Integrand size = 29, antiderivative size = 14 \[ \int \frac {-1+2 x^6}{\sqrt {1+x^6} \left (1-x^2+x^6\right )} \, dx=-\text {arctanh}\left (\frac {x}{\sqrt {1+x^6}}\right ) \]
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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 = 3.17 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00 \[ \int \frac {-1+2 x^6}{\sqrt {1+x^6} \left (1-x^2+x^6\right )} \, dx=-\text {arctanh}\left (\frac {x}{\sqrt {1+x^6}}\right ) \]
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Time = 1.19 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.07
method | result | size |
pseudoelliptic | \(-\operatorname {arctanh}\left (\frac {\sqrt {x^{6}+1}}{x}\right )\) | \(15\) |
trager | \(\frac {\ln \left (-\frac {-x^{6}+2 x \sqrt {x^{6}+1}-x^{2}-1}{x^{6}-x^{2}+1}\right )}{2}\) | \(40\) |
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Leaf count of result is larger than twice the leaf count of optimal. 34 vs. \(2 (12) = 24\).
Time = 0.27 (sec) , antiderivative size = 34, normalized size of antiderivative = 2.43 \[ \int \frac {-1+2 x^6}{\sqrt {1+x^6} \left (1-x^2+x^6\right )} \, dx=\frac {1}{2} \, \log \left (\frac {x^{6} + x^{2} - 2 \, \sqrt {x^{6} + 1} x + 1}{x^{6} - x^{2} + 1}\right ) \]
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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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