Optimal. Leaf size=23 \[ \tan ^{-1}(x+1)-\frac {x (x+2)}{2 \left (x^2+2 x+2\right )} \]
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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 = {722, 617, 204} \[ \tan ^{-1}(x+1)-\frac {x (x+2)}{2 \left (x^2+2 x+2\right )} \]
Antiderivative was successfully verified.
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Rule 204
Rule 617
Rule 722
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 )}-\operatorname {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 \[ \frac {1}{x^2+2 x+2}+\tan ^{-1}(x+1) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.40, size = 26, normalized size = 1.13 \[ \frac {{\left (x^{2} + 2 \, x + 2\right )} \arctan \left (x + 1\right ) + 1}{x^{2} + 2 \, x + 2} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.01, size = 15, normalized size = 0.65 \[ \frac {1}{x^{2} + 2 \, x + 2} + \arctan \left (x + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 16, normalized size = 0.70 \[ \arctan \left (x +1\right )+\frac {1}{x^{2}+2 x +2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.22, size = 15, normalized size = 0.65 \[ \frac {1}{x^{2} + 2 \, x + 2} + \arctan \left (x + 1\right ) \]
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 \[ \mathrm {atan}\left (x+1\right )+\frac {1}{x^2+2\,x+2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.11, size = 14, normalized size = 0.61 \[ \operatorname {atan}{\left (x + 1 \right )} + \frac {1}{x^{2} + 2 x + 2} \]
Verification of antiderivative is not currently implemented for this CAS.
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