Integrand size = 19, antiderivative size = 22 \[ \int \frac {2+x+x^3}{1+2 x^2+x^4} \, dx=\frac {x}{1+x^2}+\arctan (x)+\frac {1}{2} \log \left (1+x^2\right ) \]
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Time = 0.01 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {28, 1828, 649, 209, 266} \[ \int \frac {2+x+x^3}{1+2 x^2+x^4} \, dx=\arctan (x)+\frac {x}{x^2+1}+\frac {1}{2} \log \left (x^2+1\right ) \]
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Rule 28
Rule 209
Rule 266
Rule 649
Rule 1828
Rubi steps \begin{align*} \text {integral}& = \int \frac {2+x+x^3}{\left (1+x^2\right )^2} \, dx \\ & = \frac {x}{1+x^2}-\frac {1}{2} \int \frac {-2-2 x}{1+x^2} \, dx \\ & = \frac {x}{1+x^2}+\int \frac {1}{1+x^2} \, dx+\int \frac {x}{1+x^2} \, dx \\ & = \frac {x}{1+x^2}+\tan ^{-1}(x)+\frac {1}{2} \log \left (1+x^2\right ) \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00 \[ \int \frac {2+x+x^3}{1+2 x^2+x^4} \, dx=\frac {x}{1+x^2}+\arctan (x)+\frac {1}{2} \log \left (1+x^2\right ) \]
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Time = 0.06 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.95
method | result | size |
default | \(\frac {x}{x^{2}+1}+\arctan \left (x \right )+\frac {\ln \left (x^{2}+1\right )}{2}\) | \(21\) |
risch | \(\frac {x}{x^{2}+1}+\arctan \left (x \right )+\frac {\ln \left (x^{2}+1\right )}{2}\) | \(21\) |
parallelrisch | \(\frac {i \ln \left (x +i\right ) x^{2}-i \ln \left (x -i\right ) x^{2}+\ln \left (x +i\right ) x^{2}+\ln \left (x -i\right ) x^{2}+i \ln \left (x +i\right )-i \ln \left (x -i\right )+\ln \left (x +i\right )+\ln \left (x -i\right )+2 x}{2 x^{2}+2}\) | \(80\) |
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Time = 0.27 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.55 \[ \int \frac {2+x+x^3}{1+2 x^2+x^4} \, dx=\frac {2 \, {\left (x^{2} + 1\right )} \arctan \left (x\right ) + {\left (x^{2} + 1\right )} \log \left (x^{2} + 1\right ) + 2 \, x}{2 \, {\left (x^{2} + 1\right )}} \]
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Time = 0.05 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.77 \[ \int \frac {2+x+x^3}{1+2 x^2+x^4} \, dx=\frac {x}{x^{2} + 1} + \frac {\log {\left (x^{2} + 1 \right )}}{2} + \operatorname {atan}{\left (x \right )} \]
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Time = 0.27 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.91 \[ \int \frac {2+x+x^3}{1+2 x^2+x^4} \, dx=\frac {x}{x^{2} + 1} + \arctan \left (x\right ) + \frac {1}{2} \, \log \left (x^{2} + 1\right ) \]
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Time = 0.29 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.91 \[ \int \frac {2+x+x^3}{1+2 x^2+x^4} \, dx=\frac {x}{x^{2} + 1} + \arctan \left (x\right ) + \frac {1}{2} \, \log \left (x^{2} + 1\right ) \]
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Time = 0.02 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.91 \[ \int \frac {2+x+x^3}{1+2 x^2+x^4} \, dx=\frac {\ln \left (x^2+1\right )}{2}+\mathrm {atan}\left (x\right )+\frac {x}{x^2+1} \]
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