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

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

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Rubi [A]  time = 0.01, antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {724, 204} \begin {gather*} \tan ^{-1}\left (\frac {3-2 x}{\sqrt {x^2+2 x-4}}\right ) \end {gather*}

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 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 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]

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*}

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Mathematica [A]  time = 0.01, size = 22, normalized size = 1.16 \begin {gather*} \tan ^{-1}\left (\frac {6-4 x}{2 \sqrt {x^2+2 x-4}}\right ) \end {gather*}

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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IntegrateAlgebraic [A]  time = 0.11, size = 20, normalized size = 1.05 \begin {gather*} -2 \tan ^{-1}\left (\sqrt {x^2+2 x-4}-x+1\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-2*ArcTan[1 - x + Sqrt[-4 + 2*x + x^2]]

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fricas [A]  time = 0.39, size = 18, normalized size = 0.95 \begin {gather*} -2 \, \arctan \left (-x + \sqrt {x^{2} + 2 \, x - 4} + 1\right ) \end {gather*}

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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giac [A]  time = 0.17, size = 18, normalized size = 0.95 \begin {gather*} -2 \, \arctan \left (-x + \sqrt {x^{2} + 2 \, x - 4} + 1\right ) \end {gather*}

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)

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maple [A]  time = 0.06, size = 23, normalized size = 1.21 \begin {gather*} -\arctan \left (\frac {4 x -6}{2 \sqrt {4 x +\left (x -1\right )^{2}-5}}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.99, size = 27, normalized size = 1.42 \begin {gather*} -\arcsin \left (\frac {2 \, \sqrt {5} x}{5 \, {\left | x - 1 \right |}} - \frac {3 \, \sqrt {5}}{5 \, {\left | x - 1 \right |}}\right ) \end {gather*}

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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mupad [F]  time = 0.00, size = -1, normalized size = -0.05 \begin {gather*} -\int \frac {1}{\left (x-1\right )\,\sqrt {x^2+2\,x-4}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - \int \frac {1}{x \sqrt {x^{2} + 2 x - 4} - \sqrt {x^{2} + 2 x - 4}}\, dx \end {gather*}

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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