Integrand size = 31, antiderivative size = 17 \[ \int \frac {-1+2 x^2+2 x^4}{\left (1+2 x^2\right ) \sqrt {-1+x^6}} \, dx=\text {arctanh}\left (\frac {x \left (-1+x^2\right )}{\sqrt {-1+x^6}}\right ) \]
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\[ \int \frac {-1+2 x^2+2 x^4}{\left (1+2 x^2\right ) \sqrt {-1+x^6}} \, dx=\int \frac {-1+2 x^2+2 x^4}{\left (1+2 x^2\right ) \sqrt {-1+x^6}} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {1}{2 \sqrt {-1+x^6}}+\frac {x^2}{\sqrt {-1+x^6}}-\frac {3}{2 \left (1+2 x^2\right ) \sqrt {-1+x^6}}\right ) \, dx \\ & = \frac {1}{2} \int \frac {1}{\sqrt {-1+x^6}} \, dx-\frac {3}{2} \int \frac {1}{\left (1+2 x^2\right ) \sqrt {-1+x^6}} \, dx+\int \frac {x^2}{\sqrt {-1+x^6}} \, 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 )}{4 \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}}+\frac {1}{3} \text {Subst}\left (\int \frac {1}{\sqrt {-1+x^2}} \, dx,x,x^3\right )-\frac {3}{2} \int \left (\frac {i}{2 \left (i-\sqrt {2} x\right ) \sqrt {-1+x^6}}+\frac {i}{2 \left (i+\sqrt {2} x\right ) \sqrt {-1+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 )}{4 \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}}-\frac {3}{4} i \int \frac {1}{\left (i-\sqrt {2} x\right ) \sqrt {-1+x^6}} \, dx-\frac {3}{4} i \int \frac {1}{\left (i+\sqrt {2} x\right ) \sqrt {-1+x^6}} \, dx+\frac {1}{3} \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {x^3}{\sqrt {-1+x^6}}\right ) \\ & = \frac {1}{3} \text {arctanh}\left (\frac {x^3}{\sqrt {-1+x^6}}\right )+\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 )}{4 \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}}-\frac {3}{4} i \int \frac {1}{\left (i-\sqrt {2} x\right ) \sqrt {-1+x^6}} \, dx-\frac {3}{4} i \int \frac {1}{\left (i+\sqrt {2} x\right ) \sqrt {-1+x^6}} \, dx \\ \end{align*}
Time = 7.35 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.00 \[ \int \frac {-1+2 x^2+2 x^4}{\left (1+2 x^2\right ) \sqrt {-1+x^6}} \, dx=\text {arctanh}\left (\frac {x \left (-1+x^2\right )}{\sqrt {-1+x^6}}\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. \(31\) vs. \(2(15)=30\).
Time = 1.48 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.88
method | result | size |
trager | \(-\frac {\ln \left (-\frac {-2 x^{4}+2 x \sqrt {x^{6}-1}-1}{2 x^{2}+1}\right )}{2}\) | \(32\) |
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Leaf count of result is larger than twice the leaf count of optimal. 72 vs. \(2 (15) = 30\).
Time = 0.26 (sec) , antiderivative size = 72, normalized size of antiderivative = 4.24 \[ \int \frac {-1+2 x^2+2 x^4}{\left (1+2 x^2\right ) \sqrt {-1+x^6}} \, dx=\frac {1}{3} \, \log \left (x^{3} + \sqrt {x^{6} - 1}\right ) + \frac {1}{6} \, \log \left (-\frac {10 \, x^{6} + 6 \, x^{4} + 12 \, x^{2} + 6 \, \sqrt {x^{6} - 1} {\left (x^{3} - x\right )} - 1}{8 \, x^{6} + 12 \, x^{4} + 6 \, x^{2} + 1}\right ) \]
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\[ \int \frac {-1+2 x^2+2 x^4}{\left (1+2 x^2\right ) \sqrt {-1+x^6}} \, dx=\int \frac {2 x^{4} + 2 x^{2} - 1}{\sqrt {\left (x - 1\right ) \left (x + 1\right ) \left (x^{2} - x + 1\right ) \left (x^{2} + x + 1\right )} \left (2 x^{2} + 1\right )}\, dx \]
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\[ \int \frac {-1+2 x^2+2 x^4}{\left (1+2 x^2\right ) \sqrt {-1+x^6}} \, dx=\int { \frac {2 \, x^{4} + 2 \, x^{2} - 1}{\sqrt {x^{6} - 1} {\left (2 \, x^{2} + 1\right )}} \,d x } \]
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\[ \int \frac {-1+2 x^2+2 x^4}{\left (1+2 x^2\right ) \sqrt {-1+x^6}} \, dx=\int { \frac {2 \, x^{4} + 2 \, x^{2} - 1}{\sqrt {x^{6} - 1} {\left (2 \, x^{2} + 1\right )}} \,d x } \]
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Timed out. \[ \int \frac {-1+2 x^2+2 x^4}{\left (1+2 x^2\right ) \sqrt {-1+x^6}} \, dx=\int \frac {2\,x^4+2\,x^2-1}{\sqrt {x^6-1}\,\left (2\,x^2+1\right )} \,d x \]
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