Integrand size = 14, antiderivative size = 14 \[ \int \frac {1+x}{\left (2+2 x+x^2\right )^3} \, dx=-\frac {1}{4 \left (2+2 x+x^2\right )^2} \]
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Time = 0.02 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {643} \[ \int \frac {1+x}{\left (2+2 x+x^2\right )^3} \, dx=-\frac {1}{4 \left (x^2+2 x+2\right )^2} \]
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Rule 643
Rubi steps \begin{align*} \text {integral}& = -\frac {1}{4 \left (2+2 x+x^2\right )^2} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00 \[ \int \frac {1+x}{\left (2+2 x+x^2\right )^3} \, dx=-\frac {1}{4 \left (2+2 x+x^2\right )^2} \]
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Time = 0.26 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.93
method | result | size |
gosper | \(-\frac {1}{4 \left (x^{2}+2 x +2\right )^{2}}\) | \(13\) |
default | \(-\frac {1}{4 \left (x^{2}+2 x +2\right )^{2}}\) | \(13\) |
norman | \(-\frac {1}{4 \left (x^{2}+2 x +2\right )^{2}}\) | \(13\) |
risch | \(-\frac {1}{4 \left (x^{2}+2 x +2\right )^{2}}\) | \(13\) |
parallelrisch | \(-\frac {1}{4 \left (x^{2}+2 x +2\right )^{2}}\) | \(13\) |
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none
Time = 0.24 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.57 \[ \int \frac {1+x}{\left (2+2 x+x^2\right )^3} \, dx=-\frac {1}{4 \, {\left (x^{4} + 4 \, x^{3} + 8 \, x^{2} + 8 \, x + 4\right )}} \]
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Time = 0.04 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.57 \[ \int \frac {1+x}{\left (2+2 x+x^2\right )^3} \, dx=- \frac {1}{4 x^{4} + 16 x^{3} + 32 x^{2} + 32 x + 16} \]
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none
Time = 0.19 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.86 \[ \int \frac {1+x}{\left (2+2 x+x^2\right )^3} \, dx=-\frac {1}{4 \, {\left (x^{2} + 2 \, x + 2\right )}^{2}} \]
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none
Time = 0.27 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.86 \[ \int \frac {1+x}{\left (2+2 x+x^2\right )^3} \, dx=-\frac {1}{4 \, {\left (x^{2} + 2 \, x + 2\right )}^{2}} \]
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Time = 0.05 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.86 \[ \int \frac {1+x}{\left (2+2 x+x^2\right )^3} \, dx=-\frac {1}{4\,{\left (x^2+2\,x+2\right )}^2} \]
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