Integrand size = 29, antiderivative size = 29 \[ \int \frac {1+2 x^6}{\left (-1+x^6\right ) \sqrt {-1-2 x^2+x^6}} \, dx=-\frac {\arctan \left (\frac {\sqrt {2} x}{\sqrt {-1-2 x^2+x^6}}\right )}{\sqrt {2}} \]
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\[ \int \frac {1+2 x^6}{\left (-1+x^6\right ) \sqrt {-1-2 x^2+x^6}} \, dx=\int \frac {1+2 x^6}{\left (-1+x^6\right ) \sqrt {-1-2 x^2+x^6}} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {2}{\sqrt {-1-2 x^2+x^6}}+\frac {3}{\left (-1+x^6\right ) \sqrt {-1-2 x^2+x^6}}\right ) \, dx \\ & = 2 \int \frac {1}{\sqrt {-1-2 x^2+x^6}} \, dx+3 \int \frac {1}{\left (-1+x^6\right ) \sqrt {-1-2 x^2+x^6}} \, dx \\ & = 2 \int \frac {1}{\sqrt {-1-2 x^2+x^6}} \, dx+3 \int \left (\frac {1}{3 \left (-1+x^2\right ) \sqrt {-1-2 x^2+x^6}}+\frac {-2+x}{6 \left (1-x+x^2\right ) \sqrt {-1-2 x^2+x^6}}+\frac {-2-x}{6 \left (1+x+x^2\right ) \sqrt {-1-2 x^2+x^6}}\right ) \, dx \\ & = \frac {1}{2} \int \frac {-2+x}{\left (1-x+x^2\right ) \sqrt {-1-2 x^2+x^6}} \, dx+\frac {1}{2} \int \frac {-2-x}{\left (1+x+x^2\right ) \sqrt {-1-2 x^2+x^6}} \, dx+2 \int \frac {1}{\sqrt {-1-2 x^2+x^6}} \, dx+\int \frac {1}{\left (-1+x^2\right ) \sqrt {-1-2 x^2+x^6}} \, dx \\ & = \frac {1}{2} \int \left (\frac {1+i \sqrt {3}}{\left (-1-i \sqrt {3}+2 x\right ) \sqrt {-1-2 x^2+x^6}}+\frac {1-i \sqrt {3}}{\left (-1+i \sqrt {3}+2 x\right ) \sqrt {-1-2 x^2+x^6}}\right ) \, dx+\frac {1}{2} \int \left (\frac {-1+i \sqrt {3}}{\left (1-i \sqrt {3}+2 x\right ) \sqrt {-1-2 x^2+x^6}}+\frac {-1-i \sqrt {3}}{\left (1+i \sqrt {3}+2 x\right ) \sqrt {-1-2 x^2+x^6}}\right ) \, dx+2 \int \frac {1}{\sqrt {-1-2 x^2+x^6}} \, dx+\int \left (\frac {1}{2 (-1+x) \sqrt {-1-2 x^2+x^6}}-\frac {1}{2 (1+x) \sqrt {-1-2 x^2+x^6}}\right ) \, dx \\ & = \frac {1}{2} \int \frac {1}{(-1+x) \sqrt {-1-2 x^2+x^6}} \, dx-\frac {1}{2} \int \frac {1}{(1+x) \sqrt {-1-2 x^2+x^6}} \, dx+2 \int \frac {1}{\sqrt {-1-2 x^2+x^6}} \, dx+\frac {1}{2} \left (-1-i \sqrt {3}\right ) \int \frac {1}{\left (1+i \sqrt {3}+2 x\right ) \sqrt {-1-2 x^2+x^6}} \, dx+\frac {1}{2} \left (1-i \sqrt {3}\right ) \int \frac {1}{\left (-1+i \sqrt {3}+2 x\right ) \sqrt {-1-2 x^2+x^6}} \, dx+\frac {1}{2} \left (-1+i \sqrt {3}\right ) \int \frac {1}{\left (1-i \sqrt {3}+2 x\right ) \sqrt {-1-2 x^2+x^6}} \, dx+\frac {1}{2} \left (1+i \sqrt {3}\right ) \int \frac {1}{\left (-1-i \sqrt {3}+2 x\right ) \sqrt {-1-2 x^2+x^6}} \, dx \\ \end{align*}
Time = 0.74 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.00 \[ \int \frac {1+2 x^6}{\left (-1+x^6\right ) \sqrt {-1-2 x^2+x^6}} \, dx=-\frac {\arctan \left (\frac {\sqrt {2} x}{\sqrt {-1-2 x^2+x^6}}\right )}{\sqrt {2}} \]
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Time = 2.86 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.93
method | result | size |
pseudoelliptic | \(\frac {\sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {x^{6}-2 x^{2}-1}}{2 x}\right )}{2}\) | \(27\) |
trager | \(\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+2\right ) \ln \left (\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+2\right ) x^{6}-4 \operatorname {RootOf}\left (\textit {\_Z}^{2}+2\right ) x^{2}-4 \sqrt {x^{6}-2 x^{2}-1}\, x -\operatorname {RootOf}\left (\textit {\_Z}^{2}+2\right )}{\left (x -1\right ) \left (1+x \right ) \left (x^{2}+x +1\right ) \left (x^{2}-x +1\right )}\right )}{4}\) | \(84\) |
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Time = 0.28 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.24 \[ \int \frac {1+2 x^6}{\left (-1+x^6\right ) \sqrt {-1-2 x^2+x^6}} \, dx=-\frac {1}{4} \, \sqrt {2} \arctan \left (\frac {2 \, \sqrt {2} \sqrt {x^{6} - 2 \, x^{2} - 1} x}{x^{6} - 4 \, x^{2} - 1}\right ) \]
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\[ \int \frac {1+2 x^6}{\left (-1+x^6\right ) \sqrt {-1-2 x^2+x^6}} \, dx=\int \frac {2 x^{6} + 1}{\sqrt {\left (x^{2} + 1\right ) \left (x^{4} - x^{2} - 1\right )} \left (x - 1\right ) \left (x + 1\right ) \left (x^{2} - x + 1\right ) \left (x^{2} + x + 1\right )}\, dx \]
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\[ \int \frac {1+2 x^6}{\left (-1+x^6\right ) \sqrt {-1-2 x^2+x^6}} \, dx=\int { \frac {2 \, x^{6} + 1}{\sqrt {x^{6} - 2 \, x^{2} - 1} {\left (x^{6} - 1\right )}} \,d x } \]
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\[ \int \frac {1+2 x^6}{\left (-1+x^6\right ) \sqrt {-1-2 x^2+x^6}} \, dx=\int { \frac {2 \, x^{6} + 1}{\sqrt {x^{6} - 2 \, x^{2} - 1} {\left (x^{6} - 1\right )}} \,d x } \]
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Timed out. \[ \int \frac {1+2 x^6}{\left (-1+x^6\right ) \sqrt {-1-2 x^2+x^6}} \, dx=\int \frac {2\,x^6+1}{\left (x^6-1\right )\,\sqrt {x^6-2\,x^2-1}} \,d x \]
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