Optimal. Leaf size=33 \[ -\frac{\tan ^{-1}\left (\frac{2-3 x}{\sqrt{3} \sqrt{-3 x^2+4 x-2}}\right )}{\sqrt{3}} \]
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Rubi [A] time = 0.0077835, antiderivative size = 33, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {621, 204} \[ -\frac{\tan ^{-1}\left (\frac{2-3 x}{\sqrt{3} \sqrt{-3 x^2+4 x-2}}\right )}{\sqrt{3}} \]
Antiderivative was successfully verified.
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Rule 621
Rule 204
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{-2+4 x-3 x^2}} \, dx &=2 \operatorname{Subst}\left (\int \frac{1}{-12-x^2} \, dx,x,\frac{4-6 x}{\sqrt{-2+4 x-3 x^2}}\right )\\ &=-\frac{\tan ^{-1}\left (\frac{2-3 x}{\sqrt{3} \sqrt{-2+4 x-3 x^2}}\right )}{\sqrt{3}}\\ \end{align*}
Mathematica [A] time = 0.0072827, size = 28, normalized size = 0.85 \[ -\frac{\tan ^{-1}\left (\frac{2-3 x}{\sqrt{-9 x^2+12 x-6}}\right )}{\sqrt{3}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.057, size = 26, normalized size = 0.8 \begin{align*}{\frac{\sqrt{3}}{3}\arctan \left ({\sqrt{3} \left ( x-{\frac{2}{3}} \right ){\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 [C] time = 1.82374, size = 22, normalized size = 0.67 \begin{align*} -\frac{1}{3} i \, \sqrt{3} \operatorname{arsinh}\left (\frac{1}{2} \, \sqrt{2}{\left (3 \, x - 2\right )}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [C] time = 1.92105, size = 190, normalized size = 5.76 \begin{align*} -\frac{1}{6} i \, \sqrt{3} \log \left (\frac{2 i \, \sqrt{3} \sqrt{-3 \, x^{2} + 4 \, x - 2} - 6 \, x + 4}{x}\right ) + \frac{1}{6} i \, \sqrt{3} \log \left (\frac{-2 i \, \sqrt{3} \sqrt{-3 \, x^{2} + 4 \, x - 2} - 6 \, x + 4}{x}\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}{\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 [C] time = 1.2423, size = 22, normalized size = 0.67 \begin{align*} -\frac{1}{3} i \, \sqrt{3} \arcsin \left (\frac{1}{2} \, \sqrt{2}{\left (3 i \, x - 2 i\right )}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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