4+x^6+x^8} \, dx \\ \end{align*}
Time = 0.35 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.83 \[ \int \frac {\left (1+x^4\right ) \sqrt {-1-x^2+x^4}}{1-x^2-3 x^4+x^6+x^8} \, dx=\frac {1}{10} \left (-\left (\left (-5+\sqrt {5}\right ) \arctan \left (\frac {\left (-1+\sqrt {5}\right ) x}{2 \sqrt {-1-x^2+x^4}}\right )\right )-\left (5+\sqrt {5}\right ) \arctan \left (\frac {\left (1+\sqrt {5}\right ) x}{2 \sqrt {-1-x^2+x^4}}\right )\right ) \]
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Time = 3.74 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.30
method | result | size |
default | \(\frac {\left (\frac {4 \left (\sqrt {5}-3\right ) \sqrt {5}\, \arctan \left (\frac {4 \sqrt {x^{4}-x^{2}-1}\, \sqrt {2}}{x \left (2 \sqrt {10}-2 \sqrt {2}\right )}\right )}{5 \left (2 \sqrt {10}-2 \sqrt {2}\right )}+\frac {4 \left (3+\sqrt {5}\right ) \sqrt {5}\, \arctan \left (\frac {4 \sqrt {x^{4}-x^{2}-1}\, \sqrt {2}}{x \left (2 \sqrt {10}+2 \sqrt {2}\right )}\right )}{5 \left (2 \sqrt {10}+2 \sqrt {2}\right )}\right ) \sqrt {2}}{2}\) | \(121\) |
elliptic | \(\frac {\left (\frac {4 \left (\sqrt {5}-3\right ) \sqrt {5}\, \arctan \left (\frac {4 \sqrt {x^{4}-x^{2}-1}\, \sqrt {2}}{x \left (2 \sqrt {10}-2 \sqrt {2}\right )}\right )}{5 \left (2 \sqrt {10}-2 \sqrt {2}\right )}+\frac {4 \left (3+\sqrt {5}\right ) \sqrt {5}\, \arctan \left (\frac {4 \sqrt {x^{4}-x^{2}-1}\, \sqrt {2}}{x \left (2 \sqrt {10}+2 \sqrt {2}\right )}\right )}{5 \left (2 \sqrt {10}+2 \sqrt {2}\right )}\right ) \sqrt {2}}{2}\) | \(121\) |
pseudoelliptic | \(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (-3+4 i+\textit {\_Z}^{4}+2 i \textit {\_Z}^{3}+\left (2-4 i\right ) \textit {\_Z}^{2}+\left (4-2 i\right ) \textit {\_Z} \right )}{\sum }\frac {\left (1+2 i+i \textit {\_R}^{3}-\textit {\_R}^{2}+\left (2+i\right ) \textit {\_R} \right ) \ln \left (\frac {-\textit {\_R} x -x^{2}+\sqrt {x^{4}-x^{2}-1}-i}{x}\right )}{2 \textit {\_R}^{3}+3 i \textit {\_R}^{2}+\left (2-4 i\right ) \textit {\_R} +2-i}\right )}{2}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (-3+4 i+\textit {\_Z}^{4}-2 i \textit {\_Z}^{3}+\left (2-4 i\right ) \textit {\_Z}^{2}+\left (-4+2 i\right ) \textit {\_Z} \right )}{\sum }\frac {\ln \left (\frac {-\textit {\_R} x -x^{2}+\sqrt {x^{4}-x^{2}-1}-i}{x}\right ) \left (i \textit {\_R}^{3}+\textit {\_R}^{2}+\left (2+i\right ) \textit {\_R} -1-2 i\right )}{-2 \textit {\_R}^{3}+3 i \textit {\_R}^{2}+\left (-2+4 i\right ) \textit {\_R} +2-i}\right )}{2}\) | \(206\) |
trager | \(20 \operatorname {RootOf}\left (400 \textit {\_Z}^{4}+60 \textit {\_Z}^{2}+1\right )^{3} \ln \left (\frac {1400 x^{2} \operatorname {RootOf}\left (400 \textit {\_Z}^{4}+60 \textit {\_Z}^{2}+1\right )^{5}-70 \operatorname {RootOf}\left (400 \textit {\_Z}^{4}+60 \textit {\_Z}^{2}+1\right )^{3} x^{4}+460 \operatorname {RootOf}\left (400 \textit {\_Z}^{4}+60 \textit {\_Z}^{2}+1\right )^{3} x^{2}-9 \operatorname {RootOf}\left (400 \textit {\_Z}^{4}+60 \textit {\_Z}^{2}+1\right ) x^{4}+30 \operatorname {RootOf}\left (400 \textit {\_Z}^{4}+60 \textit {\_Z}^{2}+1\right )^{2} \sqrt {x^{4}-x^{2}-1}\, x +70 \operatorname {RootOf}\left (400 \textit {\_Z}^{4}+60 \textit {\_Z}^{2}+1\right )^{3}+36 \operatorname {RootOf}\left (400 \textit {\_Z}^{4}+60 \textit {\_Z}^{2}+1\right ) x^{2}+4 x \sqrt {x^{4}-x^{2}-1}+9 \operatorname {RootOf}\left (400 \textit {\_Z}^{4}+60 \textit {\_Z}^{2}+1\right )}{20 x^{2} \operatorname {RootOf}\left (400 \textit {\_Z}^{4}+60 \textit {\_Z}^{2}+1\right )^{2}+x^{4}+2 x^{2}-1}\right )+\operatorname {RootOf}\left (400 \textit {\_Z}^{4}+60 \textit {\_Z}^{2}+1\right ) \ln \left (-\frac {-1200 x^{2} \operatorname {RootOf}\left (400 \textit {\_Z}^{4}+60 \textit {\_Z}^{2}+1\right )^{5}-60 \operatorname {RootOf}\left (400 \textit {\_Z}^{4}+60 \textit {\_Z}^{2}+1\right )^{3} x^{4}+20 \operatorname {RootOf}\left (400 \textit {\_Z}^{4}+60 \textit {\_Z}^{2}+1\right )^{3} x^{2}-2 \operatorname {RootOf}\left (400 \textit {\_Z}^{4}+60 \textit {\_Z}^{2}+1\right ) x^{4}+60 \operatorname {RootOf}\left (400 \textit {\_Z}^{4}+60 \textit {\_Z}^{2}+1\right )^{2} \sqrt {x^{4}-x^{2}-1}\, x +60 \operatorname {RootOf}\left (400 \textit {\_Z}^{4}+60 \textit {\_Z}^{2}+1\right )^{3}+2 \operatorname {RootOf}\left (400 \textit {\_Z}^{4}+60 \textit {\_Z}^{2}+1\right ) x^{2}+x \sqrt {x^{4}-x^{2}-1}+2 \operatorname {RootOf}\left (400 \textit {\_Z}^{4}+60 \textit {\_Z}^{2}+1\right )}{20 x^{2} \operatorname {RootOf}\left (400 \textit {\_Z}^{4}+60 \textit {\_Z}^{2}+1\right )^{2}-x^{4}+x^{2}+1}\right )+3 \operatorname {RootOf}\left (400 \textit {\_Z}^{4}+60 \textit {\_Z}^{2}+1\right ) \ln \left (\frac {1400 x^{2} \operatorname {RootOf}\left (400 \textit {\_Z}^{4}+60 \textit {\_Z}^{2}+1\right )^{5}-70 \operatorname {RootOf}\left (400 \textit {\_Z}^{4}+60 \textit {\_Z}^{2}+1\right )^{3} x^{4}+460 \operatorname {RootOf}\left (400 \textit {\_Z}^{4}+60 \textit {\_Z}^{2}+1\right )^{3} x^{2}-9 \operatorname {RootOf}\left (400 \textit {\_Z}^{4}+60 \textit {\_Z}^{2}+1\right ) x^{4}+30 \operatorname {RootOf}\left (400 \textit {\_Z}^{4}+60 \textit {\_Z}^{2}+1\right )^{2} \sqrt {x^{4}-x^{2}-1}\, x +70 \operatorname {RootOf}\left (400 \textit {\_Z}^{4}+60 \textit {\_Z}^{2}+1\right )^{3}+36 \operatorname {RootOf}\left (400 \textit {\_Z}^{4}+60 \textit {\_Z}^{2}+1\right ) x^{2}+4 x \sqrt {x^{4}-x^{2}-1}+9 \operatorname {RootOf}\left (400 \textit {\_Z}^{4}+60 \textit {\_Z}^{2}+1\right )}{20 x^{2} \operatorname {RootOf}\left (400 \textit {\_Z}^{4}+60 \textit {\_Z}^{2}+1\right )^{2}+x^{4}+2 x^{2}-1}\right )\) | \(672\) |
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Leaf count of result is larger than twice the leaf count of optimal. 575 vs. \(2 (61) = 122\).
Time = 0.35 (sec) , antiderivative size = 575, normalized size of antiderivative = 6.18 \[ \int \frac {\left (1+x^4\right ) \sqrt {-1-x^2+x^4}}{1-x^2-3 x^4+x^6+x^8} \, dx=-\frac {1}{40} \, \sqrt {10} \sqrt {\sqrt {5} - 3} \log \left (\frac {\sqrt {10} {\left (5 \, x^{8} - 25 \, x^{6} + 5 \, x^{4} + 25 \, x^{2} - \sqrt {5} {\left (3 \, x^{8} - 11 \, x^{6} - x^{4} + 11 \, x^{2} + 3\right )} + 5\right )} \sqrt {\sqrt {5} - 3} + 20 \, {\left (7 \, x^{5} - 4 \, x^{3} - \sqrt {5} {\left (3 \, x^{5} - 2 \, x^{3} - 3 \, x\right )} - 7 \, x\right )} \sqrt {x^{4} - x^{2} - 1}}{x^{8} + x^{6} - 3 \, x^{4} - x^{2} + 1}\right ) + \frac {1}{40} \, \sqrt {10} \sqrt {\sqrt {5} - 3} \log \left (-\frac {\sqrt {10} {\left (5 \, x^{8} - 25 \, x^{6} + 5 \, x^{4} + 25 \, x^{2} - \sqrt {5} {\left (3 \, x^{8} - 11 \, x^{6} - x^{4} + 11 \, x^{2} + 3\right )} + 5\right )} \sqrt {\sqrt {5} - 3} - 20 \, {\left (7 \, x^{5} - 4 \, x^{3} - \sqrt {5} {\left (3 \, x^{5} - 2 \, x^{3} - 3 \, x\right )} - 7 \, x\right )} \sqrt {x^{4} - x^{2} - 1}}{x^{8} + x^{6} - 3 \, x^{4} - x^{2} + 1}\right ) - \frac {1}{40} \, \sqrt {10} \sqrt {-\sqrt {5} - 3} \log \left (\frac {\sqrt {10} {\left (5 \, x^{8} - 25 \, x^{6} + 5 \, x^{4} + 25 \, x^{2} + \sqrt {5} {\left (3 \, x^{8} - 11 \, x^{6} - x^{4} + 11 \, x^{2} + 3\right )} + 5\right )} \sqrt {-\sqrt {5} - 3} + 20 \, {\left (7 \, x^{5} - 4 \, x^{3} + \sqrt {5} {\left (3 \, x^{5} - 2 \, x^{3} - 3 \, x\right )} - 7 \, x\right )} \sqrt {x^{4} - x^{2} - 1}}{x^{8} + x^{6} - 3 \, x^{4} - x^{2} + 1}\right ) + \frac {1}{40} \, \sqrt {10} \sqrt {-\sqrt {5} - 3} \log \left (-\frac {\sqrt {10} {\left (5 \, x^{8} - 25 \, x^{6} + 5 \, x^{4} + 25 \, x^{2} + \sqrt {5} {\left (3 \, x^{8} - 11 \, x^{6} - x^{4} + 11 \, x^{2} + 3\right )} + 5\right )} \sqrt {-\sqrt {5} - 3} - 20 \, {\left (7 \, x^{5} - 4 \, x^{3} + \sqrt {5} {\left (3 \, x^{5} - 2 \, x^{3} - 3 \, x\right )} - 7 \, x\right )} \sqrt {x^{4} - x^{2} - 1}}{x^{8} + x^{6} - 3 \, x^{4} - x^{2} + 1}\right ) \]
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Timed out. \[ \int \frac {\left (1+x^4\right ) \sqrt {-1-x^2+x^4}}{1-x^2-3 x^4+x^6+x^8} \, dx=\text {Timed out} \]
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\[ \int \frac {\left (1+x^4\right ) \sqrt {-1-x^2+x^4}}{1-x^2-3 x^4+x^6+x^8} \, dx=\int { \frac {\sqrt {x^{4} - x^{2} - 1} {\left (x^{4} + 1\right )}}{x^{8} + x^{6} - 3 \, x^{4} - x^{2} + 1} \,d x } \]
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\[ \int \frac {\left (1+x^4\right ) \sqrt {-1-x^2+x^4}}{1-x^2-3 x^4+x^6+x^8} \, dx=\int { \frac {\sqrt {x^{4} - x^{2} - 1} {\left (x^{4} + 1\right )}}{x^{8} + x^{6} - 3 \, x^{4} - x^{2} + 1} \,d x } \]
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Timed out. \[ \int \frac {\left (1+x^4\right ) \sqrt {-1-x^2+x^4}}{1-x^2-3 x^4+x^6+x^8} \, dx=\int \frac {\left (x^4+1\right )\,\sqrt {x^4-x^2-1}}{x^8+x^6-3\,x^4-x^2+1} \,d x \]
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