Integrand size = 18, antiderivative size = 43 \[ \int \frac {1+x^2}{1+4 x^2+x^4} \, dx=\frac {\arctan \left (\frac {x}{\sqrt {2-\sqrt {3}}}\right )}{\sqrt {6}}+\frac {\arctan \left (\frac {x}{\sqrt {2+\sqrt {3}}}\right )}{\sqrt {6}} \]
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Time = 0.03 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {1177, 209} \[ \int \frac {1+x^2}{1+4 x^2+x^4} \, dx=\frac {\arctan \left (\frac {x}{\sqrt {2-\sqrt {3}}}\right )}{\sqrt {6}}+\frac {\arctan \left (\frac {x}{\sqrt {2+\sqrt {3}}}\right )}{\sqrt {6}} \]
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Rule 209
Rule 1177
Rubi steps \begin{align*} \text {integral}& = \frac {1}{6} \left (3-\sqrt {3}\right ) \int \frac {1}{2-\sqrt {3}+x^2} \, dx+\frac {1}{6} \left (3+\sqrt {3}\right ) \int \frac {1}{2+\sqrt {3}+x^2} \, dx \\ & = \frac {\tan ^{-1}\left (\frac {x}{\sqrt {2-\sqrt {3}}}\right )}{\sqrt {6}}+\frac {\tan ^{-1}\left (\frac {x}{\sqrt {2+\sqrt {3}}}\right )}{\sqrt {6}} \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.88 \[ \int \frac {1+x^2}{1+4 x^2+x^4} \, dx=\frac {\left (-1+\sqrt {3}\right ) \arctan \left (\frac {x}{\sqrt {2-\sqrt {3}}}\right )}{2 \sqrt {3 \left (2-\sqrt {3}\right )}}+\frac {\left (1+\sqrt {3}\right ) \arctan \left (\frac {x}{\sqrt {2+\sqrt {3}}}\right )}{2 \sqrt {3 \left (2+\sqrt {3}\right )}} \]
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Time = 0.10 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.81
method | result | size |
risch | \(\frac {\sqrt {6}\, \arctan \left (\frac {x \sqrt {6}}{6}\right )}{6}+\frac {\sqrt {6}\, \arctan \left (\frac {x^{3} \sqrt {6}}{6}+\frac {5 x \sqrt {6}}{6}\right )}{6}\) | \(35\) |
default | \(\frac {\left (1+\sqrt {3}\right ) \sqrt {3}\, \arctan \left (\frac {2 x}{\sqrt {6}+\sqrt {2}}\right )}{3 \sqrt {6}+3 \sqrt {2}}+\frac {\left (\sqrt {3}-1\right ) \sqrt {3}\, \arctan \left (\frac {2 x}{\sqrt {6}-\sqrt {2}}\right )}{3 \sqrt {6}-3 \sqrt {2}}\) | \(70\) |
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none
Time = 0.24 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.72 \[ \int \frac {1+x^2}{1+4 x^2+x^4} \, dx=\frac {1}{6} \, \sqrt {6} \arctan \left (\frac {1}{6} \, \sqrt {6} {\left (x^{3} + 5 \, x\right )}\right ) + \frac {1}{6} \, \sqrt {6} \arctan \left (\frac {1}{6} \, \sqrt {6} x\right ) \]
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Time = 0.05 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.95 \[ \int \frac {1+x^2}{1+4 x^2+x^4} \, dx=\frac {\sqrt {6} \cdot \left (2 \operatorname {atan}{\left (\frac {\sqrt {6} x}{6} \right )} + 2 \operatorname {atan}{\left (\frac {\sqrt {6} x^{3}}{6} + \frac {5 \sqrt {6} x}{6} \right )}\right )}{12} \]
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\[ \int \frac {1+x^2}{1+4 x^2+x^4} \, dx=\int { \frac {x^{2} + 1}{x^{4} + 4 \, x^{2} + 1} \,d x } \]
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Time = 0.28 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.60 \[ \int \frac {1+x^2}{1+4 x^2+x^4} \, dx=\frac {1}{12} \, \sqrt {6} {\left (\pi \mathrm {sgn}\left (x\right ) + 2 \, \arctan \left (\frac {\sqrt {6} {\left (x^{2} - 1\right )}}{6 \, x}\right )\right )} \]
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Time = 0.09 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.67 \[ \int \frac {1+x^2}{1+4 x^2+x^4} \, dx=\frac {\sqrt {6}\,\left (\mathrm {atan}\left (\frac {\sqrt {6}\,x^3}{6}+\frac {5\,\sqrt {6}\,x}{6}\right )+\mathrm {atan}\left (\frac {\sqrt {6}\,x}{6}\right )\right )}{6} \]
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