Integrand size = 22, antiderivative size = 43 \[ \int \frac {1+3 x^2}{-1-2 x^2-9 x^4} \, dx=\frac {\arctan \left (\frac {1-3 x}{\sqrt {2}}\right )}{2 \sqrt {2}}-\frac {\arctan \left (\frac {1+3 x}{\sqrt {2}}\right )}{2 \sqrt {2}} \]
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Time = 0.02 (sec) , antiderivative size = 43, 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} \[ \int \frac {1+3 x^2}{-1-2 x^2-9 x^4} \, dx=\frac {\arctan \left (\frac {1-3 x}{\sqrt {2}}\right )}{2 \sqrt {2}}-\frac {\arctan \left (\frac {3 x+1}{\sqrt {2}}\right )}{2 \sqrt {2}} \]
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Rule 210
Rule 632
Rule 1175
Rubi steps \begin{align*} \text {integral}& = -\left (\frac {1}{6} \int \frac {1}{\frac {1}{3}-\frac {2 x}{3}+x^2} \, dx\right )-\frac {1}{6} \int \frac {1}{\frac {1}{3}+\frac {2 x}{3}+x^2} \, dx \\ & = \frac {1}{3} \text {Subst}\left (\int \frac {1}{-\frac {8}{9}-x^2} \, dx,x,-\frac {2}{3}+2 x\right )+\frac {1}{3} \text {Subst}\left (\int \frac {1}{-\frac {8}{9}-x^2} \, dx,x,\frac {2}{3}+2 x\right ) \\ & = \frac {\tan ^{-1}\left (\frac {1-3 x}{\sqrt {2}}\right )}{2 \sqrt {2}}-\frac {\tan ^{-1}\left (\frac {1+3 x}{\sqrt {2}}\right )}{2 \sqrt {2}} \\ \end{align*}
Result contains complex when optimal does not.
Time = 0.02 (sec) , antiderivative size = 99, normalized size of antiderivative = 2.30 \[ \int \frac {1+3 x^2}{-1-2 x^2-9 x^4} \, dx=-\frac {\left (-i+\sqrt {2}\right ) \arctan \left (\frac {3 x}{\sqrt {1-2 i \sqrt {2}}}\right )}{2 \sqrt {2 \left (1-2 i \sqrt {2}\right )}}-\frac {\left (i+\sqrt {2}\right ) \arctan \left (\frac {3 x}{\sqrt {1+2 i \sqrt {2}}}\right )}{2 \sqrt {2 \left (1+2 i \sqrt {2}\right )}} \]
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Time = 0.04 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.79
method | result | size |
default | \(-\frac {\sqrt {2}\, \arctan \left (\frac {\left (6 x +2\right ) \sqrt {2}}{4}\right )}{4}-\frac {\sqrt {2}\, \arctan \left (\frac {\left (6 x -2\right ) \sqrt {2}}{4}\right )}{4}\) | \(34\) |
risch | \(-\frac {\sqrt {2}\, \arctan \left (\frac {3 x \sqrt {2}}{4}\right )}{4}-\frac {\sqrt {2}\, \arctan \left (\frac {9 x^{3} \sqrt {2}}{4}+\frac {5 x \sqrt {2}}{4}\right )}{4}\) | \(35\) |
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Time = 0.26 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.77 \[ \int \frac {1+3 x^2}{-1-2 x^2-9 x^4} \, dx=-\frac {1}{4} \, \sqrt {2} \arctan \left (\frac {1}{4} \, \sqrt {2} {\left (9 \, x^{3} + 5 \, x\right )}\right ) - \frac {1}{4} \, \sqrt {2} \arctan \left (\frac {3}{4} \, \sqrt {2} x\right ) \]
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Time = 0.07 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.07 \[ \int \frac {1+3 x^2}{-1-2 x^2-9 x^4} \, dx=- \frac {\sqrt {2} \cdot \left (2 \operatorname {atan}{\left (\frac {3 \sqrt {2} x}{4} \right )} + 2 \operatorname {atan}{\left (\frac {9 \sqrt {2} x^{3}}{4} + \frac {5 \sqrt {2} x}{4} \right )}\right )}{8} \]
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Time = 0.34 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.77 \[ \int \frac {1+3 x^2}{-1-2 x^2-9 x^4} \, dx=-\frac {1}{4} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (3 \, x + 1\right )}\right ) - \frac {1}{4} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (3 \, x - 1\right )}\right ) \]
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Time = 0.28 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.77 \[ \int \frac {1+3 x^2}{-1-2 x^2-9 x^4} \, dx=-\frac {1}{4} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (3 \, x + 1\right )}\right ) - \frac {1}{4} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (3 \, x - 1\right )}\right ) \]
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Time = 0.00 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.67 \[ \int \frac {1+3 x^2}{-1-2 x^2-9 x^4} \, dx=-\frac {\sqrt {2}\,\left (\mathrm {atan}\left (\frac {9\,\sqrt {2}\,x^3}{4}+\frac {5\,\sqrt {2}\,x}{4}\right )+\mathrm {atan}\left (\frac {3\,\sqrt {2}\,x}{4}\right )\right )}{4} \]
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