Optimal. Leaf size=54 \[ -\frac {1}{3} \sqrt {-2+4 x-3 x^2}-\frac {2 \tan ^{-1}\left (\frac {2-3 x}{\sqrt {3} \sqrt {-2+4 x-3 x^2}}\right )}{3 \sqrt {3}} \]
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Rubi [A]
time = 0.01, antiderivative size = 54, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {654, 635, 210}
\begin {gather*} -\frac {2 \text {ArcTan}\left (\frac {2-3 x}{\sqrt {3} \sqrt {-3 x^2+4 x-2}}\right )}{3 \sqrt {3}}-\frac {1}{3} \sqrt {-3 x^2+4 x-2} \end {gather*}
Antiderivative was successfully verified.
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Rule 210
Rule 635
Rule 654
Rubi steps
\begin {align*} \int \frac {x}{\sqrt {-2+4 x-3 x^2}} \, dx &=-\frac {1}{3} \sqrt {-2+4 x-3 x^2}+\frac {2}{3} \int \frac {1}{\sqrt {-2+4 x-3 x^2}} \, dx\\ &=-\frac {1}{3} \sqrt {-2+4 x-3 x^2}+\frac {4}{3} \text {Subst}\left (\int \frac {1}{-12-x^2} \, dx,x,\frac {4-6 x}{\sqrt {-2+4 x-3 x^2}}\right )\\ &=-\frac {1}{3} \sqrt {-2+4 x-3 x^2}-\frac {2 \tan ^{-1}\left (\frac {2-3 x}{\sqrt {3} \sqrt {-2+4 x-3 x^2}}\right )}{3 \sqrt {3}}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 0.09, size = 54, normalized size = 1.00 \begin {gather*} \frac {1}{9} \left (-3 \sqrt {-2+4 x-3 x^2}+2 i \sqrt {3} \log \left (2 i-3 i x+\sqrt {-6+12 x-9 x^2}\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.89, size = 41, normalized size = 0.76
method | result | size |
default | \(-\frac {\sqrt {-3 x^{2}+4 x -2}}{3}+\frac {2 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (x -\frac {2}{3}\right )}{\sqrt {-3 x^{2}+4 x -2}}\right )}{9}\) | \(41\) |
risch | \(\frac {3 x^{2}-4 x +2}{3 \sqrt {-3 x^{2}+4 x -2}}+\frac {2 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (x -\frac {2}{3}\right )}{\sqrt {-3 x^{2}+4 x -2}}\right )}{9}\) | \(51\) |
trager | \(-\frac {\sqrt {-3 x^{2}+4 x -2}}{3}-\frac {2 \RootOf \left (\textit {\_Z}^{2}+3\right ) \ln \left (3 x \RootOf \left (\textit {\_Z}^{2}+3\right )-2 \RootOf \left (\textit {\_Z}^{2}+3\right )+3 \sqrt {-3 x^{2}+4 x -2}\right )}{9}\) | \(57\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [C] Result contains complex when optimal does not.
time = 0.51, size = 31, normalized size = 0.57 \begin {gather*} -\frac {2}{9} i \, \sqrt {3} \operatorname {arsinh}\left (\frac {1}{2} \, \sqrt {2} {\left (3 \, x - 2\right )}\right ) - \frac {1}{3} \, \sqrt {-3 \, x^{2} + 4 \, x - 2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [C] Result contains complex when optimal does not.
time = 3.72, size = 81, normalized size = 1.50 \begin {gather*} \frac {1}{9} i \, \sqrt {3} \log \left (-\frac {2 \, {\left (i \, \sqrt {3} \sqrt {-3 \, x^{2} + 4 \, x - 2} + 3 \, x - 2\right )}}{x}\right ) - \frac {1}{9} i \, \sqrt {3} \log \left (-\frac {2 \, {\left (-i \, \sqrt {3} \sqrt {-3 \, x^{2} + 4 \, x - 2} + 3 \, x - 2\right )}}{x}\right ) - \frac {1}{3} \, \sqrt {-3 \, x^{2} + 4 \, x - 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 {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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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.10, size = 46, normalized size = 0.85 \begin {gather*} -\frac {\sqrt {-3\,x^2+4\,x-2}}{3}-\frac {\sqrt {3}\,\ln \left (\sqrt {-3\,x^2+4\,x-2}+\frac {\sqrt {3}\,\left (3\,x-2\right )\,1{}\mathrm {i}}{3}\right )\,2{}\mathrm {i}}{9} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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