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 = 0.01, antiderivative size = 33, normalized size of antiderivative = 1.00, 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 204
Rule 621
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*}
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Mathematica [A] time = 0.01, 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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fricas [C] time = 0.93, size = 65, normalized size = 1.97 \[ -\frac {1}{6} i \, \sqrt {3} \log \left (\frac {2 i \, \sqrt {3} \sqrt {-3 \, x^{2} + 4 \, x - 2} - 6 \, x + 4}{x}\right ) + \frac {1}{6} i \, \sqrt {3} \log \left (\frac {-2 i \, \sqrt {3} \sqrt {-3 \, x^{2} + 4 \, x - 2} - 6 \, x + 4}{x}\right ) \]
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 \[ \int \frac {1}{\sqrt {-3 \, x^{2} + 4 \, x - 2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 26, normalized size = 0.79 \[ \frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (x -\frac {2}{3}\right )}{\sqrt {-3 x^{2}+4 x -2}}\right )}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [C] time = 2.91, size = 16, normalized size = 0.48 \[ -\frac {1}{3} i \, \sqrt {3} \operatorname {arsinh}\left (\frac {1}{2} \, \sqrt {2} {\left (3 \, x - 2\right )}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.14, size = 17, normalized size = 0.52 \[ -\frac {\sqrt {3}\,\mathrm {asin}\left (\sqrt {2}\,\left (\frac {3\,x}{2}-1\right )\,1{}\mathrm {i}\right )}{3} \]
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 \[ \int \frac {1}{\sqrt {- 3 x^{2} + 4 x - 2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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