Integrand size = 69, antiderivative size = 123 \[ \int \frac {\left (-1-x-x^2+x^4\right ) \left (2+x+2 x^4\right )}{\sqrt {-1-x+x^4} \left (4+8 x+3 x^2-x^3-7 x^4-8 x^5+x^6+4 x^8\right )} \, dx=-(-1)^{3/4} \sqrt {\frac {1}{10} \left (-7 i+\sqrt {15}\right )} \arctan \left (\frac {\sqrt {\frac {1}{2}-\frac {i \sqrt {15}}{2}} x}{2 \sqrt {-1-x+x^4}}\right )+\frac {\sqrt [4]{-1} \sqrt {7 i+\sqrt {15}} \arctan \left (\frac {\sqrt {\frac {1}{2}+\frac {i \sqrt {15}}{2}} x}{2 \sqrt {-1-x+x^4}}\right )}{\sqrt {10}} \]
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\[ \int \frac {\left (-1-x-x^2+x^4\right ) \left (2+x+2 x^4\right )}{\sqrt {-1-x+x^4} \left (4+8 x+3 x^2-x^3-7 x^4-8 x^5+x^6+4 x^8\right )} \, dx=\int \frac {\left (-1-x-x^2+x^4\right ) \left (2+x+2 x^4\right )}{\sqrt {-1-x+x^4} \left (4+8 x+3 x^2-x^3-7 x^4-8 x^5+x^6+4 x^8\right )} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {1}{2 \sqrt {-1-x+x^4}}-\frac {8+14 x+9 x^2+x^3-7 x^4-6 x^5+5 x^6}{2 \sqrt {-1-x+x^4} \left (4+8 x+3 x^2-x^3-7 x^4-8 x^5+x^6+4 x^8\right )}\right ) \, dx \\ & = \frac {1}{2} \int \frac {1}{\sqrt {-1-x+x^4}} \, dx-\frac {1}{2} \int \frac {8+14 x+9 x^2+x^3-7 x^4-6 x^5+5 x^6}{\sqrt {-1-x+x^4} \left (4+8 x+3 x^2-x^3-7 x^4-8 x^5+x^6+4 x^8\right )} \, dx \\ & = \frac {1}{2} \int \frac {1}{\sqrt {-1-x+x^4}} \, dx-\frac {1}{2} \int \left (\frac {8}{\sqrt {-1-x+x^4} \left (4+8 x+3 x^2-x^3-7 x^4-8 x^5+x^6+4 x^8\right )}+\frac {14 x}{\sqrt {-1-x+x^4} \left (4+8 x+3 x^2-x^3-7 x^4-8 x^5+x^6+4 x^8\right )}+\frac {9 x^2}{\sqrt {-1-x+x^4} \left (4+8 x+3 x^2-x^3-7 x^4-8 x^5+x^6+4 x^8\right )}+\frac {x^3}{\sqrt {-1-x+x^4} \left (4+8 x+3 x^2-x^3-7 x^4-8 x^5+x^6+4 x^8\right )}-\frac {7 x^4}{\sqrt {-1-x+x^4} \left (4+8 x+3 x^2-x^3-7 x^4-8 x^5+x^6+4 x^8\right )}-\frac {6 x^5}{\sqrt {-1-x+x^4} \left (4+8 x+3 x^2-x^3-7 x^4-8 x^5+x^6+4 x^8\right )}+\frac {5 x^6}{\sqrt {-1-x+x^4} \left (4+8 x+3 x^2-x^3-7 x^4-8 x^5+x^6+4 x^8\right )}\right ) \, dx \\ & = \frac {1}{2} \int \frac {1}{\sqrt {-1-x+x^4}} \, dx-\frac {1}{2} \int \frac {x^3}{\sqrt {-1-x+x^4} \left (4+8 x+3 x^2-x^3-7 x^4-8 x^5+x^6+4 x^8\right )} \, dx-\frac {5}{2} \int \frac {x^6}{\sqrt {-1-x+x^4} \left (4+8 x+3 x^2-x^3-7 x^4-8 x^5+x^6+4 x^8\right )} \, dx+3 \int \frac {x^5}{\sqrt {-1-x+x^4} \left (4+8 x+3 x^2-x^3-7 x^4-8 x^5+x^6+4 x^8\right )} \, dx+\frac {7}{2} \int \frac {x^4}{\sqrt {-1-x+x^4} \left (4+8 x+3 x^2-x^3-7 x^4-8 x^5+x^6+4 x^8\right )} \, dx-4 \int \frac {1}{\sqrt {-1-x+x^4} \left (4+8 x+3 x^2-x^3-7 x^4-8 x^5+x^6+4 x^8\right )} \, dx-\frac {9}{2} \int \frac {x^2}{\sqrt {-1-x+x^4} \left (4+8 x+3 x^2-x^3-7 x^4-8 x^5+x^6+4 x^8\right )} \, dx-7 \int \frac {x}{\sqrt {-1-x+x^4} \left (4+8 x+3 x^2-x^3-7 x^4-8 x^5+x^6+4 x^8\right )} \, dx \\ \end{align*}
Time = 2.27 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.90 \[ \int \frac {\left (-1-x-x^2+x^4\right ) \left (2+x+2 x^4\right )}{\sqrt {-1-x+x^4} \left (4+8 x+3 x^2-x^3-7 x^4-8 x^5+x^6+4 x^8\right )} \, dx=\frac {1}{10} \left ((1-i) \sqrt {5 \left (-7 i+\sqrt {15}\right )} \arctan \left (\frac {\sqrt {\frac {1}{8}-\frac {i \sqrt {15}}{8}} x}{\sqrt {-1-x+x^4}}\right )+(1+i) \sqrt {5 \left (7 i+\sqrt {15}\right )} \arctan \left (\frac {\sqrt {\frac {1}{8}+\frac {i \sqrt {15}}{8}} x}{\sqrt {-1-x+x^4}}\right )\right ) \]
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Time = 8.75 (sec) , antiderivative size = 139, normalized size of antiderivative = 1.13
method | result | size |
default | \(\frac {\left (-\ln \left (\frac {2 x^{4}+\sqrt {x^{4}-x -1}\, \sqrt {3}\, x +x^{2}-2 x -2}{x^{2}}\right )+\ln \left (\frac {2 x^{4}-\sqrt {x^{4}-x -1}\, \sqrt {3}\, x +x^{2}-2 x -2}{x^{2}}\right )\right ) \sqrt {3}}{4}-\frac {\sqrt {5}\, \left (\arctan \left (\frac {\left (x \sqrt {3}+4 \sqrt {x^{4}-x -1}\right ) \sqrt {5}}{5 x}\right )-\arctan \left (\frac {\left (x \sqrt {3}-4 \sqrt {x^{4}-x -1}\right ) \sqrt {5}}{5 x}\right )\right )}{10}\) | \(139\) |
pseudoelliptic | \(\frac {\left (-\ln \left (\frac {2 x^{4}+\sqrt {x^{4}-x -1}\, \sqrt {3}\, x +x^{2}-2 x -2}{x^{2}}\right )+\ln \left (\frac {2 x^{4}-\sqrt {x^{4}-x -1}\, \sqrt {3}\, x +x^{2}-2 x -2}{x^{2}}\right )\right ) \sqrt {3}}{4}-\frac {\sqrt {5}\, \left (\arctan \left (\frac {\left (x \sqrt {3}+4 \sqrt {x^{4}-x -1}\right ) \sqrt {5}}{5 x}\right )-\arctan \left (\frac {\left (x \sqrt {3}-4 \sqrt {x^{4}-x -1}\right ) \sqrt {5}}{5 x}\right )\right )}{10}\) | \(139\) |
trager | \(-\operatorname {RootOf}\left (100 \textit {\_Z}^{4}-35 \textit {\_Z}^{2}+4\right ) \ln \left (\frac {50 \operatorname {RootOf}\left (100 \textit {\_Z}^{4}-35 \textit {\_Z}^{2}+4\right )^{4} x^{2}+10 \operatorname {RootOf}\left (100 \textit {\_Z}^{4}-35 \textit {\_Z}^{2}+4\right )^{2} x^{4}-25 x^{2} \operatorname {RootOf}\left (100 \textit {\_Z}^{4}-35 \textit {\_Z}^{2}+4\right )^{2}-3 x^{4}-10 \operatorname {RootOf}\left (100 \textit {\_Z}^{4}-35 \textit {\_Z}^{2}+4\right )^{2} x -5 \operatorname {RootOf}\left (100 \textit {\_Z}^{4}-35 \textit {\_Z}^{2}+4\right ) \sqrt {x^{4}-x -1}\, x -10 \operatorname {RootOf}\left (100 \textit {\_Z}^{4}-35 \textit {\_Z}^{2}+4\right )^{2}+3 x^{2}+3 x +3}{5 x^{2} \operatorname {RootOf}\left (100 \textit {\_Z}^{4}-35 \textit {\_Z}^{2}+4\right )^{2}-x^{4}-x^{2}+x +1}\right )-5 \ln \left (-\frac {400 \operatorname {RootOf}\left (100 \textit {\_Z}^{4}-35 \textit {\_Z}^{2}+4\right )^{5} x^{2}-80 \operatorname {RootOf}\left (100 \textit {\_Z}^{4}-35 \textit {\_Z}^{2}+4\right )^{3} x^{4}+120 \operatorname {RootOf}\left (100 \textit {\_Z}^{4}-35 \textit {\_Z}^{2}+4\right )^{3} x^{2}-36 \operatorname {RootOf}\left (100 \textit {\_Z}^{4}-35 \textit {\_Z}^{2}+4\right ) x^{4}+80 \operatorname {RootOf}\left (100 \textit {\_Z}^{4}-35 \textit {\_Z}^{2}+4\right )^{3} x -160 \operatorname {RootOf}\left (100 \textit {\_Z}^{4}-35 \textit {\_Z}^{2}+4\right )^{2} \sqrt {x^{4}-x -1}\, x +80 \operatorname {RootOf}\left (100 \textit {\_Z}^{4}-35 \textit {\_Z}^{2}+4\right )^{3}-27 \operatorname {RootOf}\left (100 \textit {\_Z}^{4}-35 \textit {\_Z}^{2}+4\right ) x^{2}+36 \operatorname {RootOf}\left (100 \textit {\_Z}^{4}-35 \textit {\_Z}^{2}+4\right ) x +56 \sqrt {x^{4}-x -1}\, x +36 \operatorname {RootOf}\left (100 \textit {\_Z}^{4}-35 \textit {\_Z}^{2}+4\right )}{20 x^{2} \operatorname {RootOf}\left (100 \textit {\_Z}^{4}-35 \textit {\_Z}^{2}+4\right )^{2}+4 x^{4}-3 x^{2}-4 x -4}\right ) \operatorname {RootOf}\left (100 \textit {\_Z}^{4}-35 \textit {\_Z}^{2}+4\right )^{3}+\frac {7 \ln \left (-\frac {400 \operatorname {RootOf}\left (100 \textit {\_Z}^{4}-35 \textit {\_Z}^{2}+4\right )^{5} x^{2}-80 \operatorname {RootOf}\left (100 \textit {\_Z}^{4}-35 \textit {\_Z}^{2}+4\right )^{3} x^{4}+120 \operatorname {RootOf}\left (100 \textit {\_Z}^{4}-35 \textit {\_Z}^{2}+4\right )^{3} x^{2}-36 \operatorname {RootOf}\left (100 \textit {\_Z}^{4}-35 \textit {\_Z}^{2}+4\right ) x^{4}+80 \operatorname {RootOf}\left (100 \textit {\_Z}^{4}-35 \textit {\_Z}^{2}+4\right )^{3} x -160 \operatorname {RootOf}\left (100 \textit {\_Z}^{4}-35 \textit {\_Z}^{2}+4\right )^{2} \sqrt {x^{4}-x -1}\, x +80 \operatorname {RootOf}\left (100 \textit {\_Z}^{4}-35 \textit {\_Z}^{2}+4\right )^{3}-27 \operatorname {RootOf}\left (100 \textit {\_Z}^{4}-35 \textit {\_Z}^{2}+4\right ) x^{2}+36 \operatorname {RootOf}\left (100 \textit {\_Z}^{4}-35 \textit {\_Z}^{2}+4\right ) x +56 \sqrt {x^{4}-x -1}\, x +36 \operatorname {RootOf}\left (100 \textit {\_Z}^{4}-35 \textit {\_Z}^{2}+4\right )}{20 x^{2} \operatorname {RootOf}\left (100 \textit {\_Z}^{4}-35 \textit {\_Z}^{2}+4\right )^{2}+4 x^{4}-3 x^{2}-4 x -4}\right ) \operatorname {RootOf}\left (100 \textit {\_Z}^{4}-35 \textit {\_Z}^{2}+4\right )}{4}\) | \(710\) |
elliptic | \(\text {Expression too large to display}\) | \(8121\) |
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Leaf count of result is larger than twice the leaf count of optimal. 807 vs. \(2 (76) = 152\).
Time = 0.55 (sec) , antiderivative size = 807, normalized size of antiderivative = 6.56 \[ \int \frac {\left (-1-x-x^2+x^4\right ) \left (2+x+2 x^4\right )}{\sqrt {-1-x+x^4} \left (4+8 x+3 x^2-x^3-7 x^4-8 x^5+x^6+4 x^8\right )} \, dx=-\frac {1}{40} \, \sqrt {10} \sqrt {\sqrt {5} \sqrt {-3} + 7} \log \left (-\frac {\sqrt {10} {\left (20 \, x^{8} + 15 \, x^{6} - 40 \, x^{5} - 45 \, x^{4} - 15 \, x^{3} - \sqrt {5} \sqrt {-3} {\left (4 \, x^{8} - 5 \, x^{6} - 8 \, x^{5} - 9 \, x^{4} + 5 \, x^{3} + 9 \, x^{2} + 8 \, x + 4\right )} + 5 \, x^{2} + 40 \, x + 20\right )} \sqrt {\sqrt {5} \sqrt {-3} + 7} + 40 \, {\left (7 \, x^{5} - x^{3} + \sqrt {5} \sqrt {-3} {\left (x^{5} + x^{3} - x^{2} - x\right )} - 7 \, x^{2} - 7 \, x\right )} \sqrt {x^{4} - x - 1}}{4 \, x^{8} + x^{6} - 8 \, x^{5} - 7 \, x^{4} - x^{3} + 3 \, x^{2} + 8 \, x + 4}\right ) + \frac {1}{40} \, \sqrt {10} \sqrt {\sqrt {5} \sqrt {-3} + 7} \log \left (\frac {\sqrt {10} {\left (20 \, x^{8} + 15 \, x^{6} - 40 \, x^{5} - 45 \, x^{4} - 15 \, x^{3} - \sqrt {5} \sqrt {-3} {\left (4 \, x^{8} - 5 \, x^{6} - 8 \, x^{5} - 9 \, x^{4} + 5 \, x^{3} + 9 \, x^{2} + 8 \, x + 4\right )} + 5 \, x^{2} + 40 \, x + 20\right )} \sqrt {\sqrt {5} \sqrt {-3} + 7} - 40 \, {\left (7 \, x^{5} - x^{3} + \sqrt {5} \sqrt {-3} {\left (x^{5} + x^{3} - x^{2} - x\right )} - 7 \, x^{2} - 7 \, x\right )} \sqrt {x^{4} - x - 1}}{4 \, x^{8} + x^{6} - 8 \, x^{5} - 7 \, x^{4} - x^{3} + 3 \, x^{2} + 8 \, x + 4}\right ) - \frac {1}{40} \, \sqrt {10} \sqrt {-\sqrt {5} \sqrt {-3} + 7} \log \left (-\frac {\sqrt {10} {\left (20 \, x^{8} + 15 \, x^{6} - 40 \, x^{5} - 45 \, x^{4} - 15 \, x^{3} + \sqrt {5} \sqrt {-3} {\left (4 \, x^{8} - 5 \, x^{6} - 8 \, x^{5} - 9 \, x^{4} + 5 \, x^{3} + 9 \, x^{2} + 8 \, x + 4\right )} + 5 \, x^{2} + 40 \, x + 20\right )} \sqrt {-\sqrt {5} \sqrt {-3} + 7} + 40 \, {\left (7 \, x^{5} - x^{3} - \sqrt {5} \sqrt {-3} {\left (x^{5} + x^{3} - x^{2} - x\right )} - 7 \, x^{2} - 7 \, x\right )} \sqrt {x^{4} - x - 1}}{4 \, x^{8} + x^{6} - 8 \, x^{5} - 7 \, x^{4} - x^{3} + 3 \, x^{2} + 8 \, x + 4}\right ) + \frac {1}{40} \, \sqrt {10} \sqrt {-\sqrt {5} \sqrt {-3} + 7} \log \left (\frac {\sqrt {10} {\left (20 \, x^{8} + 15 \, x^{6} - 40 \, x^{5} - 45 \, x^{4} - 15 \, x^{3} + \sqrt {5} \sqrt {-3} {\left (4 \, x^{8} - 5 \, x^{6} - 8 \, x^{5} - 9 \, x^{4} + 5 \, x^{3} + 9 \, x^{2} + 8 \, x + 4\right )} + 5 \, x^{2} + 40 \, x + 20\right )} \sqrt {-\sqrt {5} \sqrt {-3} + 7} - 40 \, {\left (7 \, x^{5} - x^{3} - \sqrt {5} \sqrt {-3} {\left (x^{5} + x^{3} - x^{2} - x\right )} - 7 \, x^{2} - 7 \, x\right )} \sqrt {x^{4} - x - 1}}{4 \, x^{8} + x^{6} - 8 \, x^{5} - 7 \, x^{4} - x^{3} + 3 \, x^{2} + 8 \, x + 4}\right ) \]
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\[ \int \frac {\left (-1-x-x^2+x^4\right ) \left (2+x+2 x^4\right )}{\sqrt {-1-x+x^4} \left (4+8 x+3 x^2-x^3-7 x^4-8 x^5+x^6+4 x^8\right )} \, dx=\int \frac {\left (x + 1\right ) \left (x^{3} - x^{2} - 1\right ) \left (2 x^{4} + x + 2\right )}{\sqrt {x^{4} - x - 1} \cdot \left (4 x^{8} + x^{6} - 8 x^{5} - 7 x^{4} - x^{3} + 3 x^{2} + 8 x + 4\right )}\, dx \]
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\[ \int \frac {\left (-1-x-x^2+x^4\right ) \left (2+x+2 x^4\right )}{\sqrt {-1-x+x^4} \left (4+8 x+3 x^2-x^3-7 x^4-8 x^5+x^6+4 x^8\right )} \, dx=\int { \frac {{\left (2 \, x^{4} + x + 2\right )} {\left (x^{4} - x^{2} - x - 1\right )}}{{\left (4 \, x^{8} + x^{6} - 8 \, x^{5} - 7 \, x^{4} - x^{3} + 3 \, x^{2} + 8 \, x + 4\right )} \sqrt {x^{4} - x - 1}} \,d x } \]
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\[ \int \frac {\left (-1-x-x^2+x^4\right ) \left (2+x+2 x^4\right )}{\sqrt {-1-x+x^4} \left (4+8 x+3 x^2-x^3-7 x^4-8 x^5+x^6+4 x^8\right )} \, dx=\int { \frac {{\left (2 \, x^{4} + x + 2\right )} {\left (x^{4} - x^{2} - x - 1\right )}}{{\left (4 \, x^{8} + x^{6} - 8 \, x^{5} - 7 \, x^{4} - x^{3} + 3 \, x^{2} + 8 \, x + 4\right )} \sqrt {x^{4} - x - 1}} \,d x } \]
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Timed out. \[ \int \frac {\left (-1-x-x^2+x^4\right ) \left (2+x+2 x^4\right )}{\sqrt {-1-x+x^4} \left (4+8 x+3 x^2-x^3-7 x^4-8 x^5+x^6+4 x^8\right )} \, dx=\int -\frac {\left (2\,x^4+x+2\right )\,\left (-x^4+x^2+x+1\right )}{\sqrt {x^4-x-1}\,\left (4\,x^8+x^6-8\,x^5-7\,x^4-x^3+3\,x^2+8\,x+4\right )} \,d x \]
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