Integrand size = 22, antiderivative size = 13 \[ \int \frac {2+x+x^2+x^3}{2+3 x^2+x^4} \, dx=\arctan (x)+\frac {1}{2} \log \left (2+x^2\right ) \]
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Time = 0.02 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {1687, 1163, 209, 1261, 640, 31} \[ \int \frac {2+x+x^2+x^3}{2+3 x^2+x^4} \, dx=\arctan (x)+\frac {1}{2} \log \left (x^2+2\right ) \]
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Rule 31
Rule 209
Rule 640
Rule 1163
Rule 1261
Rule 1687
Rubi steps \begin{align*} \text {integral}& = \int \frac {x \left (1+x^2\right )}{2+3 x^2+x^4} \, dx+\int \frac {2+x^2}{2+3 x^2+x^4} \, dx \\ & = \frac {1}{2} \text {Subst}\left (\int \frac {1+x}{2+3 x+x^2} \, dx,x,x^2\right )+\int \frac {1}{1+x^2} \, dx \\ & = \tan ^{-1}(x)+\frac {1}{2} \text {Subst}\left (\int \frac {1}{2+x} \, dx,x,x^2\right ) \\ & = \tan ^{-1}(x)+\frac {1}{2} \log \left (2+x^2\right ) \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.00 \[ \int \frac {2+x+x^2+x^3}{2+3 x^2+x^4} \, dx=\arctan (x)+\frac {1}{2} \log \left (2+x^2\right ) \]
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Time = 0.07 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.92
method | result | size |
default | \(\arctan \left (x \right )+\frac {\ln \left (x^{2}+2\right )}{2}\) | \(12\) |
risch | \(\arctan \left (x \right )+\frac {\ln \left (x^{2}+2\right )}{2}\) | \(12\) |
parallelrisch | \(\frac {i \ln \left (x +i\right )}{2}-\frac {i \ln \left (x -i\right )}{2}+\frac {\ln \left (x^{2}+2\right )}{2}\) | \(26\) |
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Time = 0.26 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.85 \[ \int \frac {2+x+x^2+x^3}{2+3 x^2+x^4} \, dx=\arctan \left (x\right ) + \frac {1}{2} \, \log \left (x^{2} + 2\right ) \]
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Time = 0.05 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.77 \[ \int \frac {2+x+x^2+x^3}{2+3 x^2+x^4} \, dx=\frac {\log {\left (x^{2} + 2 \right )}}{2} + \operatorname {atan}{\left (x \right )} \]
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Time = 0.26 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.85 \[ \int \frac {2+x+x^2+x^3}{2+3 x^2+x^4} \, dx=\arctan \left (x\right ) + \frac {1}{2} \, \log \left (x^{2} + 2\right ) \]
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Time = 0.32 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.85 \[ \int \frac {2+x+x^2+x^3}{2+3 x^2+x^4} \, dx=\arctan \left (x\right ) + \frac {1}{2} \, \log \left (x^{2} + 2\right ) \]
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Time = 0.02 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.85 \[ \int \frac {2+x+x^2+x^3}{2+3 x^2+x^4} \, dx=\frac {\ln \left (x^2+2\right )}{2}+\mathrm {atan}\left (x\right ) \]
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