### 3.2435 $$\int \frac{1}{x \sqrt{-2+4 x-3 x^2}} \, dx$$

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

[Out]

-(ArcTan[(Sqrt[2]*(1 - x))/Sqrt[-2 + 4*x - 3*x^2]]/Sqrt[2])

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Rubi [A]  time = 0.011465, antiderivative size = 33, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 18, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.111, Rules used = {724, 204} $-\frac{\tan ^{-1}\left (\frac{\sqrt{2} (1-x)}{\sqrt{-3 x^2+4 x-2}}\right )}{\sqrt{2}}$

Antiderivative was successfully veriﬁed.

[In]

Int[1/(x*Sqrt[-2 + 4*x - 3*x^2]),x]

[Out]

-(ArcTan[(Sqrt[2]*(1 - x))/Sqrt[-2 + 4*x - 3*x^2]]/Sqrt[2])

Rule 724

Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Symbol] :> Dist[-2, Subst[Int[1/(4*c*d
^2 - 4*b*d*e + 4*a*e^2 - x^2), x], x, (2*a*e - b*d - (2*c*d - b*e)*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a,
b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[2*c*d - b*e, 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}{x \sqrt{-2+4 x-3 x^2}} \, dx &=-\left (2 \operatorname{Subst}\left (\int \frac{1}{-8-x^2} \, dx,x,\frac{-4+4 x}{\sqrt{-2+4 x-3 x^2}}\right )\right )\\ &=-\frac{\tan ^{-1}\left (\frac{\sqrt{2} (1-x)}{\sqrt{-2+4 x-3 x^2}}\right )}{\sqrt{2}}\\ \end{align*}

Mathematica [A]  time = 0.0077565, size = 27, normalized size = 0.82 $\frac{\tan ^{-1}\left (\frac{x-1}{\sqrt{-\frac{3 x^2}{2}+2 x-1}}\right )}{\sqrt{2}}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

ArcTan[(-1 + x)/Sqrt[-1 + 2*x - (3*x^2)/2]]/Sqrt[2]

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Maple [A]  time = 0.043, size = 29, normalized size = 0.9 \begin{align*}{\frac{\sqrt{2}}{2}\arctan \left ({\frac{ \left ( -4+4\,x \right ) \sqrt{2}}{4}{\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/x/(-3*x^2+4*x-2)^(1/2),x)

[Out]

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

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Maxima [C]  time = 1.46616, size = 34, normalized size = 1.03 \begin{align*} \frac{1}{2} i \, \sqrt{2} \operatorname{arsinh}\left (\frac{\sqrt{2} x}{{\left | x \right |}} - \frac{\sqrt{2}}{{\left | x \right |}}\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B]  time = 2.00073, size = 178, normalized size = 5.39 \begin{align*} \frac{1}{4} \, \sqrt{-2} \log \left (\frac{\sqrt{-2} \sqrt{-3 \, x^{2} + 4 \, x - 2} + 2 \, x - 2}{x}\right ) - \frac{1}{4} \, \sqrt{-2} \log \left (-\frac{\sqrt{-2} \sqrt{-3 \, x^{2} + 4 \, x - 2} - 2 \, x + 2}{x}\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [C]  time = 1.24401, size = 158, normalized size = 4.79 \begin{align*} -\frac{2 i \, \sqrt{3} \log \left (192 i \, \sqrt{6} + 192 i \, \sqrt{2} + \frac{384 \,{\left (\sqrt{3} \sqrt{-3 \, x^{2} + 4 \, x - 2} + i \, \sqrt{2}\right )}}{3 \, x - 2}\right )}{\sqrt{6} + \sqrt{2}} + \frac{2 i \, \sqrt{3} \log \left (-192 i \, \sqrt{6} + 192 i \, \sqrt{2} + \frac{384 \,{\left (\sqrt{3} \sqrt{-3 \, x^{2} + 4 \, x - 2} + i \, \sqrt{2}\right )}}{3 \, x - 2}\right )}{\sqrt{6} - \sqrt{2}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*I*sqrt(3)*log(192*I*sqrt(6) + 192*I*sqrt(2) + 384*(sqrt(3)*sqrt(-3*x^2 + 4*x - 2) + I*sqrt(2))/(3*x - 2))/(
sqrt(6) + sqrt(2)) + 2*I*sqrt(3)*log(-192*I*sqrt(6) + 192*I*sqrt(2) + 384*(sqrt(3)*sqrt(-3*x^2 + 4*x - 2) + I*
sqrt(2))/(3*x - 2))/(sqrt(6) - sqrt(2))