Integrand size = 63, antiderivative size = 59 \[ \int \frac {\sqrt {-2+x^2-2 x^3+2 x^4} \left (2-x^3+2 x^4\right )}{\left (-1-x^3+x^4\right ) \left (-2-x^2-2 x^3+2 x^4\right )} \, dx=2 \text {arctanh}\left (\frac {x}{\sqrt {-2+x^2-2 x^3+2 x^4}}\right )-2 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} x}{\sqrt {-2+x^2-2 x^3+2 x^4}}\right ) \]
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\[ \int \frac {\sqrt {-2+x^2-2 x^3+2 x^4} \left (2-x^3+2 x^4\right )}{\left (-1-x^3+x^4\right ) \left (-2-x^2-2 x^3+2 x^4\right )} \, dx=\int \frac {\sqrt {-2+x^2-2 x^3+2 x^4} \left (2-x^3+2 x^4\right )}{\left (-1-x^3+x^4\right ) \left (-2-x^2-2 x^3+2 x^4\right )} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {x (-3+4 x) \sqrt {-2+x^2-2 x^3+2 x^4}}{-1-x^3+x^4}+\frac {2 \left (-1-3 x+4 x^2\right ) \sqrt {-2+x^2-2 x^3+2 x^4}}{-2-x^2-2 x^3+2 x^4}\right ) \, dx \\ & = 2 \int \frac {\left (-1-3 x+4 x^2\right ) \sqrt {-2+x^2-2 x^3+2 x^4}}{-2-x^2-2 x^3+2 x^4} \, dx-\int \frac {x (-3+4 x) \sqrt {-2+x^2-2 x^3+2 x^4}}{-1-x^3+x^4} \, dx \\ & = 2 \int \left (\frac {\sqrt {-2+x^2-2 x^3+2 x^4}}{2+x^2+2 x^3-2 x^4}-\frac {3 x \sqrt {-2+x^2-2 x^3+2 x^4}}{-2-x^2-2 x^3+2 x^4}+\frac {4 x^2 \sqrt {-2+x^2-2 x^3+2 x^4}}{-2-x^2-2 x^3+2 x^4}\right ) \, dx-\int \left (-\frac {3 x \sqrt {-2+x^2-2 x^3+2 x^4}}{-1-x^3+x^4}+\frac {4 x^2 \sqrt {-2+x^2-2 x^3+2 x^4}}{-1-x^3+x^4}\right ) \, dx \\ & = 2 \int \frac {\sqrt {-2+x^2-2 x^3+2 x^4}}{2+x^2+2 x^3-2 x^4} \, dx+3 \int \frac {x \sqrt {-2+x^2-2 x^3+2 x^4}}{-1-x^3+x^4} \, dx-4 \int \frac {x^2 \sqrt {-2+x^2-2 x^3+2 x^4}}{-1-x^3+x^4} \, dx-6 \int \frac {x \sqrt {-2+x^2-2 x^3+2 x^4}}{-2-x^2-2 x^3+2 x^4} \, dx+8 \int \frac {x^2 \sqrt {-2+x^2-2 x^3+2 x^4}}{-2-x^2-2 x^3+2 x^4} \, dx \\ \end{align*}
Time = 0.18 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.95 \[ \int \frac {\sqrt {-2+x^2-2 x^3+2 x^4} \left (2-x^3+2 x^4\right )}{\left (-1-x^3+x^4\right ) \left (-2-x^2-2 x^3+2 x^4\right )} \, dx=-2 \sqrt {2} \text {arctanh}\left (\frac {x}{\sqrt {-1+\frac {x^2}{2}-x^3+x^4}}\right )+2 \text {arctanh}\left (\frac {x}{\sqrt {-2+x^2-2 x^3+2 x^4}}\right ) \]
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Time = 10.01 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.44
method | result | size |
default | \(-2 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {2 x^{4}-2 x^{3}+x^{2}-2}\, \sqrt {2}}{2 x}\right )+\ln \left (\frac {\sqrt {2 x^{4}-2 x^{3}+x^{2}-2}+x}{x}\right )-\ln \left (\frac {\sqrt {2 x^{4}-2 x^{3}+x^{2}-2}-x}{x}\right )\) | \(85\) |
pseudoelliptic | \(-2 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {2 x^{4}-2 x^{3}+x^{2}-2}\, \sqrt {2}}{2 x}\right )+\ln \left (\frac {\sqrt {2 x^{4}-2 x^{3}+x^{2}-2}+x}{x}\right )-\ln \left (\frac {\sqrt {2 x^{4}-2 x^{3}+x^{2}-2}-x}{x}\right )\) | \(85\) |
trager | \(-\ln \left (-\frac {-x^{4}+x^{3}+\sqrt {2 x^{4}-2 x^{3}+x^{2}-2}\, x -x^{2}+1}{x^{4}-x^{3}-1}\right )+\operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right ) \ln \left (-\frac {-2 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right ) x^{4}+2 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right ) x^{3}-3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right ) x^{2}+4 \sqrt {2 x^{4}-2 x^{3}+x^{2}-2}\, x +2 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right )}{2 x^{4}-2 x^{3}-x^{2}-2}\right )\) | \(144\) |
elliptic | \(\text {Expression too large to display}\) | \(1160375\) |
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Leaf count of result is larger than twice the leaf count of optimal. 168 vs. \(2 (51) = 102\).
Time = 0.34 (sec) , antiderivative size = 168, normalized size of antiderivative = 2.85 \[ \int \frac {\sqrt {-2+x^2-2 x^3+2 x^4} \left (2-x^3+2 x^4\right )}{\left (-1-x^3+x^4\right ) \left (-2-x^2-2 x^3+2 x^4\right )} \, dx=\frac {1}{2} \, \sqrt {2} \log \left (-\frac {4 \, x^{8} - 8 \, x^{7} + 32 \, x^{6} - 28 \, x^{5} + 9 \, x^{4} + 8 \, x^{3} - 4 \, \sqrt {2} {\left (2 \, x^{5} - 2 \, x^{4} + 3 \, x^{3} - 2 \, x\right )} \sqrt {2 \, x^{4} - 2 \, x^{3} + x^{2} - 2} - 28 \, x^{2} + 4}{4 \, x^{8} - 8 \, x^{7} + 4 \, x^{5} - 7 \, x^{4} + 8 \, x^{3} + 4 \, x^{2} + 4}\right ) + \log \left (-\frac {x^{4} - x^{3} + x^{2} + \sqrt {2 \, x^{4} - 2 \, x^{3} + x^{2} - 2} x - 1}{x^{4} - x^{3} - 1}\right ) \]
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Timed out. \[ \int \frac {\sqrt {-2+x^2-2 x^3+2 x^4} \left (2-x^3+2 x^4\right )}{\left (-1-x^3+x^4\right ) \left (-2-x^2-2 x^3+2 x^4\right )} \, dx=\text {Timed out} \]
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\[ \int \frac {\sqrt {-2+x^2-2 x^3+2 x^4} \left (2-x^3+2 x^4\right )}{\left (-1-x^3+x^4\right ) \left (-2-x^2-2 x^3+2 x^4\right )} \, dx=\int { \frac {{\left (2 \, x^{4} - x^{3} + 2\right )} \sqrt {2 \, x^{4} - 2 \, x^{3} + x^{2} - 2}}{{\left (2 \, x^{4} - 2 \, x^{3} - x^{2} - 2\right )} {\left (x^{4} - x^{3} - 1\right )}} \,d x } \]
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\[ \int \frac {\sqrt {-2+x^2-2 x^3+2 x^4} \left (2-x^3+2 x^4\right )}{\left (-1-x^3+x^4\right ) \left (-2-x^2-2 x^3+2 x^4\right )} \, dx=\int { \frac {{\left (2 \, x^{4} - x^{3} + 2\right )} \sqrt {2 \, x^{4} - 2 \, x^{3} + x^{2} - 2}}{{\left (2 \, x^{4} - 2 \, x^{3} - x^{2} - 2\right )} {\left (x^{4} - x^{3} - 1\right )}} \,d x } \]
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Timed out. \[ \int \frac {\sqrt {-2+x^2-2 x^3+2 x^4} \left (2-x^3+2 x^4\right )}{\left (-1-x^3+x^4\right ) \left (-2-x^2-2 x^3+2 x^4\right )} \, dx=\int \frac {\left (2\,x^4-x^3+2\right )\,\sqrt {2\,x^4-2\,x^3+x^2-2}}{\left (-x^4+x^3+1\right )\,\left (-2\,x^4+2\,x^3+x^2+2\right )} \,d x \]
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