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 = 0.01, antiderivative size = 33, normalized size of antiderivative = 1.00, 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} \begin {gather*} -\frac {\tan ^{-1}\left (\frac {\sqrt {2} (1-x)}{\sqrt {-3 x^2+4 x-2}}\right )}{\sqrt {2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 204
Rule 724
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*}
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Mathematica [A] time = 0.01, size = 27, normalized size = 0.82 \begin {gather*} \frac {\tan ^{-1}\left (\frac {x-1}{\sqrt {-\frac {3 x^2}{2}+2 x-1}}\right )}{\sqrt {2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [C] time = 0.09, size = 43, normalized size = 1.30 \begin {gather*} -i \sqrt {2} \tanh ^{-1}\left (\sqrt {\frac {3}{2}} x+\frac {i \sqrt {-3 x^2+4 x-2}}{\sqrt {2}}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.39, size = 64, normalized size = 1.94 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt {-3 \, x^{2} + 4 \, x - 2} x}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 29, normalized size = 0.88 \begin {gather*} \frac {\sqrt {2}\, \arctan \left (\frac {\left (4 x -4\right ) \sqrt {2}}{4 \sqrt {-3 x^{2}+4 x -2}}\right )}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [C] time = 1.92, size = 25, normalized size = 0.76 \begin {gather*} \frac {1}{2} i \, \sqrt {2} \operatorname {arsinh}\left (\frac {\sqrt {2} x}{{\left | x \right |}} - \frac {\sqrt {2}}{{\left | x \right |}}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.26, size = 34, normalized size = 1.03 \begin {gather*} \frac {\sqrt {2}\,\ln \left (\frac {2\,x-2+\sqrt {2}\,\sqrt {-3\,x^2+4\,x-2}\,1{}\mathrm {i}}{x}\right )\,1{}\mathrm {i}}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{x \sqrt {- 3 x^{2} + 4 x - 2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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