Integrand size = 33, antiderivative size = 21 \[ \int \frac {1+2 x^4}{\left (-1+2 x^4\right ) \sqrt {-1-x^2+2 x^4}} \, dx=-\arctan \left (\frac {x}{\sqrt {-1-x^2+2 x^4}}\right ) \]
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Time = 0.11 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.061, Rules used = {2137, 210} \[ \int \frac {1+2 x^4}{\left (-1+2 x^4\right ) \sqrt {-1-x^2+2 x^4}} \, dx=-\arctan \left (\frac {x}{\sqrt {2 x^4-x^2-1}}\right ) \]
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Rule 210
Rule 2137
Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,\frac {x}{\sqrt {-1-x^2+2 x^4}}\right ) \\ & = -\arctan \left (\frac {x}{\sqrt {-1-x^2+2 x^4}}\right ) \\ \end{align*}
Time = 0.29 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.00 \[ \int \frac {1+2 x^4}{\left (-1+2 x^4\right ) \sqrt {-1-x^2+2 x^4}} \, dx=-\arctan \left (\frac {x}{\sqrt {-1-x^2+2 x^4}}\right ) \]
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Time = 2.82 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.95
method | result | size |
default | \(\arctan \left (\frac {\sqrt {2 x^{4}-x^{2}-1}}{x}\right )\) | \(20\) |
elliptic | \(\arctan \left (\frac {\sqrt {2 x^{4}-x^{2}-1}}{x}\right )\) | \(20\) |
pseudoelliptic | \(\arctan \left (\frac {\sqrt {2 x^{4}-x^{2}-1}}{x}\right )\) | \(20\) |
trager | \(-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \ln \left (-\frac {2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{4}-2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{2}+2 x \sqrt {2 x^{4}-x^{2}-1}-\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )}{2 x^{4}-1}\right )}{2}\) | \(69\) |
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none
Time = 0.28 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.62 \[ \int \frac {1+2 x^4}{\left (-1+2 x^4\right ) \sqrt {-1-x^2+2 x^4}} \, dx=-\frac {1}{2} \, \arctan \left (\frac {2 \, \sqrt {2 \, x^{4} - x^{2} - 1} x}{2 \, x^{4} - 2 \, x^{2} - 1}\right ) \]
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\[ \int \frac {1+2 x^4}{\left (-1+2 x^4\right ) \sqrt {-1-x^2+2 x^4}} \, dx=\int \frac {2 x^{4} + 1}{\sqrt {\left (x - 1\right ) \left (x + 1\right ) \left (2 x^{2} + 1\right )} \left (2 x^{4} - 1\right )}\, dx \]
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\[ \int \frac {1+2 x^4}{\left (-1+2 x^4\right ) \sqrt {-1-x^2+2 x^4}} \, dx=\int { \frac {2 \, x^{4} + 1}{\sqrt {2 \, x^{4} - x^{2} - 1} {\left (2 \, x^{4} - 1\right )}} \,d x } \]
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\[ \int \frac {1+2 x^4}{\left (-1+2 x^4\right ) \sqrt {-1-x^2+2 x^4}} \, dx=\int { \frac {2 \, x^{4} + 1}{\sqrt {2 \, x^{4} - x^{2} - 1} {\left (2 \, x^{4} - 1\right )}} \,d x } \]
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Timed out. \[ \int \frac {1+2 x^4}{\left (-1+2 x^4\right ) \sqrt {-1-x^2+2 x^4}} \, dx=\int \frac {2\,x^4+1}{\left (2\,x^4-1\right )\,\sqrt {2\,x^4-x^2-1}} \,d x \]
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