Integrand size = 21, antiderivative size = 21 \[ \int \frac {-3+2 x-3 x^2+x^3}{1+x^2} \, dx=-3 x+\frac {x^2}{2}+\frac {1}{2} \log \left (1+x^2\right ) \]
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Time = 0.01 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {1824, 266} \[ \int \frac {-3+2 x-3 x^2+x^3}{1+x^2} \, dx=\frac {x^2}{2}+\frac {1}{2} \log \left (x^2+1\right )-3 x \]
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Rule 266
Rule 1824
Rubi steps \begin{align*} \text {integral}& = \int \left (-3+x+\frac {x}{1+x^2}\right ) \, dx \\ & = -3 x+\frac {x^2}{2}+\int \frac {x}{1+x^2} \, dx \\ & = -3 x+\frac {x^2}{2}+\frac {1}{2} \log \left (1+x^2\right ) \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.00 \[ \int \frac {-3+2 x-3 x^2+x^3}{1+x^2} \, dx=-3 x+\frac {x^2}{2}+\frac {1}{2} \log \left (1+x^2\right ) \]
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Time = 0.78 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.86
method | result | size |
default | \(-3 x +\frac {x^{2}}{2}+\frac {\ln \left (x^{2}+1\right )}{2}\) | \(18\) |
norman | \(-3 x +\frac {x^{2}}{2}+\frac {\ln \left (x^{2}+1\right )}{2}\) | \(18\) |
meijerg | \(-3 x +\frac {x^{2}}{2}+\frac {\ln \left (x^{2}+1\right )}{2}\) | \(18\) |
risch | \(-3 x +\frac {x^{2}}{2}+\frac {\ln \left (x^{2}+1\right )}{2}\) | \(18\) |
parallelrisch | \(-3 x +\frac {x^{2}}{2}+\frac {\ln \left (x^{2}+1\right )}{2}\) | \(18\) |
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Time = 0.25 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.81 \[ \int \frac {-3+2 x-3 x^2+x^3}{1+x^2} \, dx=\frac {1}{2} \, x^{2} - 3 \, x + \frac {1}{2} \, \log \left (x^{2} + 1\right ) \]
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Time = 0.04 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.71 \[ \int \frac {-3+2 x-3 x^2+x^3}{1+x^2} \, dx=\frac {x^{2}}{2} - 3 x + \frac {\log {\left (x^{2} + 1 \right )}}{2} \]
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Time = 0.28 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.81 \[ \int \frac {-3+2 x-3 x^2+x^3}{1+x^2} \, dx=\frac {1}{2} \, x^{2} - 3 \, x + \frac {1}{2} \, \log \left (x^{2} + 1\right ) \]
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none
Time = 0.29 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.81 \[ \int \frac {-3+2 x-3 x^2+x^3}{1+x^2} \, dx=\frac {1}{2} \, x^{2} - 3 \, x + \frac {1}{2} \, \log \left (x^{2} + 1\right ) \]
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Time = 0.02 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.81 \[ \int \frac {-3+2 x-3 x^2+x^3}{1+x^2} \, dx=\frac {\ln \left (x^2+1\right )}{2}-3\,x+\frac {x^2}{2} \]
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