Optimal. Leaf size=51 \[ \frac{1}{10} \left (5+3 \sqrt{5}\right ) \log \left (2 x-\sqrt{5}+3\right )+\frac{1}{10} \left (5-3 \sqrt{5}\right ) \log \left (2 x+\sqrt{5}+3\right ) \]
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Rubi [A] time = 0.0186711, antiderivative size = 51, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {632, 31} \[ \frac{1}{10} \left (5+3 \sqrt{5}\right ) \log \left (2 x-\sqrt{5}+3\right )+\frac{1}{10} \left (5-3 \sqrt{5}\right ) \log \left (2 x+\sqrt{5}+3\right ) \]
Antiderivative was successfully verified.
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Rule 632
Rule 31
Rubi steps
\begin{align*} \int \frac{3+x}{1+3 x+x^2} \, dx &=-\left (\frac{1}{10} \left (-5+3 \sqrt{5}\right ) \int \frac{1}{\frac{3}{2}+\frac{\sqrt{5}}{2}+x} \, dx\right )+\frac{1}{10} \left (5+3 \sqrt{5}\right ) \int \frac{1}{\frac{3}{2}-\frac{\sqrt{5}}{2}+x} \, dx\\ &=\frac{1}{10} \left (5+3 \sqrt{5}\right ) \log \left (3-\sqrt{5}+2 x\right )+\frac{1}{10} \left (5-3 \sqrt{5}\right ) \log \left (3+\sqrt{5}+2 x\right )\\ \end{align*}
Mathematica [A] time = 0.0250529, size = 49, normalized size = 0.96 \[ \frac{1}{10} \left (5+3 \sqrt{5}\right ) \log \left (-2 x+\sqrt{5}-3\right )+\frac{1}{10} \left (5-3 \sqrt{5}\right ) \log \left (2 x+\sqrt{5}+3\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 29, normalized size = 0.6 \begin{align*}{\frac{\ln \left ({x}^{2}+3\,x+1 \right ) }{2}}-{\frac{3\,\sqrt{5}}{5}{\it Artanh} \left ({\frac{ \left ( 3+2\,x \right ) \sqrt{5}}{5}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.53988, size = 53, normalized size = 1.04 \begin{align*} \frac{3}{10} \, \sqrt{5} \log \left (\frac{2 \, x - \sqrt{5} + 3}{2 \, x + \sqrt{5} + 3}\right ) + \frac{1}{2} \, \log \left (x^{2} + 3 \, x + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.62943, size = 132, normalized size = 2.59 \begin{align*} \frac{3}{10} \, \sqrt{5} \log \left (\frac{2 \, x^{2} - \sqrt{5}{\left (2 \, x + 3\right )} + 6 \, x + 7}{x^{2} + 3 \, x + 1}\right ) + \frac{1}{2} \, \log \left (x^{2} + 3 \, x + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.111567, size = 49, normalized size = 0.96 \begin{align*} \left (\frac{1}{2} + \frac{3 \sqrt{5}}{10}\right ) \log{\left (x - \frac{\sqrt{5}}{2} + \frac{3}{2} \right )} + \left (\frac{1}{2} - \frac{3 \sqrt{5}}{10}\right ) \log{\left (x + \frac{\sqrt{5}}{2} + \frac{3}{2} \right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.19062, size = 57, normalized size = 1.12 \begin{align*} \frac{3}{10} \, \sqrt{5} \log \left (\frac{{\left | 2 \, x - \sqrt{5} + 3 \right |}}{{\left | 2 \, x + \sqrt{5} + 3 \right |}}\right ) + \frac{1}{2} \, \log \left ({\left | x^{2} + 3 \, x + 1 \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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