### 3.2403 $$\int \frac{-1+x}{\sqrt{3-4 x+x^2}} \, dx$$

Optimal. Leaf size=34 $\sqrt{x^2-4 x+3}-\tanh ^{-1}\left (\frac{2-x}{\sqrt{x^2-4 x+3}}\right )$

[Out]

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

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Rubi [A]  time = 0.009237, antiderivative size = 34, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 16, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.188, Rules used = {640, 621, 206} $\sqrt{x^2-4 x+3}-\tanh ^{-1}\left (\frac{2-x}{\sqrt{x^2-4 x+3}}\right )$

Antiderivative was successfully veriﬁed.

[In]

Int[(-1 + x)/Sqrt[3 - 4*x + x^2],x]

[Out]

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

Rule 640

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

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 206

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

Rubi steps

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

Mathematica [A]  time = 0.0191737, size = 30, normalized size = 0.88 $\sqrt{x^2-4 x+3}+\tanh ^{-1}\left (\frac{x-2}{\sqrt{x^2-4 x+3}}\right )$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

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Maple [A]  time = 0.043, size = 26, normalized size = 0.8 \begin{align*} \ln \left ( x-2+\sqrt{{x}^{2}-4\,x+3} \right ) +\sqrt{{x}^{2}-4\,x+3} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.994683, size = 39, normalized size = 1.15 \begin{align*} \sqrt{x^{2} - 4 \, x + 3} + \log \left (2 \, x + 2 \, \sqrt{x^{2} - 4 \, x + 3} - 4\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 2.29784, size = 77, normalized size = 2.26 \begin{align*} \sqrt{x^{2} - 4 \, x + 3} - \log \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+x)/(x^2-4*x+3)^(1/2),x, algorithm="fricas")

[Out]

sqrt(x^2 - 4*x + 3) - log(-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{x - 1}{\sqrt{\left (x - 3\right ) \left (x - 1\right )}}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.38208, size = 41, normalized size = 1.21 \begin{align*} \sqrt{x^{2} - 4 \, x + 3} - \log \left ({\left | -x + \sqrt{x^{2} - 4 \, x + 3} + 2 \right |}\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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