Optimal. Leaf size=26 \[ -\frac {2+x}{2 \left (2+2 x+x^2\right )}-\frac {1}{2} \tan ^{-1}(1+x) \]
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Rubi [A]
time = 0.01, antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {652, 631, 210}
\begin {gather*} -\frac {1}{2} \text {ArcTan}(x+1)-\frac {x+2}{2 \left (x^2+2 x+2\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 210
Rule 631
Rule 652
Rubi steps
\begin {align*} \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} \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*}
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Mathematica [A]
time = 0.01, size = 28, normalized size = 1.08 \begin {gather*} \frac {-2-x}{2 \left (2+2 x+x^2\right )}-\frac {1}{2} \tan ^{-1}(1+x) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.87, size = 25, normalized size = 0.96
method | result | size |
risch | \(\frac {-\frac {x}{2}-1}{x^{2}+2 x +2}-\frac {\arctan \left (x +1\right )}{2}\) | \(24\) |
default | \(\frac {-2 x -4}{4 x^{2}+8 x +8}-\frac {\arctan \left (x +1\right )}{2}\) | \(25\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.51, size = 22, normalized size = 0.85 \begin {gather*} -\frac {x + 2}{2 \, {\left (x^{2} + 2 \, x + 2\right )}} - \frac {1}{2} \, \arctan \left (x + 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 4.01, size = 28, normalized size = 1.08 \begin {gather*} -\frac {{\left (x^{2} + 2 \, x + 2\right )} \arctan \left (x + 1\right ) + x + 2}{2 \, {\left (x^{2} + 2 \, x + 2\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.04, size = 20, normalized size = 0.77 \begin {gather*} \frac {- x - 2}{2 x^{2} + 4 x + 4} - \frac {\operatorname {atan}{\left (x + 1 \right )}}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.13, size = 22, normalized size = 0.85 \begin {gather*} -\frac {x + 2}{2 \, {\left (x^{2} + 2 \, x + 2\right )}} - \frac {1}{2} \, \arctan \left (x + 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.04, size = 24, normalized size = 0.92 \begin {gather*} -\frac {\mathrm {atan}\left (x+1\right )}{2}-\frac {\frac {x}{2}+1}{x^2+2\,x+2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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