Optimal. Leaf size=31 \[ -\frac{\tanh ^{-1}\left (\frac{\sqrt{2} (x+1)}{\sqrt{3 x^2+4 x+2}}\right )}{\sqrt{2}} \]
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Rubi [A] time = 0.0117503, antiderivative size = 31, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {724, 206} \[ -\frac{\tanh ^{-1}\left (\frac{\sqrt{2} (x+1)}{\sqrt{3 x^2+4 x+2}}\right )}{\sqrt{2}} \]
Antiderivative was successfully verified.
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Rule 724
Rule 206
Rubi steps
\begin{align*} \int \frac{1}{x \sqrt{2+4 x+3 x^2}} \, dx &=-\left (2 \operatorname{Subst}\left (\int \frac{1}{8-x^2} \, dx,x,\frac{4+4 x}{\sqrt{2+4 x+3 x^2}}\right )\right )\\ &=-\frac{\tanh ^{-1}\left (\frac{\sqrt{2} (1+x)}{\sqrt{2+4 x+3 x^2}}\right )}{\sqrt{2}}\\ \end{align*}
Mathematica [A] time = 0.0082974, size = 28, normalized size = 0.9 \[ -\frac{\tanh ^{-1}\left (\frac{x+1}{\sqrt{\frac{3 x^2}{2}+2 x+1}}\right )}{\sqrt{2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.042, size = 29, normalized size = 0.9 \begin{align*} -{\frac{\sqrt{2}}{2}{\it Artanh} \left ({\frac{ \left ( 4+4\,x \right ) \sqrt{2}}{4}{\frac{1}{\sqrt{3\,{x}^{2}+4\,x+2}}}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.56766, size = 32, normalized size = 1.03 \begin{align*} -\frac{1}{2} \, \sqrt{2} \operatorname{arsinh}\left (\frac{\sqrt{2} x}{{\left | x \right |}} + \frac{\sqrt{2}}{{\left | x \right |}}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.00629, size = 111, normalized size = 3.58 \begin{align*} \frac{1}{4} \, \sqrt{2} \log \left (\frac{2 \, \sqrt{2} \sqrt{3 \, x^{2} + 4 \, x + 2}{\left (x + 1\right )} - 5 \, x^{2} - 8 \, x - 4}{x^{2}}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{x \sqrt{3 x^{2} + 4 x + 2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.12745, size = 81, normalized size = 2.61 \begin{align*} -\frac{1}{2} \, \sqrt{2} \log \left (-\sqrt{3} x + \sqrt{2} + \sqrt{3 \, x^{2} + 4 \, x + 2}\right ) + \frac{1}{2} \, \sqrt{2} \log \left ({\left | -\sqrt{3} x - \sqrt{2} + \sqrt{3 \, x^{2} + 4 \, x + 2} \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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