### 3.2404 $$\int \frac{1}{(1-x) \sqrt{-4+2 x+x^2}} \, dx$$

Optimal. Leaf size=19 $\tan ^{-1}\left (\frac{3-2 x}{\sqrt{x^2+2 x-4}}\right )$

[Out]

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

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Rubi [A]  time = 0.0118567, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 20, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.1, Rules used = {724, 204} $\tan ^{-1}\left (\frac{3-2 x}{\sqrt{x^2+2 x-4}}\right )$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

ArcTan[(3 - 2*x)/Sqrt[-4 + 2*x + x^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}{(1-x) \sqrt{-4+2 x+x^2}} \, dx &=-\left (2 \operatorname{Subst}\left (\int \frac{1}{-4-x^2} \, dx,x,\frac{6-4 x}{\sqrt{-4+2 x+x^2}}\right )\right )\\ &=\tan ^{-1}\left (\frac{3-2 x}{\sqrt{-4+2 x+x^2}}\right )\\ \end{align*}

Mathematica [A]  time = 0.0059554, size = 22, normalized size = 1.16 $\tan ^{-1}\left (\frac{6-4 x}{2 \sqrt{x^2+2 x-4}}\right )$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

ArcTan[(6 - 4*x)/(2*Sqrt[-4 + 2*x + x^2])]

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Maple [A]  time = 0.044, size = 23, normalized size = 1.2 \begin{align*} -\arctan \left ({\frac{-6+4\,x}{2}{\frac{1}{\sqrt{ \left ( -1+x \right ) ^{2}-5+4\,x}}}} \right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-arctan(1/2*(-6+4*x)/((-1+x)^2-5+4*x)^(1/2))

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Maxima [A]  time = 1.49182, size = 36, normalized size = 1.89 \begin{align*} -\arcsin \left (\frac{2 \, \sqrt{5} x}{5 \,{\left | x - 1 \right |}} - \frac{3 \, \sqrt{5}}{5 \,{\left | x - 1 \right |}}\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-arcsin(2/5*sqrt(5)*x/abs(x - 1) - 3/5*sqrt(5)/abs(x - 1))

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Fricas [A]  time = 2.03564, size = 55, normalized size = 2.89 \begin{align*} -2 \, \arctan \left (-x + \sqrt{x^{2} + 2 \, x - 4} + 1\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*arctan(-x + sqrt(x^2 + 2*x - 4) + 1)

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.38457, size = 24, normalized size = 1.26 \begin{align*} -2 \, \arctan \left (-x + \sqrt{x^{2} + 2 \, x - 4} + 1\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*arctan(-x + sqrt(x^2 + 2*x - 4) + 1)