Integrand size = 35, antiderivative size = 63 \[ \int \frac {\sqrt {-1+x^2+x^5} \left (2+3 x^5\right )}{1+x^4-2 x^5+x^{10}} \, dx=-\sqrt {1+i} \arctan \left (\frac {\sqrt {-1-i} x}{\sqrt {-1+x^2+x^5}}\right )-\sqrt {1-i} \arctan \left (\frac {\sqrt {-1+i} x}{\sqrt {-1+x^2+x^5}}\right ) \]
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\[ \int \frac {\sqrt {-1+x^2+x^5} \left (2+3 x^5\right )}{1+x^4-2 x^5+x^{10}} \, dx=\int \frac {\sqrt {-1+x^2+x^5} \left (2+3 x^5\right )}{1+x^4-2 x^5+x^{10}} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {2 \sqrt {-1+x^2+x^5}}{1+x^4-2 x^5+x^{10}}+\frac {3 x^5 \sqrt {-1+x^2+x^5}}{1+x^4-2 x^5+x^{10}}\right ) \, dx \\ & = 2 \int \frac {\sqrt {-1+x^2+x^5}}{1+x^4-2 x^5+x^{10}} \, dx+3 \int \frac {x^5 \sqrt {-1+x^2+x^5}}{1+x^4-2 x^5+x^{10}} \, dx \\ \end{align*}
Time = 1.52 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt {-1+x^2+x^5} \left (2+3 x^5\right )}{1+x^4-2 x^5+x^{10}} \, dx=-\sqrt {1+i} \arctan \left (\frac {\sqrt {-1-i} x}{\sqrt {-1+x^2+x^5}}\right )-\sqrt {1-i} \arctan \left (\frac {\sqrt {-1+i} x}{\sqrt {-1+x^2+x^5}}\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. \(269\) vs. \(2(51)=102\).
Time = 14.96 (sec) , antiderivative size = 270, normalized size of antiderivative = 4.29
method | result | size |
pseudoelliptic | \(\frac {-\sqrt {2}\, \ln \left (\frac {x^{5}+\sqrt {2}\, x^{2}+\sqrt {x^{5}+x^{2}-1}\, \sqrt {2+2 \sqrt {2}}\, x +x^{2}-1}{x^{2}}\right )+\sqrt {2}\, \ln \left (\frac {x^{5}+\sqrt {2}\, x^{2}-\sqrt {x^{5}+x^{2}-1}\, \sqrt {2+2 \sqrt {2}}\, x +x^{2}-1}{x^{2}}\right )-2 \arctan \left (\frac {\sqrt {2+2 \sqrt {2}}\, x -2 \sqrt {x^{5}+x^{2}-1}}{x \sqrt {-2+2 \sqrt {2}}}\right )+\ln \left (\frac {x^{5}+\sqrt {2}\, x^{2}+\sqrt {x^{5}+x^{2}-1}\, \sqrt {2+2 \sqrt {2}}\, x +x^{2}-1}{x^{2}}\right )+2 \arctan \left (\frac {\sqrt {2+2 \sqrt {2}}\, x +2 \sqrt {x^{5}+x^{2}-1}}{x \sqrt {-2+2 \sqrt {2}}}\right )-\ln \left (\frac {x^{5}+\sqrt {2}\, x^{2}-\sqrt {x^{5}+x^{2}-1}\, \sqrt {2+2 \sqrt {2}}\, x +x^{2}-1}{x^{2}}\right )}{2 \sqrt {-2+2 \sqrt {2}}}\) | \(270\) |
trager | \(\operatorname {RootOf}\left (8 \textit {\_Z}^{4}+4 \textit {\_Z}^{2}+1\right ) \ln \left (-\frac {-2 \operatorname {RootOf}\left (8 \textit {\_Z}^{4}+4 \textit {\_Z}^{2}+1\right )^{3} x^{5}+8 \operatorname {RootOf}\left (8 \textit {\_Z}^{4}+4 \textit {\_Z}^{2}+1\right )^{5} x^{2}-\operatorname {RootOf}\left (8 \textit {\_Z}^{4}+4 \textit {\_Z}^{2}+1\right ) x^{5}+2 \operatorname {RootOf}\left (8 \textit {\_Z}^{4}+4 \textit {\_Z}^{2}+1\right )^{3} x^{2}+4 \sqrt {x^{5}+x^{2}-1}\, \operatorname {RootOf}\left (8 \textit {\_Z}^{4}+4 \textit {\_Z}^{2}+1\right )^{2} x +2 \operatorname {RootOf}\left (8 \textit {\_Z}^{4}+4 \textit {\_Z}^{2}+1\right )^{3}-\operatorname {RootOf}\left (8 \textit {\_Z}^{4}+4 \textit {\_Z}^{2}+1\right ) x^{2}+\sqrt {x^{5}+x^{2}-1}\, x +\operatorname {RootOf}\left (8 \textit {\_Z}^{4}+4 \textit {\_Z}^{2}+1\right )}{x^{5}+4 \operatorname {RootOf}\left (8 \textit {\_Z}^{4}+4 \textit {\_Z}^{2}+1\right )^{2} x^{2}+x^{2}-1}\right )+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+4 \operatorname {RootOf}\left (8 \textit {\_Z}^{4}+4 \textit {\_Z}^{2}+1\right )^{2}+2\right ) \ln \left (-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+4 \operatorname {RootOf}\left (8 \textit {\_Z}^{4}+4 \textit {\_Z}^{2}+1\right )^{2}+2\right ) \operatorname {RootOf}\left (8 \textit {\_Z}^{4}+4 \textit {\_Z}^{2}+1\right )^{2} x^{5}+4 \operatorname {RootOf}\left (\textit {\_Z}^{2}+4 \operatorname {RootOf}\left (8 \textit {\_Z}^{4}+4 \textit {\_Z}^{2}+1\right )^{2}+2\right ) \operatorname {RootOf}\left (8 \textit {\_Z}^{4}+4 \textit {\_Z}^{2}+1\right )^{4} x^{2}+3 \operatorname {RootOf}\left (\textit {\_Z}^{2}+4 \operatorname {RootOf}\left (8 \textit {\_Z}^{4}+4 \textit {\_Z}^{2}+1\right )^{2}+2\right ) \operatorname {RootOf}\left (8 \textit {\_Z}^{4}+4 \textit {\_Z}^{2}+1\right )^{2} x^{2}-4 \sqrt {x^{5}+x^{2}-1}\, \operatorname {RootOf}\left (8 \textit {\_Z}^{4}+4 \textit {\_Z}^{2}+1\right )^{2} x -\operatorname {RootOf}\left (8 \textit {\_Z}^{4}+4 \textit {\_Z}^{2}+1\right )^{2} \operatorname {RootOf}\left (\textit {\_Z}^{2}+4 \operatorname {RootOf}\left (8 \textit {\_Z}^{4}+4 \textit {\_Z}^{2}+1\right )^{2}+2\right )-\sqrt {x^{5}+x^{2}-1}\, x}{-x^{5}+4 \operatorname {RootOf}\left (8 \textit {\_Z}^{4}+4 \textit {\_Z}^{2}+1\right )^{2} x^{2}+x^{2}+1}\right )}{2}\) | \(486\) |
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 301 vs. \(2 (43) = 86\).
Time = 0.29 (sec) , antiderivative size = 301, normalized size of antiderivative = 4.78 \[ \int \frac {\sqrt {-1+x^2+x^5} \left (2+3 x^5\right )}{1+x^4-2 x^5+x^{10}} \, dx=\frac {1}{4} \, \sqrt {-i - 1} \log \left (\frac {x^{10} + \left (2 i + 2\right ) \, x^{7} - 2 \, x^{5} + \left (2 i - 1\right ) \, x^{4} - 2 \, \sqrt {-i - 1} {\left (i \, x^{6} - x^{3} - i \, x\right )} \sqrt {x^{5} + x^{2} - 1} - \left (2 i + 2\right ) \, x^{2} + 1}{x^{10} - 2 \, x^{5} + x^{4} + 1}\right ) - \frac {1}{4} \, \sqrt {-i - 1} \log \left (\frac {x^{10} + \left (2 i + 2\right ) \, x^{7} - 2 \, x^{5} + \left (2 i - 1\right ) \, x^{4} - 2 \, \sqrt {-i - 1} {\left (-i \, x^{6} + x^{3} + i \, x\right )} \sqrt {x^{5} + x^{2} - 1} - \left (2 i + 2\right ) \, x^{2} + 1}{x^{10} - 2 \, x^{5} + x^{4} + 1}\right ) - \frac {1}{4} \, \sqrt {i - 1} \log \left (\frac {x^{10} - \left (2 i - 2\right ) \, x^{7} - 2 \, x^{5} - \left (2 i + 1\right ) \, x^{4} - 2 \, \sqrt {i - 1} {\left (i \, x^{6} + x^{3} - i \, x\right )} \sqrt {x^{5} + x^{2} - 1} + \left (2 i - 2\right ) \, x^{2} + 1}{x^{10} - 2 \, x^{5} + x^{4} + 1}\right ) + \frac {1}{4} \, \sqrt {i - 1} \log \left (\frac {x^{10} - \left (2 i - 2\right ) \, x^{7} - 2 \, x^{5} - \left (2 i + 1\right ) \, x^{4} - 2 \, \sqrt {i - 1} {\left (-i \, x^{6} - x^{3} + i \, x\right )} \sqrt {x^{5} + x^{2} - 1} + \left (2 i - 2\right ) \, x^{2} + 1}{x^{10} - 2 \, x^{5} + x^{4} + 1}\right ) \]
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\[ \int \frac {\sqrt {-1+x^2+x^5} \left (2+3 x^5\right )}{1+x^4-2 x^5+x^{10}} \, dx=\int \frac {\left (3 x^{5} + 2\right ) \sqrt {x^{5} + x^{2} - 1}}{x^{10} - 2 x^{5} + x^{4} + 1}\, dx \]
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\[ \int \frac {\sqrt {-1+x^2+x^5} \left (2+3 x^5\right )}{1+x^4-2 x^5+x^{10}} \, dx=\int { \frac {{\left (3 \, x^{5} + 2\right )} \sqrt {x^{5} + x^{2} - 1}}{x^{10} - 2 \, x^{5} + x^{4} + 1} \,d x } \]
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\[ \int \frac {\sqrt {-1+x^2+x^5} \left (2+3 x^5\right )}{1+x^4-2 x^5+x^{10}} \, dx=\int { \frac {{\left (3 \, x^{5} + 2\right )} \sqrt {x^{5} + x^{2} - 1}}{x^{10} - 2 \, x^{5} + x^{4} + 1} \,d x } \]
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Timed out. \[ \int \frac {\sqrt {-1+x^2+x^5} \left (2+3 x^5\right )}{1+x^4-2 x^5+x^{10}} \, dx=\int \frac {\left (3\,x^5+2\right )\,\sqrt {x^5+x^2-1}}{x^{10}-2\,x^5+x^4+1} \,d x \]
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