Optimal. Leaf size=39 \[ \frac{1-x}{4 \left (x^2+2 x+3\right )}+\frac{3 \tan ^{-1}\left (\frac{x+1}{\sqrt{2}}\right )}{4 \sqrt{2}} \]
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Rubi [A] time = 0.0233002, antiderivative size = 39, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.235, Rules used = {1660, 12, 618, 204} \[ \frac{1-x}{4 \left (x^2+2 x+3\right )}+\frac{3 \tan ^{-1}\left (\frac{x+1}{\sqrt{2}}\right )}{4 \sqrt{2}} \]
Antiderivative was successfully verified.
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Rule 1660
Rule 12
Rule 618
Rule 204
Rubi steps
\begin{align*} \int \frac{1+x+x^2}{\left (3+2 x+x^2\right )^2} \, dx &=\frac{1-x}{4 \left (3+2 x+x^2\right )}+\frac{1}{8} \int \frac{6}{3+2 x+x^2} \, dx\\ &=\frac{1-x}{4 \left (3+2 x+x^2\right )}+\frac{3}{4} \int \frac{1}{3+2 x+x^2} \, dx\\ &=\frac{1-x}{4 \left (3+2 x+x^2\right )}-\frac{3}{2} \operatorname{Subst}\left (\int \frac{1}{-8-x^2} \, dx,x,2+2 x\right )\\ &=\frac{1-x}{4 \left (3+2 x+x^2\right )}+\frac{3 \tan ^{-1}\left (\frac{1+x}{\sqrt{2}}\right )}{4 \sqrt{2}}\\ \end{align*}
Mathematica [A] time = 0.0246723, size = 39, normalized size = 1. \[ \frac{1-x}{4 \left (x^2+2 x+3\right )}+\frac{3 \tan ^{-1}\left (\frac{x+1}{\sqrt{2}}\right )}{4 \sqrt{2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.048, size = 34, normalized size = 0.9 \begin{align*}{\frac{1}{{x}^{2}+2\,x+3} \left ( -{\frac{x}{4}}+{\frac{1}{4}} \right ) }+{\frac{3\,\sqrt{2}}{8}\arctan \left ({\frac{ \left ( 2\,x+2 \right ) \sqrt{2}}{4}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.56463, size = 41, normalized size = 1.05 \begin{align*} \frac{3}{8} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (x + 1\right )}\right ) - \frac{x - 1}{4 \,{\left (x^{2} + 2 \, x + 3\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.68439, size = 117, normalized size = 3. \begin{align*} \frac{3 \, \sqrt{2}{\left (x^{2} + 2 \, x + 3\right )} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (x + 1\right )}\right ) - 2 \, x + 2}{8 \,{\left (x^{2} + 2 \, x + 3\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.122881, size = 37, normalized size = 0.95 \begin{align*} - \frac{x - 1}{4 x^{2} + 8 x + 12} + \frac{3 \sqrt{2} \operatorname{atan}{\left (\frac{\sqrt{2} x}{2} + \frac{\sqrt{2}}{2} \right )}}{8} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.24636, size = 41, normalized size = 1.05 \begin{align*} \frac{3}{8} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (x + 1\right )}\right ) - \frac{x - 1}{4 \,{\left (x^{2} + 2 \, x + 3\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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