Integrand size = 22, antiderivative size = 48 \[ \int \frac {1+2 x^2}{1+2 x^2+4 x^4} \, dx=-\frac {\arctan \left (\frac {1-2 \sqrt {2} x}{\sqrt {3}}\right )}{\sqrt {6}}+\frac {\arctan \left (\frac {1+2 \sqrt {2} x}{\sqrt {3}}\right )}{\sqrt {6}} \]
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Time = 0.03 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {1175, 632, 210} \[ \int \frac {1+2 x^2}{1+2 x^2+4 x^4} \, dx=\frac {\arctan \left (\frac {2 \sqrt {2} x+1}{\sqrt {3}}\right )}{\sqrt {6}}-\frac {\arctan \left (\frac {1-2 \sqrt {2} x}{\sqrt {3}}\right )}{\sqrt {6}} \]
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Rule 210
Rule 632
Rule 1175
Rubi steps \begin{align*} \text {integral}& = \frac {1}{4} \int \frac {1}{\frac {1}{2}-\frac {x}{\sqrt {2}}+x^2} \, dx+\frac {1}{4} \int \frac {1}{\frac {1}{2}+\frac {x}{\sqrt {2}}+x^2} \, dx \\ & = -\left (\frac {1}{2} \text {Subst}\left (\int \frac {1}{-\frac {3}{2}-x^2} \, dx,x,-\frac {1}{\sqrt {2}}+2 x\right )\right )-\frac {1}{2} \text {Subst}\left (\int \frac {1}{-\frac {3}{2}-x^2} \, dx,x,\frac {1}{\sqrt {2}}+2 x\right ) \\ & = -\frac {\tan ^{-1}\left (\frac {1-2 \sqrt {2} x}{\sqrt {3}}\right )}{\sqrt {6}}+\frac {\tan ^{-1}\left (\frac {1+2 \sqrt {2} x}{\sqrt {3}}\right )}{\sqrt {6}} \\ \end{align*}
Result contains complex when optimal does not.
Time = 0.07 (sec) , antiderivative size = 99, normalized size of antiderivative = 2.06 \[ \int \frac {1+2 x^2}{1+2 x^2+4 x^4} \, dx=\frac {\left (-i+\sqrt {3}\right ) \arctan \left (\frac {2 x}{\sqrt {1-i \sqrt {3}}}\right )}{2 \sqrt {3 \left (1-i \sqrt {3}\right )}}+\frac {\left (i+\sqrt {3}\right ) \arctan \left (\frac {2 x}{\sqrt {1+i \sqrt {3}}}\right )}{2 \sqrt {3 \left (1+i \sqrt {3}\right )}} \]
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Time = 0.08 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.73
method | result | size |
risch | \(\frac {\sqrt {6}\, \arctan \left (\frac {x \sqrt {6}}{3}\right )}{6}+\frac {\sqrt {6}\, \arctan \left (\frac {2 x^{3} \sqrt {6}}{3}+\frac {2 x \sqrt {6}}{3}\right )}{6}\) | \(35\) |
default | \(\frac {\sqrt {6}\, \arctan \left (\frac {\left (4 x -\sqrt {2}\right ) \sqrt {6}}{6}\right )}{6}+\frac {\sqrt {6}\, \arctan \left (\frac {\left (4 x +\sqrt {2}\right ) \sqrt {6}}{6}\right )}{6}\) | \(40\) |
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Time = 0.25 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.60 \[ \int \frac {1+2 x^2}{1+2 x^2+4 x^4} \, dx=\frac {1}{6} \, \sqrt {6} \arctan \left (\frac {2}{3} \, \sqrt {6} {\left (x^{3} + x\right )}\right ) + \frac {1}{6} \, \sqrt {6} \arctan \left (\frac {1}{3} \, \sqrt {6} x\right ) \]
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Time = 0.05 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.88 \[ \int \frac {1+2 x^2}{1+2 x^2+4 x^4} \, dx=\frac {\sqrt {6} \cdot \left (2 \operatorname {atan}{\left (\frac {\sqrt {6} x}{3} \right )} + 2 \operatorname {atan}{\left (\frac {2 \sqrt {6} x^{3}}{3} + \frac {2 \sqrt {6} x}{3} \right )}\right )}{12} \]
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\[ \int \frac {1+2 x^2}{1+2 x^2+4 x^4} \, dx=\int { \frac {2 \, x^{2} + 1}{4 \, x^{4} + 2 \, x^{2} + 1} \,d x } \]
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Time = 0.28 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.94 \[ \int \frac {1+2 x^2}{1+2 x^2+4 x^4} \, dx=\frac {1}{6} \, \sqrt {6} \arctan \left (\frac {4}{3} \, \sqrt {3} \left (\frac {1}{4}\right )^{\frac {3}{4}} {\left (2 \, x + \left (\frac {1}{4}\right )^{\frac {1}{4}}\right )}\right ) + \frac {1}{6} \, \sqrt {6} \arctan \left (\frac {4}{3} \, \sqrt {3} \left (\frac {1}{4}\right )^{\frac {3}{4}} {\left (2 \, x - \left (\frac {1}{4}\right )^{\frac {1}{4}}\right )}\right ) \]
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Time = 0.09 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.60 \[ \int \frac {1+2 x^2}{1+2 x^2+4 x^4} \, dx=\frac {\sqrt {6}\,\left (\mathrm {atan}\left (\frac {2\,\sqrt {6}\,x^3}{3}+\frac {2\,\sqrt {6}\,x}{3}\right )+\mathrm {atan}\left (\frac {\sqrt {6}\,x}{3}\right )\right )}{6} \]
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