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^2}{3 \left (1-x^2+x^4\right ) \sqrt {1-2 x^2+x^6}}\right ) \, 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-\int \frac {2-x^2}{\left (1-x^2+x^4\right ) \sqrt {1-2 x^2+x^6}} \, dx \\ & = 2 \int \frac {1}{\sqrt {1-2 x^2+x^6}} \, dx-\int \left (\frac {i}{2 (i-x) \sqrt {1-2 x^2+x^6}}+\frac {i}{2 (i+x) \sqrt {1-2 x^2+x^6}}\right ) \, dx-\int \left (\frac {-1-i \sqrt {3}}{\left (-1-i \sqrt {3}+2 x^2\right ) \sqrt {1-2 x^2+x^6}}+\frac {-1+i \sqrt {3}}{\left (-1+i \sqrt {3}+2 x^2\right ) \sqrt {1-2 x^2+x^6}}\right ) \, dx \\ & = -\left (\frac {1}{2} i \int \frac {1}{(i-x) \sqrt {1-2 x^2+x^6}} \, dx\right )-\frac {1}{2} i \int \frac {1}{(i+x) \sqrt {1-2 x^2+x^6}} \, dx+2 \int \frac {1}{\sqrt {1-2 x^2+x^6}} \, dx-\left (-1-i \sqrt {3}\right ) \int \frac {1}{\left (-1-i \sqrt {3}+2 x^2\right ) \sqrt {1-2 x^2+x^6}} \, dx-\left (-1+i \sqrt {3}\right ) \int \frac {1}{\left (-1+i \sqrt {3}+2 x^2\right ) \sqrt {1-2 x^2+x^6}} \, dx \\ & = -\left (\frac {1}{2} i \int \frac {1}{(i-x) \sqrt {1-2 x^2+x^6}} \, dx\right )-\frac {1}{2} i \int \frac {1}{(i+x) \sqrt {1-2 x^2+x^6}} \, dx+2 \int \frac {1}{\sqrt {1-2 x^2+x^6}} \, dx-\left (-1-i \sqrt {3}\right ) \int \left (\frac {\sqrt {1+i \sqrt {3}}}{2 \left (-1-i \sqrt {3}\right ) \left (\sqrt {1+i \sqrt {3}}-\sqrt {2} x\right ) \sqrt {1-2 x^2+x^6}}+\frac {\sqrt {1+i \sqrt {3}}}{2 \left (-1-i \sqrt {3}\right ) \left (\sqrt {1+i \sqrt {3}}+\sqrt {2} x\right ) \sqrt {1-2 x^2+x^6}}\right ) \, dx-\left (-1+i \sqrt {3}\right ) \int \left (\frac {\sqrt {1-i \sqrt {3}}}{2 \left (-1+i \sqrt {3}\right ) \left (\sqrt {1-i \sqrt {3}}-\sqrt {2} x\right ) \sqrt {1-2 x^2+x^6}}+\frac {\sqrt {1-i \sqrt {3}}}{2 \left (-1+i \sqrt {3}\right ) \left (\sqrt {1-i \sqrt {3}}+\sqrt {2} x\right ) \sqrt {1-2 x^2+x^6}}\right ) \, dx \\ & = -\left (\frac {1}{2} i \int \frac {1}{(i-x) \sqrt {1-2 x^2+x^6}} \, dx\right )-\frac {1}{2} i \int \frac {1}{(i+x) \sqrt {1-2 x^2+x^6}} \, dx+2 \int \frac {1}{\sqrt {1-2 x^2+x^6}} \, dx-\frac {1}{2} \sqrt {1-i \sqrt {3}} \int \frac {1}{\left (\sqrt {1-i \sqrt {3}}-\sqrt {2} x\right ) \sqrt {1-2 x^2+x^6}} \, dx-\frac {1}{2} \sqrt {1-i \sqrt {3}} \int \frac {1}{\left (\sqrt {1-i \sqrt {3}}+\sqrt {2} x\right ) \sqrt {1-2 x^2+x^6}} \, dx-\frac {1}{2} \sqrt {1+i \sqrt {3}} \int \frac {1}{\left (\sqrt {1+i \sqrt {3}}-\sqrt {2} x\right ) \sqrt {1-2 x^2+x^6}} \, dx-\frac {1}{2} \sqrt {1+i \sqrt {3}} \int \frac {1}{\left (\sqrt {1+i \sqrt {3}}+\sqrt {2} x\right ) \sqrt {1-2 x^2+x^6}} \, dx \\ \end{align*}
Time = 0.73 (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 = 2.90 (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 | \(-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+2\right ) \ln \left (-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+2\right ) x^{6}-4 \operatorname {RootOf}\left (\textit {\_Z}^{2}+2\right ) x^{2}+4 \sqrt {x^{6}-2 x^{2}+1}\, x +\operatorname {RootOf}\left (\textit {\_Z}^{2}+2\right )}{\left (x^{2}+1\right ) \left (x^{4}-x^{2}+1\right )}\right )}{4}\) | \(74\) |
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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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\[ \int \frac {-1+2 x^6}{\left (1+x^6\right ) \sqrt {1-2 x^2+x^6}} \, dx=\int \frac {2 x^{6} - 1}{\sqrt {\left (x - 1\right ) \left (x + 1\right ) \left (x^{4} + x^{2} - 1\right )} \left (x^{2} + 1\right ) \left (x^{4} - x^{2} + 1\right )}\, dx \]
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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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