Integrand size = 20, antiderivative size = 50 \[ \int \frac {1-x^2}{1+5 x^2+x^4} \, dx=-\frac {\arctan \left (\sqrt {\frac {2}{5+\sqrt {21}}} x\right )}{\sqrt {3}}+\frac {\arctan \left (\sqrt {\frac {1}{2} \left (5+\sqrt {21}\right )} x\right )}{\sqrt {3}} \]
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Time = 0.03 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {1177, 209} \[ \int \frac {1-x^2}{1+5 x^2+x^4} \, dx=\frac {\arctan \left (\sqrt {\frac {1}{2} \left (5+\sqrt {21}\right )} x\right )}{\sqrt {3}}-\frac {\arctan \left (\sqrt {\frac {2}{5+\sqrt {21}}} x\right )}{\sqrt {3}} \]
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Rule 209
Rule 1177
Rubi steps \begin{align*} \text {integral}& = \frac {1}{6} \left (-3+\sqrt {21}\right ) \int \frac {1}{\frac {5}{2}-\frac {\sqrt {21}}{2}+x^2} \, dx-\frac {1}{6} \left (3+\sqrt {21}\right ) \int \frac {1}{\frac {5}{2}+\frac {\sqrt {21}}{2}+x^2} \, dx \\ & = -\frac {\tan ^{-1}\left (\sqrt {\frac {2}{5+\sqrt {21}}} x\right )}{\sqrt {3}}+\frac {\tan ^{-1}\left (\sqrt {\frac {1}{2} \left (5+\sqrt {21}\right )} x\right )}{\sqrt {3}} \\ \end{align*}
Time = 0.07 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.74 \[ \int \frac {1-x^2}{1+5 x^2+x^4} \, dx=\frac {\left (7-\sqrt {21}\right ) \arctan \left (\sqrt {\frac {2}{5-\sqrt {21}}} x\right )}{\sqrt {42 \left (5-\sqrt {21}\right )}}+\frac {\left (-7-\sqrt {21}\right ) \arctan \left (\sqrt {\frac {2}{5+\sqrt {21}}} x\right )}{\sqrt {42 \left (5+\sqrt {21}\right )}} \]
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Time = 0.05 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.70
method | result | size |
risch | \(-\frac {\sqrt {3}\, \arctan \left (\frac {x \sqrt {3}}{3}\right )}{3}+\frac {\sqrt {3}\, \arctan \left (\frac {x^{3} \sqrt {3}}{3}+\frac {4 x \sqrt {3}}{3}\right )}{3}\) | \(35\) |
default | \(-\frac {2 \sqrt {21}\, \left (7+\sqrt {21}\right ) \arctan \left (\frac {4 x}{2 \sqrt {7}+2 \sqrt {3}}\right )}{21 \left (2 \sqrt {7}+2 \sqrt {3}\right )}-\frac {2 \left (-7+\sqrt {21}\right ) \sqrt {21}\, \arctan \left (\frac {4 x}{2 \sqrt {7}-2 \sqrt {3}}\right )}{21 \left (2 \sqrt {7}-2 \sqrt {3}\right )}\) | \(82\) |
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Time = 0.25 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.62 \[ \int \frac {1-x^2}{1+5 x^2+x^4} \, dx=\frac {1}{3} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (x^{3} + 4 \, x\right )}\right ) - \frac {1}{3} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} x\right ) \]
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Time = 0.05 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.84 \[ \int \frac {1-x^2}{1+5 x^2+x^4} \, dx=- \frac {\sqrt {3} \cdot \left (2 \operatorname {atan}{\left (\frac {\sqrt {3} x}{3} \right )} - 2 \operatorname {atan}{\left (\frac {\sqrt {3} x^{3}}{3} + \frac {4 \sqrt {3} x}{3} \right )}\right )}{6} \]
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\[ \int \frac {1-x^2}{1+5 x^2+x^4} \, dx=\int { -\frac {x^{2} - 1}{x^{4} + 5 \, x^{2} + 1} \,d x } \]
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Time = 0.28 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.52 \[ \int \frac {1-x^2}{1+5 x^2+x^4} \, dx=\frac {1}{6} \, \sqrt {3} {\left (\pi \mathrm {sgn}\left (x\right ) - 2 \, \arctan \left (\frac {\sqrt {3} {\left (x^{2} + 1\right )}}{3 \, x}\right )\right )} \]
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Time = 13.14 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.62 \[ \int \frac {1-x^2}{1+5 x^2+x^4} \, dx=\frac {\sqrt {3}\,\left (\mathrm {atan}\left (\frac {\sqrt {3}\,x^3}{3}+\frac {4\,\sqrt {3}\,x}{3}\right )-\mathrm {atan}\left (\frac {\sqrt {3}\,x}{3}\right )\right )}{3} \]
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