3.270 \(\int \frac{1}{10-12 x+9 x^2} \, dx\)

Optimal. Leaf size=21 \[ -\frac{\tan ^{-1}\left (\frac{2-3 x}{\sqrt{6}}\right )}{3 \sqrt{6}} \]

[Out]

-ArcTan[(2 - 3*x)/Sqrt[6]]/(3*Sqrt[6])

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Rubi [A]  time = 0.0170595, antiderivative size = 21, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {618, 204} \[ -\frac{\tan ^{-1}\left (\frac{2-3 x}{\sqrt{6}}\right )}{3 \sqrt{6}} \]

Antiderivative was successfully verified.

[In]

Int[(10 - 12*x + 9*x^2)^(-1),x]

[Out]

-ArcTan[(2 - 3*x)/Sqrt[6]]/(3*Sqrt[6])

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}{10-12 x+9 x^2} \, dx &=-\left (2 \operatorname{Subst}\left (\int \frac{1}{-216-x^2} \, dx,x,-12+18 x\right )\right )\\ &=-\frac{\tan ^{-1}\left (\frac{2-3 x}{\sqrt{6}}\right )}{3 \sqrt{6}}\\ \end{align*}

Mathematica [A]  time = 0.008068, size = 21, normalized size = 1. \[ \frac{\tan ^{-1}\left (\frac{3 x-2}{\sqrt{6}}\right )}{3 \sqrt{6}} \]

Antiderivative was successfully verified.

[In]

Integrate[(10 - 12*x + 9*x^2)^(-1),x]

[Out]

ArcTan[(-2 + 3*x)/Sqrt[6]]/(3*Sqrt[6])

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Maple [A]  time = 0.003, size = 17, normalized size = 0.8 \begin{align*}{\frac{\sqrt{6}}{18}\arctan \left ({\frac{ \left ( 18\,x-12 \right ) \sqrt{6}}{36}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(9*x^2-12*x+10),x)

[Out]

1/18*6^(1/2)*arctan(1/36*(18*x-12)*6^(1/2))

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Maxima [A]  time = 1.41101, size = 22, normalized size = 1.05 \begin{align*} \frac{1}{18} \, \sqrt{6} \arctan \left (\frac{1}{6} \, \sqrt{6}{\left (3 \, x - 2\right )}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(9*x^2-12*x+10),x, algorithm="maxima")

[Out]

1/18*sqrt(6)*arctan(1/6*sqrt(6)*(3*x - 2))

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Fricas [A]  time = 1.83148, size = 59, normalized size = 2.81 \begin{align*} \frac{1}{18} \, \sqrt{6} \arctan \left (\frac{1}{6} \, \sqrt{6}{\left (3 \, x - 2\right )}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(9*x^2-12*x+10),x, algorithm="fricas")

[Out]

1/18*sqrt(6)*arctan(1/6*sqrt(6)*(3*x - 2))

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Sympy [A]  time = 0.104321, size = 22, normalized size = 1.05 \begin{align*} \frac{\sqrt{6} \operatorname{atan}{\left (\frac{\sqrt{6} x}{2} - \frac{\sqrt{6}}{3} \right )}}{18} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(9*x**2-12*x+10),x)

[Out]

sqrt(6)*atan(sqrt(6)*x/2 - sqrt(6)/3)/18

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Giac [A]  time = 1.0824, size = 22, normalized size = 1.05 \begin{align*} \frac{1}{18} \, \sqrt{6} \arctan \left (\frac{1}{6} \, \sqrt{6}{\left (3 \, x - 2\right )}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(9*x^2-12*x+10),x, algorithm="giac")

[Out]

1/18*sqrt(6)*arctan(1/6*sqrt(6)*(3*x - 2))