Integrand size = 34, antiderivative size = 30 \[ \int \frac {\sqrt {1-x^6} \left (1+2 x^6\right )}{x^2 \left (-1-x^2+x^6\right )} \, dx=\frac {\sqrt {1-x^6}}{x}+\arctan \left (\frac {x}{\sqrt {1-x^6}}\right ) \]
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\[ \int \frac {\sqrt {1-x^6} \left (1+2 x^6\right )}{x^2 \left (-1-x^2+x^6\right )} \, dx=\int \frac {\sqrt {1-x^6} \left (1+2 x^6\right )}{x^2 \left (-1-x^2+x^6\right )} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {\sqrt {1-x^6}}{x^2}+\frac {\left (-1+3 x^4\right ) \sqrt {1-x^6}}{-1-x^2+x^6}\right ) \, dx \\ & = -\int \frac {\sqrt {1-x^6}}{x^2} \, dx+\int \frac {\left (-1+3 x^4\right ) \sqrt {1-x^6}}{-1-x^2+x^6} \, dx \\ & = \frac {\sqrt {1-x^6}}{x}+3 \int \frac {x^4}{\sqrt {1-x^6}} \, dx+\int \left (\frac {\sqrt {1-x^6}}{1+x^2-x^6}+\frac {3 x^4 \sqrt {1-x^6}}{-1-x^2+x^6}\right ) \, dx \\ & = \frac {\sqrt {1-x^6}}{x}-\frac {3}{2} \int \frac {-1+\sqrt {3}-2 x^4}{\sqrt {1-x^6}} \, dx+3 \int \frac {x^4 \sqrt {1-x^6}}{-1-x^2+x^6} \, dx-\frac {1}{2} \left (3 \left (1-\sqrt {3}\right )\right ) \int \frac {1}{\sqrt {1-x^6}} \, dx+\int \frac {\sqrt {1-x^6}}{1+x^2-x^6} \, dx \\ & = \frac {\sqrt {1-x^6}}{x}+\frac {3 \left (1+\sqrt {3}\right ) x \sqrt {1-x^6}}{2 \left (1-\left (1+\sqrt {3}\right ) x^2\right )}-\frac {3 \sqrt [4]{3} x \left (1-x^2\right ) \sqrt {\frac {1+x^2+x^4}{\left (1-\left (1+\sqrt {3}\right ) x^2\right )^2}} E\left (\arccos \left (\frac {1-\left (1-\sqrt {3}\right ) x^2}{1-\left (1+\sqrt {3}\right ) x^2}\right )|\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{2 \sqrt {-\frac {x^2 \left (1-x^2\right )}{\left (1-\left (1+\sqrt {3}\right ) x^2\right )^2}} \sqrt {1-x^6}}-\frac {3^{3/4} \left (1-\sqrt {3}\right ) x \left (1-x^2\right ) \sqrt {\frac {1+x^2+x^4}{\left (1-\left (1+\sqrt {3}\right ) x^2\right )^2}} \operatorname {EllipticF}\left (\arccos \left (\frac {1-\left (1-\sqrt {3}\right ) x^2}{1-\left (1+\sqrt {3}\right ) x^2}\right ),\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{4 \sqrt {-\frac {x^2 \left (1-x^2\right )}{\left (1-\left (1+\sqrt {3}\right ) x^2\right )^2}} \sqrt {1-x^6}}+3 \int \frac {x^4 \sqrt {1-x^6}}{-1-x^2+x^6} \, dx+\int \frac {\sqrt {1-x^6}}{1+x^2-x^6} \, dx \\ \end{align*}
Time = 2.82 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt {1-x^6} \left (1+2 x^6\right )}{x^2 \left (-1-x^2+x^6\right )} \, dx=\frac {\sqrt {1-x^6}}{x}+\arctan \left (\frac {x}{\sqrt {1-x^6}}\right ) \]
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Time = 3.33 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.07
method | result | size |
pseudoelliptic | \(\frac {-\arctan \left (\frac {\sqrt {-x^{6}+1}}{x}\right ) x +\sqrt {-x^{6}+1}}{x}\) | \(32\) |
trager | \(\frac {\sqrt {-x^{6}+1}}{x}+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \ln \left (\frac {-\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{6}-\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{2}+2 \sqrt {-x^{6}+1}\, x +\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )}{x^{6}-x^{2}-1}\right )}{2}\) | \(78\) |
risch | \(-\frac {x^{6}-1}{x \sqrt {-x^{6}+1}}+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \ln \left (-\frac {-\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{6}-\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{2}+2 \sqrt {-x^{6}+1}\, x +\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )}{x^{6}-x^{2}-1}\right )}{2}\) | \(85\) |
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Time = 0.52 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.40 \[ \int \frac {\sqrt {1-x^6} \left (1+2 x^6\right )}{x^2 \left (-1-x^2+x^6\right )} \, dx=-\frac {x \arctan \left (\frac {2 \, \sqrt {-x^{6} + 1} x}{x^{6} + x^{2} - 1}\right ) - 2 \, \sqrt {-x^{6} + 1}}{2 \, x} \]
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Timed out. \[ \int \frac {\sqrt {1-x^6} \left (1+2 x^6\right )}{x^2 \left (-1-x^2+x^6\right )} \, dx=\text {Timed out} \]
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\[ \int \frac {\sqrt {1-x^6} \left (1+2 x^6\right )}{x^2 \left (-1-x^2+x^6\right )} \, dx=\int { \frac {{\left (2 \, x^{6} + 1\right )} \sqrt {-x^{6} + 1}}{{\left (x^{6} - x^{2} - 1\right )} x^{2}} \,d x } \]
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\[ \int \frac {\sqrt {1-x^6} \left (1+2 x^6\right )}{x^2 \left (-1-x^2+x^6\right )} \, dx=\int { \frac {{\left (2 \, x^{6} + 1\right )} \sqrt {-x^{6} + 1}}{{\left (x^{6} - x^{2} - 1\right )} x^{2}} \,d x } \]
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Timed out. \[ \int \frac {\sqrt {1-x^6} \left (1+2 x^6\right )}{x^2 \left (-1-x^2+x^6\right )} \, dx=\int -\frac {\sqrt {1-x^6}\,\left (2\,x^6+1\right )}{x^2\,\left (-x^6+x^2+1\right )} \,d x \]
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