Integrand size = 49, antiderivative size = 55 \[ \int \frac {\sqrt {-1-x-x^2+x^4} \left (2+x+2 x^4\right )}{\left (-1-x+x^4\right ) \left (-1-x+x^2+x^4\right )} \, dx=2 \arctan \left (\frac {x}{\sqrt {-1-x-x^2+x^4}}\right )-2 \sqrt {2} \arctan \left (\frac {\sqrt {2} x}{\sqrt {-1-x-x^2+x^4}}\right ) \]
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\[ \int \frac {\sqrt {-1-x-x^2+x^4} \left (2+x+2 x^4\right )}{\left (-1-x+x^4\right ) \left (-1-x+x^2+x^4\right )} \, dx=\int \frac {\sqrt {-1-x-x^2+x^4} \left (2+x+2 x^4\right )}{\left (-1-x+x^4\right ) \left (-1-x+x^2+x^4\right )} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {\sqrt {-1-x-x^2+x^4}}{1-x}+\frac {\left (2-x+2 x^2\right ) \sqrt {-1-x-x^2+x^4}}{1+2 x+x^2+x^3}+\frac {\left (1+4 x^2-x^3\right ) \sqrt {-1-x-x^2+x^4}}{-1-x+x^4}\right ) \, dx \\ & = \int \frac {\sqrt {-1-x-x^2+x^4}}{1-x} \, dx+\int \frac {\left (2-x+2 x^2\right ) \sqrt {-1-x-x^2+x^4}}{1+2 x+x^2+x^3} \, dx+\int \frac {\left (1+4 x^2-x^3\right ) \sqrt {-1-x-x^2+x^4}}{-1-x+x^4} \, dx \\ & = \int \frac {\sqrt {-1-x-x^2+x^4}}{1-x} \, dx+\int \left (\frac {2 \sqrt {-1-x-x^2+x^4}}{1+2 x+x^2+x^3}-\frac {x \sqrt {-1-x-x^2+x^4}}{1+2 x+x^2+x^3}+\frac {2 x^2 \sqrt {-1-x-x^2+x^4}}{1+2 x+x^2+x^3}\right ) \, dx+\int \left (\frac {\sqrt {-1-x-x^2+x^4}}{-1-x+x^4}+\frac {4 x^2 \sqrt {-1-x-x^2+x^4}}{-1-x+x^4}-\frac {x^3 \sqrt {-1-x-x^2+x^4}}{-1-x+x^4}\right ) \, dx \\ & = 2 \int \frac {\sqrt {-1-x-x^2+x^4}}{1+2 x+x^2+x^3} \, dx+2 \int \frac {x^2 \sqrt {-1-x-x^2+x^4}}{1+2 x+x^2+x^3} \, dx+4 \int \frac {x^2 \sqrt {-1-x-x^2+x^4}}{-1-x+x^4} \, dx+\int \frac {\sqrt {-1-x-x^2+x^4}}{1-x} \, dx-\int \frac {x \sqrt {-1-x-x^2+x^4}}{1+2 x+x^2+x^3} \, dx+\int \frac {\sqrt {-1-x-x^2+x^4}}{-1-x+x^4} \, dx-\int \frac {x^3 \sqrt {-1-x-x^2+x^4}}{-1-x+x^4} \, dx \\ \end{align*}
Time = 0.18 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt {-1-x-x^2+x^4} \left (2+x+2 x^4\right )}{\left (-1-x+x^4\right ) \left (-1-x+x^2+x^4\right )} \, dx=2 \arctan \left (\frac {x}{\sqrt {-1-x-x^2+x^4}}\right )-2 \sqrt {2} \arctan \left (\frac {\sqrt {2} x}{\sqrt {-1-x-x^2+x^4}}\right ) \]
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Time = 21.19 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.96
method | result | size |
default | \(2 \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {x^{4}-x^{2}-x -1}}{2 x}\right )-2 \arctan \left (\frac {\sqrt {x^{4}-x^{2}-x -1}}{x}\right )\) | \(53\) |
pseudoelliptic | \(2 \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {x^{4}-x^{2}-x -1}}{2 x}\right )-2 \arctan \left (\frac {\sqrt {x^{4}-x^{2}-x -1}}{x}\right )\) | \(53\) |
trager | \(\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \ln \left (-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{4}-2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{2}+2 \sqrt {x^{4}-x^{2}-x -1}\, x -\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x -\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )}{x^{4}-x -1}\right )+\operatorname {RootOf}\left (\textit {\_Z}^{2}+2\right ) \ln \left (-\frac {-\operatorname {RootOf}\left (\textit {\_Z}^{2}+2\right ) x^{4}+3 \operatorname {RootOf}\left (\textit {\_Z}^{2}+2\right ) x^{2}+4 \sqrt {x^{4}-x^{2}-x -1}\, x +\operatorname {RootOf}\left (\textit {\_Z}^{2}+2\right ) x +\operatorname {RootOf}\left (\textit {\_Z}^{2}+2\right )}{\left (x -1\right ) \left (x^{3}+x^{2}+2 x +1\right )}\right )\) | \(162\) |
elliptic | \(\text {Expression too large to display}\) | \(788632\) |
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Time = 0.34 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.40 \[ \int \frac {\sqrt {-1-x-x^2+x^4} \left (2+x+2 x^4\right )}{\left (-1-x+x^4\right ) \left (-1-x+x^2+x^4\right )} \, dx=-\sqrt {2} \arctan \left (\frac {2 \, \sqrt {2} \sqrt {x^{4} - x^{2} - x - 1} x}{x^{4} - 3 \, x^{2} - x - 1}\right ) + \arctan \left (\frac {2 \, \sqrt {x^{4} - x^{2} - x - 1} x}{x^{4} - 2 \, x^{2} - x - 1}\right ) \]
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Timed out. \[ \int \frac {\sqrt {-1-x-x^2+x^4} \left (2+x+2 x^4\right )}{\left (-1-x+x^4\right ) \left (-1-x+x^2+x^4\right )} \, dx=\text {Timed out} \]
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\[ \int \frac {\sqrt {-1-x-x^2+x^4} \left (2+x+2 x^4\right )}{\left (-1-x+x^4\right ) \left (-1-x+x^2+x^4\right )} \, dx=\int { \frac {{\left (2 \, x^{4} + x + 2\right )} \sqrt {x^{4} - x^{2} - x - 1}}{{\left (x^{4} + x^{2} - x - 1\right )} {\left (x^{4} - x - 1\right )}} \,d x } \]
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\[ \int \frac {\sqrt {-1-x-x^2+x^4} \left (2+x+2 x^4\right )}{\left (-1-x+x^4\right ) \left (-1-x+x^2+x^4\right )} \, dx=\int { \frac {{\left (2 \, x^{4} + x + 2\right )} \sqrt {x^{4} - x^{2} - x - 1}}{{\left (x^{4} + x^{2} - x - 1\right )} {\left (x^{4} - x - 1\right )}} \,d x } \]
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Timed out. \[ \int \frac {\sqrt {-1-x-x^2+x^4} \left (2+x+2 x^4\right )}{\left (-1-x+x^4\right ) \left (-1-x+x^2+x^4\right )} \, dx=\int \frac {\left (2\,x^4+x+2\right )\,\sqrt {x^4-x^2-x-1}}{\left (-x^4+x+1\right )\,\left (-x^4-x^2+x+1\right )} \,d x \]
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