3.122 \(\int \frac{1}{\sqrt{-2+4 x-3 x^2}} \, dx\)

Optimal. Leaf size=33 \[ -\frac{\tan ^{-1}\left (\frac{2-3 x}{\sqrt{3} \sqrt{-3 x^2+4 x-2}}\right )}{\sqrt{3}} \]

[Out]

-(ArcTan[(2 - 3*x)/(Sqrt[3]*Sqrt[-2 + 4*x - 3*x^2])]/Sqrt[3])

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Rubi [A]  time = 0.0203906, antiderivative size = 33, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143 \[ -\frac{\tan ^{-1}\left (\frac{2-3 x}{\sqrt{3} \sqrt{-3 x^2+4 x-2}}\right )}{\sqrt{3}} \]

Antiderivative was successfully verified.

[In]  Int[1/Sqrt[-2 + 4*x - 3*x^2],x]

[Out]

-(ArcTan[(2 - 3*x)/(Sqrt[3]*Sqrt[-2 + 4*x - 3*x^2])]/Sqrt[3])

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Rubi in Sympy [A]  time = 1.48579, size = 34, normalized size = 1.03 \[ - \frac{\sqrt{3} \operatorname{atan}{\left (\frac{\sqrt{3} \left (- 6 x + 4\right )}{6 \sqrt{- 3 x^{2} + 4 x - 2}} \right )}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(-3*x**2+4*x-2)**(1/2),x)

[Out]

-sqrt(3)*atan(sqrt(3)*(-6*x + 4)/(6*sqrt(-3*x**2 + 4*x - 2)))/3

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Mathematica [A]  time = 0.0224558, 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.

[In]  Integrate[1/Sqrt[-2 + 4*x - 3*x^2],x]

[Out]

-(ArcTan[(2 - 3*x)/Sqrt[-6 + 12*x - 9*x^2]]/Sqrt[3])

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Maple [A]  time = 0.003, size = 26, normalized size = 0.8 \[{\frac{\sqrt{3}}{3}\arctan \left ({\sqrt{3} \left ( x-{\frac{2}{3}} \right ){\frac{1}{\sqrt{-3\,{x}^{2}+4\,x-2}}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(-3*x^2+4*x-2)^(1/2),x)

[Out]

1/3*3^(1/2)*arctan(3^(1/2)*(x-2/3)/(-3*x^2+4*x-2)^(1/2))

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Maxima [A]  time = 0.796606, size = 22, normalized size = 0.67 \[ -\frac{1}{3} i \, \sqrt{3} \operatorname{arsinh}\left (\frac{1}{2} \, \sqrt{2}{\left (3 \, x - 2\right )}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/sqrt(-3*x^2 + 4*x - 2),x, algorithm="maxima")

[Out]

-1/3*I*sqrt(3)*arcsinh(1/2*sqrt(2)*(3*x - 2))

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Fricas [A]  time = 0.223021, size = 86, normalized size = 2.61 \[ \frac{1}{6} \, \sqrt{3}{\left (-i \, \log \left (\frac{2 i \, \sqrt{3} \sqrt{-3 \, x^{2} + 4 \, x - 2} - 6 \, x + 4}{x}\right ) + i \, \log \left (\frac{-2 i \, \sqrt{3} \sqrt{-3 \, x^{2} + 4 \, x - 2} - 6 \, x + 4}{x}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/sqrt(-3*x^2 + 4*x - 2),x, algorithm="fricas")

[Out]

1/6*sqrt(3)*(-I*log((2*I*sqrt(3)*sqrt(-3*x^2 + 4*x - 2) - 6*x + 4)/x) + I*log((-
2*I*sqrt(3)*sqrt(-3*x^2 + 4*x - 2) - 6*x + 4)/x))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{- 3 x^{2} + 4 x - 2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(-3*x**2+4*x-2)**(1/2),x)

[Out]

Integral(1/sqrt(-3*x**2 + 4*x - 2), x)

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GIAC/XCAS [A]  time = 0.215377, size = 24, normalized size = 0.73 \[ -\frac{1}{3} \, \sqrt{3} i \arcsin \left (\frac{1}{2} \, \sqrt{2} i{\left (3 \, x - 2\right )}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/sqrt(-3*x^2 + 4*x - 2),x, algorithm="giac")

[Out]

-1/3*sqrt(3)*i*arcsin(1/2*sqrt(2)*i*(3*x - 2))