Optimal. Leaf size=26 \[ \frac{1}{2} \log \left (x^2+2 x+5\right )-\frac{1}{2} \tan ^{-1}\left (\frac{x+1}{2}\right ) \]
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Rubi [A] time = 0.0134394, antiderivative size = 26, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {634, 618, 204, 628} \[ \frac{1}{2} \log \left (x^2+2 x+5\right )-\frac{1}{2} \tan ^{-1}\left (\frac{x+1}{2}\right ) \]
Antiderivative was successfully verified.
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Rule 634
Rule 618
Rule 204
Rule 628
Rubi steps
\begin{align*} \int \frac{x}{5+2 x+x^2} \, dx &=\frac{1}{2} \int \frac{2+2 x}{5+2 x+x^2} \, dx-\int \frac{1}{5+2 x+x^2} \, dx\\ &=\frac{1}{2} \log \left (5+2 x+x^2\right )+2 \operatorname{Subst}\left (\int \frac{1}{-16-x^2} \, dx,x,2+2 x\right )\\ &=-\frac{1}{2} \tan ^{-1}\left (\frac{1+x}{2}\right )+\frac{1}{2} \log \left (5+2 x+x^2\right )\\ \end{align*}
Mathematica [A] time = 0.0035023, size = 26, normalized size = 1. \[ \frac{1}{2} \log \left (x^2+2 x+5\right )-\frac{1}{2} \tan ^{-1}\left (\frac{x+1}{2}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.04, size = 21, normalized size = 0.8 \begin{align*} -{\frac{1}{2}\arctan \left ({\frac{1}{2}}+{\frac{x}{2}} \right ) }+{\frac{\ln \left ({x}^{2}+2\,x+5 \right ) }{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.50598, size = 27, normalized size = 1.04 \begin{align*} -\frac{1}{2} \, \arctan \left (\frac{1}{2} \, x + \frac{1}{2}\right ) + \frac{1}{2} \, \log \left (x^{2} + 2 \, x + 5\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.44496, size = 69, normalized size = 2.65 \begin{align*} -\frac{1}{2} \, \arctan \left (\frac{1}{2} \, x + \frac{1}{2}\right ) + \frac{1}{2} \, \log \left (x^{2} + 2 \, x + 5\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.103024, size = 20, normalized size = 0.77 \begin{align*} \frac{\log{\left (x^{2} + 2 x + 5 \right )}}{2} - \frac{\operatorname{atan}{\left (\frac{x}{2} + \frac{1}{2} \right )}}{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.09128, size = 27, normalized size = 1.04 \begin{align*} -\frac{1}{2} \, \arctan \left (\frac{1}{2} \, x + \frac{1}{2}\right ) + \frac{1}{2} \, \log \left (x^{2} + 2 \, x + 5\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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