Integrand size = 12, antiderivative size = 26 \[ \int \frac {x}{\left (2+2 x+x^2\right )^2} \, dx=-\frac {2+x}{2 \left (2+2 x+x^2\right )}-\frac {1}{2} \arctan (1+x) \]
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Time = 0.00 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {652, 631, 210} \[ \int \frac {x}{\left (2+2 x+x^2\right )^2} \, dx=-\frac {1}{2} \arctan (x+1)-\frac {x+2}{2 \left (x^2+2 x+2\right )} \]
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Rule 210
Rule 631
Rule 652
Rubi steps \begin{align*} \text {integral}& = -\frac {2+x}{2 \left (2+2 x+x^2\right )}-\frac {1}{2} \int \frac {1}{2+2 x+x^2} \, dx \\ & = -\frac {2+x}{2 \left (2+2 x+x^2\right )}+\frac {1}{2} \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+x\right ) \\ & = -\frac {2+x}{2 \left (2+2 x+x^2\right )}-\frac {1}{2} \tan ^{-1}(1+x) \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.08 \[ \int \frac {x}{\left (2+2 x+x^2\right )^2} \, dx=\frac {-2-x}{2 \left (2+2 x+x^2\right )}-\frac {1}{2} \arctan (1+x) \]
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Time = 0.24 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.92
method | result | size |
risch | \(\frac {-\frac {x}{2}-1}{x^{2}+2 x +2}-\frac {\arctan \left (1+x \right )}{2}\) | \(24\) |
default | \(\frac {-2 x -4}{4 x^{2}+8 x +8}-\frac {\arctan \left (1+x \right )}{2}\) | \(25\) |
parallelrisch | \(\frac {i \ln \left (x +1-i\right ) x^{2}-i \ln \left (x +1+i\right ) x^{2}+2 i \ln \left (x +1-i\right ) x -2 i \ln \left (x +1+i\right ) x -2+2 i \ln \left (x +1-i\right )-2 i \ln \left (x +1+i\right )+x^{2}}{4 x^{2}+8 x +8}\) | \(80\) |
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Time = 0.32 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.08 \[ \int \frac {x}{\left (2+2 x+x^2\right )^2} \, dx=-\frac {{\left (x^{2} + 2 \, x + 2\right )} \arctan \left (x + 1\right ) + x + 2}{2 \, {\left (x^{2} + 2 \, x + 2\right )}} \]
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Time = 0.05 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.77 \[ \int \frac {x}{\left (2+2 x+x^2\right )^2} \, dx=\frac {- x - 2}{2 x^{2} + 4 x + 4} - \frac {\operatorname {atan}{\left (x + 1 \right )}}{2} \]
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Time = 0.29 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.85 \[ \int \frac {x}{\left (2+2 x+x^2\right )^2} \, dx=-\frac {x + 2}{2 \, {\left (x^{2} + 2 \, x + 2\right )}} - \frac {1}{2} \, \arctan \left (x + 1\right ) \]
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Time = 0.27 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.85 \[ \int \frac {x}{\left (2+2 x+x^2\right )^2} \, dx=-\frac {x + 2}{2 \, {\left (x^{2} + 2 \, x + 2\right )}} - \frac {1}{2} \, \arctan \left (x + 1\right ) \]
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Time = 0.04 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.92 \[ \int \frac {x}{\left (2+2 x+x^2\right )^2} \, dx=-\frac {\mathrm {atan}\left (x+1\right )}{2}-\frac {\frac {x}{2}+1}{x^2+2\,x+2} \]
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