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 = 0.0055748, antiderivative size = 26, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, 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 614
Rule 617
Rule 204
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*}
Mathematica [A] time = 0.007493, 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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Maple [A] time = 0.001, size = 25, normalized size = 1. \begin{align*}{\frac{2\,x+2}{4\,{x}^{2}+8\,x+8}}+{\frac{\arctan \left ( 1+x \right ) }{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.4083, size = 30, normalized size = 1.15 \begin{align*} \frac{x + 1}{2 \,{\left (x^{2} + 2 \, x + 2\right )}} + \frac{1}{2} \, \arctan \left (x + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.2187, size = 82, normalized size = 3.15 \begin{align*} \frac{{\left (x^{2} + 2 \, x + 2\right )} \arctan \left (x + 1\right ) + x + 1}{2 \,{\left (x^{2} + 2 \, x + 2\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.113337, size = 19, normalized size = 0.73 \begin{align*} \frac{x + 1}{2 x^{2} + 4 x + 4} + \frac{\operatorname{atan}{\left (x + 1 \right )}}{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.05064, size = 30, normalized size = 1.15 \begin{align*} \frac{x + 1}{2 \,{\left (x^{2} + 2 \, x + 2\right )}} + \frac{1}{2} \, \arctan \left (x + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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