Optimal. Leaf size=28 \[ -\frac{1}{2} \log \left (x^2+3\right )+\log (x)+\frac{2 \tan ^{-1}\left (\frac{x}{\sqrt{3}}\right )}{\sqrt{3}} \]
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Rubi [A] time = 0.0224832, antiderivative size = 28, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {1593, 801, 635, 203, 260} \[ -\frac{1}{2} \log \left (x^2+3\right )+\log (x)+\frac{2 \tan ^{-1}\left (\frac{x}{\sqrt{3}}\right )}{\sqrt{3}} \]
Antiderivative was successfully verified.
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Rule 1593
Rule 801
Rule 635
Rule 203
Rule 260
Rubi steps
\begin{align*} \int \frac{3+2 x}{3 x+x^3} \, dx &=\int \frac{3+2 x}{x \left (3+x^2\right )} \, dx\\ &=\int \left (\frac{1}{x}+\frac{2-x}{3+x^2}\right ) \, dx\\ &=\log (x)+\int \frac{2-x}{3+x^2} \, dx\\ &=\log (x)+2 \int \frac{1}{3+x^2} \, dx-\int \frac{x}{3+x^2} \, dx\\ &=\frac{2 \tan ^{-1}\left (\frac{x}{\sqrt{3}}\right )}{\sqrt{3}}+\log (x)-\frac{1}{2} \log \left (3+x^2\right )\\ \end{align*}
Mathematica [A] time = 0.0081235, size = 28, normalized size = 1. \[ -\frac{1}{2} \log \left (x^2+3\right )+\log (x)+\frac{2 \tan ^{-1}\left (\frac{x}{\sqrt{3}}\right )}{\sqrt{3}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 24, normalized size = 0.9 \begin{align*} \ln \left ( x \right ) -{\frac{\ln \left ({x}^{2}+3 \right ) }{2}}+{\frac{2\,\sqrt{3}}{3}\arctan \left ({\frac{x\sqrt{3}}{3}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.41322, size = 31, normalized size = 1.11 \begin{align*} \frac{2}{3} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3} x\right ) - \frac{1}{2} \, \log \left (x^{2} + 3\right ) + \log \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.90635, size = 85, normalized size = 3.04 \begin{align*} \frac{2}{3} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3} x\right ) - \frac{1}{2} \, \log \left (x^{2} + 3\right ) + \log \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.11735, size = 29, normalized size = 1.04 \begin{align*} \log{\left (x \right )} - \frac{\log{\left (x^{2} + 3 \right )}}{2} + \frac{2 \sqrt{3} \operatorname{atan}{\left (\frac{\sqrt{3} x}{3} \right )}}{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.05555, size = 32, normalized size = 1.14 \begin{align*} \frac{2}{3} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3} x\right ) - \frac{1}{2} \, \log \left (x^{2} + 3\right ) + \log \left ({\left | x \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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