3.10 \(\int \frac{1}{3-2 x+x^2} \, dx\)

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

[Out]

-(ArcTan[(1 - x)/Sqrt[2]]/Sqrt[2])

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Rubi [A]  time = 0.0100857, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {618, 204} \[ -\frac{\tan ^{-1}\left (\frac{1-x}{\sqrt{2}}\right )}{\sqrt{2}} \]

Antiderivative was successfully verified.

[In]

Int[(3 - 2*x + x^2)^(-1),x]

[Out]

-(ArcTan[(1 - x)/Sqrt[2]]/Sqrt[2])

Rule 618

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

Mathematica [A]  time = 0.0050858, size = 16, normalized size = 0.84 \[ \frac{\tan ^{-1}\left (\frac{x-1}{\sqrt{2}}\right )}{\sqrt{2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(3 - 2*x + x^2)^(-1),x]

[Out]

ArcTan[(-1 + x)/Sqrt[2]]/Sqrt[2]

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Maple [A]  time = 0.003, size = 17, normalized size = 0.9 \begin{align*}{\frac{\sqrt{2}}{2}\arctan \left ({\frac{ \left ( 2\,x-2 \right ) \sqrt{2}}{4}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.41061, size = 19, normalized size = 1. \begin{align*} \frac{1}{2} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (x - 1\right )}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*sqrt(2)*arctan(1/2*sqrt(2)*(x - 1))

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Fricas [A]  time = 1.9499, size = 55, normalized size = 2.89 \begin{align*} \frac{1}{2} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (x - 1\right )}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*sqrt(2)*arctan(1/2*sqrt(2)*(x - 1))

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Sympy [A]  time = 0.098663, size = 22, normalized size = 1.16 \begin{align*} \frac{\sqrt{2} \operatorname{atan}{\left (\frac{\sqrt{2} x}{2} - \frac{\sqrt{2}}{2} \right )}}{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sqrt(2)*atan(sqrt(2)*x/2 - sqrt(2)/2)/2

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Giac [A]  time = 1.06335, size = 19, normalized size = 1. \begin{align*} \frac{1}{2} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (x - 1\right )}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*sqrt(2)*arctan(1/2*sqrt(2)*(x - 1))