### 3.2405 $$\int \frac{1}{(-2+x) \sqrt{3-4 x+x^2}} \, dx$$

Optimal. Leaf size=13 $\tan ^{-1}\left (\sqrt{x^2-4 x+3}\right )$

[Out]

ArcTan[Sqrt[3 - 4*x + x^2]]

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Rubi [A]  time = 0.008268, antiderivative size = 13, 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 = {688, 203} $\tan ^{-1}\left (\sqrt{x^2-4 x+3}\right )$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

ArcTan[Sqrt[3 - 4*x + x^2]]

Rule 688

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

Rule 203

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

Rubi steps

\begin{align*} \int \frac{1}{(-2+x) \sqrt{3-4 x+x^2}} \, dx &=4 \operatorname{Subst}\left (\int \frac{1}{4+4 x^2} \, dx,x,\sqrt{3-4 x+x^2}\right )\\ &=\tan ^{-1}\left (\sqrt{3-4 x+x^2}\right )\\ \end{align*}

Mathematica [A]  time = 0.0037221, size = 12, normalized size = 0.92 $\tan ^{-1}\left (\sqrt{(x-2)^2-1}\right )$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

ArcTan[Sqrt[-1 + (-2 + x)^2]]

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Maple [A]  time = 0.044, size = 13, normalized size = 1. \begin{align*} -\arctan \left ({\frac{1}{\sqrt{ \left ( -2+x \right ) ^{2}-1}}} \right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-arctan(1/((-2+x)^2-1)^(1/2))

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Maxima [A]  time = 1.5334, size = 12, normalized size = 0.92 \begin{align*} -\arcsin \left (\frac{1}{{\left | x - 2 \right |}}\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-arcsin(1/abs(x - 2))

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Fricas [A]  time = 2.05789, size = 54, normalized size = 4.15 \begin{align*} 2 \, \arctan \left (-x + \sqrt{x^{2} - 4 \, x + 3} + 2\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*arctan(-x + sqrt(x^2 - 4*x + 3) + 2)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{\left (x - 3\right ) \left (x - 1\right )} \left (x - 2\right )}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.48421, size = 24, normalized size = 1.85 \begin{align*} 2 \, \arctan \left (-x + \sqrt{x^{2} - 4 \, x + 3} + 2\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*arctan(-x + sqrt(x^2 - 4*x + 3) + 2)