3.3.70 \(\int \frac {1}{10-12 x+9 x^2} \, dx\) [270]

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 = 0.01, 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 = {632, 210} \begin {gather*} -\frac {\text {ArcTan}\left (\frac {2-3 x}{\sqrt {6}}\right )}{3 \sqrt {6}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(10 - 12*x + 9*x^2)^(-1),x]

[Out]

-1/3*ArcTan[(2 - 3*x)/Sqrt[6]]/Sqrt[6]

Rule 210

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^(-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])
], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 632

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 \text {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 \begin {gather*} \frac {\tan ^{-1}\left (\frac {-2+3 x}{\sqrt {6}}\right )}{3 \sqrt {6}} \end {gather*}

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 = 0.16, size = 17, normalized size = 0.81

method	result	size
default	\(\frac {\sqrt {6}\, \arctan \left (\frac {\left (18 x -12\right ) \sqrt {6}}{36}\right )}{18}\)	\(17\)
risch	\(\frac {\sqrt {6}\, \arctan \left (\frac {\left (-2+3 x \right ) \sqrt {6}}{6}\right )}{18}\)	\(17\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 2.65, size = 16, normalized size = 0.76 \begin {gather*} \frac {1}{18} \, \sqrt {6} \arctan \left (\frac {1}{6} \, \sqrt {6} {\left (3 \, x - 2\right )}\right ) \end {gather*}

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 = 0.93, size = 16, normalized size = 0.76 \begin {gather*} \frac {1}{18} \, \sqrt {6} \arctan \left (\frac {1}{6} \, \sqrt {6} {\left (3 \, x - 2\right )}\right ) \end {gather*}

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 = 0.04, size = 22, normalized size = 1.05 \begin {gather*} \frac {\sqrt {6} \operatorname {atan}{\left (\frac {\sqrt {6} x}{2} - \frac {\sqrt {6}}{3} \right )}}{18} \end {gather*}

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 = 0.44, size = 16, normalized size = 0.76 \begin {gather*} \frac {1}{18} \, \sqrt {6} \arctan \left (\frac {1}{6} \, \sqrt {6} {\left (3 \, x - 2\right )}\right ) \end {gather*}

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

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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