Optimal. Leaf size=39 \[ -\frac{4 x+7}{6 \left (x^2+2 x+4\right )}-\frac{2 \tan ^{-1}\left (\frac{x+1}{\sqrt{3}}\right )}{3 \sqrt{3}} \]
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Rubi [A] time = 0.0141822, antiderivative size = 39, normalized size of antiderivative = 1., 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 = {638, 618, 204} \[ -\frac{4 x+7}{6 \left (x^2+2 x+4\right )}-\frac{2 \tan ^{-1}\left (\frac{x+1}{\sqrt{3}}\right )}{3 \sqrt{3}} \]
Antiderivative was successfully verified.
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Rule 638
Rule 618
Rule 204
Rubi steps
\begin{align*} \int \frac{-3+x}{\left (4+2 x+x^2\right )^2} \, dx &=-\frac{7+4 x}{6 \left (4+2 x+x^2\right )}-\frac{2}{3} \int \frac{1}{4+2 x+x^2} \, dx\\ &=-\frac{7+4 x}{6 \left (4+2 x+x^2\right )}+\frac{4}{3} \operatorname{Subst}\left (\int \frac{1}{-12-x^2} \, dx,x,2+2 x\right )\\ &=-\frac{7+4 x}{6 \left (4+2 x+x^2\right )}-\frac{2 \tan ^{-1}\left (\frac{1+x}{\sqrt{3}}\right )}{3 \sqrt{3}}\\ \end{align*}
Mathematica [A] time = 0.0224098, size = 39, normalized size = 1. \[ \frac{-4 x-7}{6 \left (x^2+2 x+4\right )}-\frac{2 \tan ^{-1}\left (\frac{x+1}{\sqrt{3}}\right )}{3 \sqrt{3}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 35, normalized size = 0.9 \begin{align*}{\frac{-8\,x-14}{12\,{x}^{2}+24\,x+48}}-{\frac{2\,\sqrt{3}}{9}\arctan \left ({\frac{ \left ( 2\,x+2 \right ) \sqrt{3}}{6}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.42095, size = 43, normalized size = 1.1 \begin{align*} -\frac{2}{9} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (x + 1\right )}\right ) - \frac{4 \, x + 7}{6 \,{\left (x^{2} + 2 \, x + 4\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.01193, size = 123, normalized size = 3.15 \begin{align*} -\frac{4 \, \sqrt{3}{\left (x^{2} + 2 \, x + 4\right )} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (x + 1\right )}\right ) + 12 \, x + 21}{18 \,{\left (x^{2} + 2 \, x + 4\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.115488, size = 41, normalized size = 1.05 \begin{align*} - \frac{4 x + 7}{6 x^{2} + 12 x + 24} - \frac{2 \sqrt{3} \operatorname{atan}{\left (\frac{\sqrt{3} x}{3} + \frac{\sqrt{3}}{3} \right )}}{9} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.0604, size = 43, normalized size = 1.1 \begin{align*} -\frac{2}{9} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (x + 1\right )}\right ) - \frac{4 \, x + 7}{6 \,{\left (x^{2} + 2 \, x + 4\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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