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]

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

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Rubi [A]  time = 0.02, antiderivative size = 21, normalized size of antiderivative = 1.00, 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 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 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]

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*}

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Mathematica [A]  time = 0.01, size = 21, normalized size = 1.00 \[ \frac {\tan ^{-1}\left (\frac {3 x-2}{\sqrt {6}}\right )}{3 \sqrt {6}} \]

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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fricas [A]  time = 0.39, size = 16, normalized size = 0.76 \[ \frac {1}{18} \, \sqrt {6} \arctan \left (\frac {1}{6} \, \sqrt {6} {\left (3 \, x - 2\right )}\right ) \]

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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giac [A]  time = 1.05, size = 16, normalized size = 0.76 \[ \frac {1}{18} \, \sqrt {6} \arctan \left (\frac {1}{6} \, \sqrt {6} {\left (3 \, x - 2\right )}\right ) \]

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))

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maple [A]  time = 0.00, size = 17, normalized size = 0.81 \[ \frac {\sqrt {6}\, \arctan \left (\frac {\left (18 x -12\right ) \sqrt {6}}{36}\right )}{18} \]

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 = 0.95, size = 16, normalized size = 0.76 \[ \frac {1}{18} \, \sqrt {6} \arctan \left (\frac {1}{6} \, \sqrt {6} {\left (3 \, x - 2\right )}\right ) \]

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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mupad [B]  time = 0.14, size = 16, normalized size = 0.76 \[ \frac {\sqrt {6}\,\mathrm {atan}\left (\frac {\sqrt {6}\,\left (3\,x-2\right )}{6}\right )}{18} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(9*x^2 - 12*x + 10),x)

[Out]

(6^(1/2)*atan((6^(1/2)*(3*x - 2))/6))/18

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sympy [A]  time = 0.12, size = 22, normalized size = 1.05 \[ \frac {\sqrt {6} \operatorname {atan}{\left (\frac {\sqrt {6} x}{2} - \frac {\sqrt {6}}{3} \right )}}{18} \]

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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