Optimal. Leaf size=12 \[ \frac{2}{x+2}+\log (x+2) \]
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Rubi [A] time = 0.005216, antiderivative size = 12, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {27, 43} \[ \frac{2}{x+2}+\log (x+2) \]
Antiderivative was successfully verified.
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Rule 27
Rule 43
Rubi steps
\begin{align*} \int \frac{x}{4+4 x+x^2} \, dx &=\int \frac{x}{(2+x)^2} \, dx\\ &=\int \left (-\frac{2}{(2+x)^2}+\frac{1}{2+x}\right ) \, dx\\ &=\frac{2}{2+x}+\log (2+x)\\ \end{align*}
Mathematica [A] time = 0.0031645, size = 12, normalized size = 1. \[ \frac{2}{x+2}+\log (x+2) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 13, normalized size = 1.1 \begin{align*} 2\, \left ( 2+x \right ) ^{-1}+\ln \left ( 2+x \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.926493, size = 16, normalized size = 1.33 \begin{align*} \frac{2}{x + 2} + \log \left (x + 2\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.68812, size = 46, normalized size = 3.83 \begin{align*} \frac{{\left (x + 2\right )} \log \left (x + 2\right ) + 2}{x + 2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.072204, size = 8, normalized size = 0.67 \begin{align*} \log{\left (x + 2 \right )} + \frac{2}{x + 2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.06902, size = 18, normalized size = 1.5 \begin{align*} \frac{2}{x + 2} + \log \left ({\left | x + 2 \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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