Integrand size = 25, antiderivative size = 17 \[ \int \frac {-1+x^2}{\left (1+x^2\right ) \sqrt {1+x^2+x^4}} \, dx=-\arctan \left (\frac {x}{\sqrt {1+x^2+x^4}}\right ) \]
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Time = 0.03 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {1712, 209} \[ \int \frac {-1+x^2}{\left (1+x^2\right ) \sqrt {1+x^2+x^4}} \, dx=-\arctan \left (\frac {x}{\sqrt {x^4+x^2+1}}\right ) \]
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Rule 209
Rule 1712
Rubi steps \begin{align*} \text {integral}& = -\text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\frac {x}{\sqrt {1+x^2+x^4}}\right ) \\ & = -\arctan \left (\frac {x}{\sqrt {1+x^2+x^4}}\right ) \\ \end{align*}
Time = 0.15 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.00 \[ \int \frac {-1+x^2}{\left (1+x^2\right ) \sqrt {1+x^2+x^4}} \, dx=-\arctan \left (\frac {x}{\sqrt {1+x^2+x^4}}\right ) \]
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Time = 3.15 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.94
method | result | size |
default | \(-\arctan \left (\frac {x}{\sqrt {x^{4}+x^{2}+1}}\right )\) | \(16\) |
elliptic | \(\arctan \left (\frac {\sqrt {x^{4}+x^{2}+1}}{x}\right )\) | \(16\) |
pseudoelliptic | \(-\arctan \left (\frac {x}{\sqrt {x^{4}+x^{2}+1}}\right )\) | \(16\) |
trager | \(\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \ln \left (\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x +\sqrt {x^{4}+x^{2}+1}}{x^{2}+1}\right )\) | \(36\) |
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none
Time = 0.28 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.88 \[ \int \frac {-1+x^2}{\left (1+x^2\right ) \sqrt {1+x^2+x^4}} \, dx=-\arctan \left (\frac {x}{\sqrt {x^{4} + x^{2} + 1}}\right ) \]
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\[ \int \frac {-1+x^2}{\left (1+x^2\right ) \sqrt {1+x^2+x^4}} \, dx=\int \frac {\left (x - 1\right ) \left (x + 1\right )}{\sqrt {\left (x^{2} - x + 1\right ) \left (x^{2} + x + 1\right )} \left (x^{2} + 1\right )}\, dx \]
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\[ \int \frac {-1+x^2}{\left (1+x^2\right ) \sqrt {1+x^2+x^4}} \, dx=\int { \frac {x^{2} - 1}{\sqrt {x^{4} + x^{2} + 1} {\left (x^{2} + 1\right )}} \,d x } \]
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\[ \int \frac {-1+x^2}{\left (1+x^2\right ) \sqrt {1+x^2+x^4}} \, dx=\int { \frac {x^{2} - 1}{\sqrt {x^{4} + x^{2} + 1} {\left (x^{2} + 1\right )}} \,d x } \]
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Timed out. \[ \int \frac {-1+x^2}{\left (1+x^2\right ) \sqrt {1+x^2+x^4}} \, dx=\int \frac {x^2-1}{\left (x^2+1\right )\,\sqrt {x^4+x^2+1}} \,d x \]
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