Optimal. Leaf size=33 \[ -\frac{\tan ^{-1}\left (\frac{\sqrt{2} (1-x)}{\sqrt{3 x^2+4 x-2}}\right )}{\sqrt{2}} \]
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Rubi [A] time = 0.0106804, antiderivative size = 33, 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, 204} \[ -\frac{\tan ^{-1}\left (\frac{\sqrt{2} (1-x)}{\sqrt{3 x^2+4 x-2}}\right )}{\sqrt{2}} \]
Antiderivative was successfully verified.
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Rule 724
Rule 204
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{\tan ^{-1}\left (\frac{\sqrt{2} (1-x)}{\sqrt{-2+4 x+3 x^2}}\right )}{\sqrt{2}}\\ \end{align*}
Mathematica [A] time = 0.0087675, size = 27, normalized size = 0.82 \[ \frac{\tan ^{-1}\left (\frac{x-1}{\sqrt{\frac{3 x^2}{2}+2 x-1}}\right )}{\sqrt{2}} \]
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}\arctan \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 = 1.54439, size = 35, normalized size = 1.06 \begin{align*} \frac{1}{2} \, \sqrt{2} \arcsin \left (\frac{\sqrt{10} x}{5 \,{\left | x \right |}} - \frac{\sqrt{10}}{5 \,{\left | x \right |}}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.06923, size = 80, normalized size = 2.42 \begin{align*} \frac{1}{2} \, \sqrt{2} \arctan \left (\frac{\sqrt{2}{\left (x - 1\right )}}{\sqrt{3 \, x^{2} + 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 [A] time = 1.13588, size = 41, normalized size = 1.24 \begin{align*} \sqrt{2} \arctan \left (-\frac{1}{2} \, \sqrt{2}{\left (\sqrt{3} x - \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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