Integrand size = 27, antiderivative size = 14 \[ \int \frac {1+2 x^6}{\sqrt {-1+x^6} \left (-1+x^2+x^6\right )} \, dx=-\arctan \left (\frac {x}{\sqrt {-1+x^6}}\right ) \]
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\[ \int \frac {1+2 x^6}{\sqrt {-1+x^6} \left (-1+x^2+x^6\right )} \, dx=\int \frac {1+2 x^6}{\sqrt {-1+x^6} \left (-1+x^2+x^6\right )} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {2}{\sqrt {-1+x^6}}+\frac {3-2 x^2}{\sqrt {-1+x^6} \left (-1+x^2+x^6\right )}\right ) \, dx \\ & = 2 \int \frac {1}{\sqrt {-1+x^6}} \, dx+\int \frac {3-2 x^2}{\sqrt {-1+x^6} \left (-1+x^2+x^6\right )} \, dx \\ & = \frac {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 )}{\sqrt [4]{3} \sqrt {-\frac {x^2 \left (1-x^2\right )}{\left (1-\left (1+\sqrt {3}\right ) x^2\right )^2}} \sqrt {-1+x^6}}+\int \left (\frac {3}{\sqrt {-1+x^6} \left (-1+x^2+x^6\right )}-\frac {2 x^2}{\sqrt {-1+x^6} \left (-1+x^2+x^6\right )}\right ) \, dx \\ & = \frac {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 )}{\sqrt [4]{3} \sqrt {-\frac {x^2 \left (1-x^2\right )}{\left (1-\left (1+\sqrt {3}\right ) x^2\right )^2}} \sqrt {-1+x^6}}-2 \int \frac {x^2}{\sqrt {-1+x^6} \left (-1+x^2+x^6\right )} \, dx+3 \int \frac {1}{\sqrt {-1+x^6} \left (-1+x^2+x^6\right )} \, dx \\ \end{align*}
Time = 3.10 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00 \[ \int \frac {1+2 x^6}{\sqrt {-1+x^6} \left (-1+x^2+x^6\right )} \, dx=-\arctan \left (\frac {x}{\sqrt {-1+x^6}}\right ) \]
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Time = 2.70 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.93
method | result | size |
pseudoelliptic | \(\arctan \left (\frac {\sqrt {x^{6}-1}}{x}\right )\) | \(13\) |
trager | \(-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \ln \left (-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{6}-\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{2}+2 x \sqrt {x^{6}-1}-\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )}{x^{6}+x^{2}-1}\right )}{2}\) | \(62\) |
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Leaf count of result is larger than twice the leaf count of optimal. 25 vs. \(2 (12) = 24\).
Time = 0.28 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.79 \[ \int \frac {1+2 x^6}{\sqrt {-1+x^6} \left (-1+x^2+x^6\right )} \, dx=-\frac {1}{2} \, \arctan \left (\frac {2 \, \sqrt {x^{6} - 1} x}{x^{6} - x^{2} - 1}\right ) \]
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Timed out. \[ \int \frac {1+2 x^6}{\sqrt {-1+x^6} \left (-1+x^2+x^6\right )} \, dx=\text {Timed out} \]
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\[ \int \frac {1+2 x^6}{\sqrt {-1+x^6} \left (-1+x^2+x^6\right )} \, dx=\int { \frac {2 \, x^{6} + 1}{{\left (x^{6} + x^{2} - 1\right )} \sqrt {x^{6} - 1}} \,d x } \]
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\[ \int \frac {1+2 x^6}{\sqrt {-1+x^6} \left (-1+x^2+x^6\right )} \, dx=\int { \frac {2 \, x^{6} + 1}{{\left (x^{6} + x^{2} - 1\right )} \sqrt {x^{6} - 1}} \,d x } \]
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Timed out. \[ \int \frac {1+2 x^6}{\sqrt {-1+x^6} \left (-1+x^2+x^6\right )} \, dx=\int \frac {2\,x^6+1}{\sqrt {x^6-1}\,\left (x^6+x^2-1\right )} \,d x \]
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