Optimal. Leaf size=23 \[ -\frac {x (2+x)}{2 \left (2+2 x+x^2\right )}+\tan ^{-1}(1+x) \]
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Rubi [A]
time = 0.01, antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {736, 631, 210}
\begin {gather*} \text {ArcTan}(x+1)-\frac {x (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 736
Rubi steps
\begin {align*} \int \frac {x^2}{\left (2+2 x+x^2\right )^2} \, dx &=-\frac {x (2+x)}{2 \left (2+2 x+x^2\right )}+\int \frac {1}{2+2 x+x^2} \, dx\\ &=-\frac {x (2+x)}{2 \left (2+2 x+x^2\right )}-\text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+x\right )\\ &=-\frac {x (2+x)}{2 \left (2+2 x+x^2\right )}+\tan ^{-1}(1+x)\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 15, normalized size = 0.65 \begin {gather*} \frac {1}{2+2 x+x^2}+\tan ^{-1}(1+x) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.10, size = 16, normalized size = 0.70
method | result | size |
default | \(\frac {1}{x^{2}+2 x +2}+\arctan \left (1+x \right )\) | \(16\) |
risch | \(\frac {1}{x^{2}+2 x +2}+\arctan \left (1+x \right )\) | \(16\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.81, size = 15, normalized size = 0.65 \begin {gather*} \frac {1}{x^{2} + 2 \, x + 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 = 1.05, size = 26, normalized size = 1.13 \begin {gather*} \frac {{\left (x^{2} + 2 \, x + 2\right )} \arctan \left (x + 1\right ) + 1}{x^{2} + 2 \, x + 2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.04, size = 14, normalized size = 0.61 \begin {gather*} \operatorname {atan}{\left (x + 1 \right )} + \frac {1}{x^{2} + 2 x + 2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.46, size = 15, normalized size = 0.65 \begin {gather*} \frac {1}{x^{2} + 2 \, x + 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 = 0.08, size = 15, normalized size = 0.65 \begin {gather*} \mathrm {atan}\left (x+1\right )+\frac {1}{x^2+2\,x+2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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