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 = 0.011465, 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 = 0.0077565, 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 = 0.043, 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 [C] time = 1.46616, size = 34, normalized size = 1.03 \begin{align*} \frac{1}{2} i \, \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 [B] time = 2.00073, size = 178, normalized size = 5.39 \begin{align*} \frac{1}{4} \, \sqrt{-2} \log \left (\frac{\sqrt{-2} \sqrt{-3 \, x^{2} + 4 \, x - 2} + 2 \, x - 2}{x}\right ) - \frac{1}{4} \, \sqrt{-2} \log \left (-\frac{\sqrt{-2} \sqrt{-3 \, x^{2} + 4 \, x - 2} - 2 \, x + 2}{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}{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 [C] time = 1.24401, size = 158, normalized size = 4.79 \begin{align*} -\frac{2 i \, \sqrt{3} \log \left (192 i \, \sqrt{6} + 192 i \, \sqrt{2} + \frac{384 \,{\left (\sqrt{3} \sqrt{-3 \, x^{2} + 4 \, x - 2} + i \, \sqrt{2}\right )}}{3 \, x - 2}\right )}{\sqrt{6} + \sqrt{2}} + \frac{2 i \, \sqrt{3} \log \left (-192 i \, \sqrt{6} + 192 i \, \sqrt{2} + \frac{384 \,{\left (\sqrt{3} \sqrt{-3 \, x^{2} + 4 \, x - 2} + i \, \sqrt{2}\right )}}{3 \, x - 2}\right )}{\sqrt{6} - \sqrt{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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