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 = 0.0077835, 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, 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.

[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])

Rule 621

Int[1/Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Dist[2, Subst[Int[1/(4*c - x^2), x], x, (b + 2*c*x)
/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 204

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

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*}

Mathematica [A]  time = 0.0072827, 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 = 0.057, size = 26, normalized size = 0.8 \begin{align*}{\frac{\sqrt{3}}{3}\arctan \left ({\sqrt{3} \left ( x-{\frac{2}{3}} \right ){\frac{1}{\sqrt{-3\,{x}^{2}+4\,x-2}}}} \right ) } \end{align*}

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 [C]  time = 1.82374, size = 22, normalized size = 0.67 \begin{align*} -\frac{1}{3} i \, \sqrt{3} \operatorname{arsinh}\left (\frac{1}{2} \, \sqrt{2}{\left (3 \, x - 2\right )}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [C]  time = 1.92105, size = 190, normalized size = 5.76 \begin{align*} -\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 ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6*I*sqrt(3)*log((2*I*sqrt(3)*sqrt(-3*x^2 + 4*x - 2) - 6*x + 4)/x) + 1/6*I*sqrt(3)*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. \begin{align*} \int \frac{1}{\sqrt{- 3 x^{2} + 4 x - 2}}\, dx \end{align*}

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 [C]  time = 1.2423, size = 22, normalized size = 0.67 \begin{align*} -\frac{1}{3} i \, \sqrt{3} \arcsin \left (\frac{1}{2} \, \sqrt{2}{\left (3 i \, x - 2 i\right )}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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