Optimal. Leaf size=26 \[ \frac {x+1}{2 \left (x^2+2 x+2\right )}+\frac {1}{2} \tan ^{-1}(x+1) \]
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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 = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {614, 617, 204} \[ \frac {x+1}{2 \left (x^2+2 x+2\right )}+\frac {1}{2} \tan ^{-1}(x+1) \]
Antiderivative was successfully verified.
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Rule 204
Rule 614
Rule 617
Rubi steps
\begin {align*} \int \frac {1}{\left (2+2 x+x^2\right )^2} \, dx &=\frac {1+x}{2 \left (2+2 x+x^2\right )}+\frac {1}{2} \int \frac {1}{2+2 x+x^2} \, dx\\ &=\frac {1+x}{2 \left (2+2 x+x^2\right )}-\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+x\right )\\ &=\frac {1+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 = 23, normalized size = 0.88 \[ \frac {1}{2} \left (\frac {x+1}{x^2+2 x+2}+\tan ^{-1}(x+1)\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.40, size = 28, normalized size = 1.08 \[ \frac {{\left (x^{2} + 2 \, x + 2\right )} \arctan \left (x + 1\right ) + x + 1}{2 \, {\left (x^{2} + 2 \, x + 2\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.97, size = 22, normalized size = 0.85 \[ \frac {x + 1}{2 \, {\left (x^{2} + 2 \, x + 2\right )}} + \frac {1}{2} \, \arctan \left (x + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 25, normalized size = 0.96 \[ \frac {\arctan \left (x +1\right )}{2}+\frac {2 x +2}{4 x^{2}+8 x +8} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.10, size = 22, normalized size = 0.85 \[ \frac {x + 1}{2 \, {\left (x^{2} + 2 \, x + 2\right )}} + \frac {1}{2} \, \arctan \left (x + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.19, size = 23, normalized size = 0.88 \[ \frac {\mathrm {atan}\left (x+1\right )}{2}+\frac {\frac {x}{2}+\frac {1}{2}}{x^2+2\,x+2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.13, size = 19, normalized size = 0.73 \[ \frac {x + 1}{2 x^{2} + 4 x + 4} + \frac {\operatorname {atan}{\left (x + 1 \right )}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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