Optimal. Leaf size=54 \[ -\frac{1}{3} \sqrt{-3 x^2+4 x-2}-\frac{2 \tan ^{-1}\left (\frac{2-3 x}{\sqrt{3} \sqrt{-3 x^2+4 x-2}}\right )}{3 \sqrt{3}} \]
[Out]
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Rubi [A] time = 0.0456785, antiderivative size = 54, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.188 \[ -\frac{1}{3} \sqrt{-3 x^2+4 x-2}-\frac{2 \tan ^{-1}\left (\frac{2-3 x}{\sqrt{3} \sqrt{-3 x^2+4 x-2}}\right )}{3 \sqrt{3}} \]
Antiderivative was successfully verified.
[In] Int[x/Sqrt[-2 + 4*x - 3*x^2],x]
[Out]
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Rubi in Sympy [A] time = 4.62918, size = 51, normalized size = 0.94 \[ - \frac{\sqrt{- 3 x^{2} + 4 x - 2}}{3} - \frac{2 \sqrt{3} \operatorname{atan}{\left (\frac{\sqrt{3} \left (- 6 x + 4\right )}{6 \sqrt{- 3 x^{2} + 4 x - 2}} \right )}}{9} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x/(-3*x**2+4*x-2)**(1/2),x)
[Out]
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Mathematica [A] time = 0.0507397, size = 49, normalized size = 0.91 \[ \frac{1}{9} \left (-3 \sqrt{-3 x^2+4 x-2}-2 \sqrt{3} \tan ^{-1}\left (\frac{2-3 x}{\sqrt{-9 x^2+12 x-6}}\right )\right ) \]
Antiderivative was successfully verified.
[In] Integrate[x/Sqrt[-2 + 4*x - 3*x^2],x]
[Out]
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Maple [A] time = 0.01, size = 41, normalized size = 0.8 \[ -{\frac{1}{3}\sqrt{-3\,{x}^{2}+4\,x-2}}+{\frac{2\,\sqrt{3}}{9}\arctan \left ({\sqrt{3} \left ( x-{\frac{2}{3}} \right ){\frac{1}{\sqrt{-3\,{x}^{2}+4\,x-2}}}} \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x/(-3*x^2+4*x-2)^(1/2),x)
[Out]
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Maxima [A] time = 0.75924, size = 42, normalized size = 0.78 \[ -\frac{2}{9} i \, \sqrt{3} \operatorname{arsinh}\left (\frac{1}{2} \, \sqrt{2}{\left (3 \, x - 2\right )}\right ) - \frac{1}{3} \, \sqrt{-3 \, x^{2} + 4 \, x - 2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x/sqrt(-3*x^2 + 4*x - 2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.221977, size = 108, normalized size = 2. \[ -\frac{1}{9} \, \sqrt{3}{\left (\sqrt{3} \sqrt{-3 \, x^{2} + 4 \, x - 2} + i \, \log \left (\frac{2 i \, \sqrt{3} \sqrt{-3 \, x^{2} + 4 \, x - 2} - 6 \, x + 4}{x}\right ) - i \, \log \left (\frac{-2 i \, \sqrt{3} \sqrt{-3 \, x^{2} + 4 \, x - 2} - 6 \, x + 4}{x}\right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x/sqrt(-3*x^2 + 4*x - 2),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x}{\sqrt{- 3 x^{2} + 4 x - 2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x/(-3*x**2+4*x-2)**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.214733, size = 45, normalized size = 0.83 \[ -\frac{2}{9} \, \sqrt{3} i \arcsin \left (\frac{1}{2} \, \sqrt{2} i{\left (3 \, x - 2\right )}\right ) - \frac{1}{3} \, \sqrt{-3 \, x^{2} + 4 \, x - 2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x/sqrt(-3*x^2 + 4*x - 2),x, algorithm="giac")
[Out]