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]

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

________________________________________________________________________________________

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.

[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 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])

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]

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 = 0.01, 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.

[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])

________________________________________________________________________________________

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.

[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)

________________________________________________________________________________________

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.

[In]

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

[Out]

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

________________________________________________________________________________________

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.

[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))

________________________________________________________________________________________

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.

[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))

________________________________________________________________________________________

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.

[In]

int(1/(4*x - 3*x^2 - 2)^(1/2),x)

[Out]

-(3^(1/2)*asin(2^(1/2)*((3*x)/2 - 1)*1i))/3

________________________________________________________________________________________

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.

[In]

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

[Out]

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

________________________________________________________________________________________