3.464 \(\int \frac{-1+x}{3-4 x+3 x^2} \, dx\)

Optimal. Leaf size=37 \[ \frac{1}{6} \log \left (3 x^2-4 x+3\right )+\frac{\tan ^{-1}\left (\frac{2-3 x}{\sqrt{5}}\right )}{3 \sqrt{5}} \]

[Out]

ArcTan[(2 - 3*x)/Sqrt[5]]/(3*Sqrt[5]) + Log[3 - 4*x + 3*x^2]/6

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Rubi [A]  time = 0.0208234, antiderivative size = 37, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {634, 618, 204, 628} \[ \frac{1}{6} \log \left (3 x^2-4 x+3\right )+\frac{\tan ^{-1}\left (\frac{2-3 x}{\sqrt{5}}\right )}{3 \sqrt{5}} \]

Antiderivative was successfully verified.

[In]

Int[(-1 + x)/(3 - 4*x + 3*x^2),x]

[Out]

ArcTan[(2 - 3*x)/Sqrt[5]]/(3*Sqrt[5]) + Log[3 - 4*x + 3*x^2]/6

Rule 634

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[(2*c*d - b*e)/(2*c), Int[1/(a +
b*x + c*x^2), x], x] + Dist[e/(2*c), Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] &
& NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] &&  !NiceSqrtQ[b^2 - 4*a*c]

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])

Rule 628

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(d*Log[RemoveContent[a + b*x +
c*x^2, x]])/b, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rubi steps

\begin{align*} \int \frac{-1+x}{3-4 x+3 x^2} \, dx &=\frac{1}{6} \int \frac{-4+6 x}{3-4 x+3 x^2} \, dx-\frac{1}{3} \int \frac{1}{3-4 x+3 x^2} \, dx\\ &=\frac{1}{6} \log \left (3-4 x+3 x^2\right )+\frac{2}{3} \operatorname{Subst}\left (\int \frac{1}{-20-x^2} \, dx,x,-4+6 x\right )\\ &=\frac{\tan ^{-1}\left (\frac{2-3 x}{\sqrt{5}}\right )}{3 \sqrt{5}}+\frac{1}{6} \log \left (3-4 x+3 x^2\right )\\ \end{align*}

Mathematica [A]  time = 0.0100448, size = 37, normalized size = 1. \[ \frac{1}{6} \log \left (3 x^2-4 x+3\right )-\frac{\tan ^{-1}\left (\frac{3 x-2}{\sqrt{5}}\right )}{3 \sqrt{5}} \]

Antiderivative was successfully verified.

[In]

Integrate[(-1 + x)/(3 - 4*x + 3*x^2),x]

[Out]

-ArcTan[(-2 + 3*x)/Sqrt[5]]/(3*Sqrt[5]) + Log[3 - 4*x + 3*x^2]/6

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Maple [A]  time = 0.004, size = 31, normalized size = 0.8 \begin{align*}{\frac{\ln \left ( 3\,{x}^{2}-4\,x+3 \right ) }{6}}-{\frac{\sqrt{5}}{15}\arctan \left ({\frac{ \left ( 6\,x-4 \right ) \sqrt{5}}{10}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x-1)/(3*x^2-4*x+3),x)

[Out]

1/6*ln(3*x^2-4*x+3)-1/15*5^(1/2)*arctan(1/10*(6*x-4)*5^(1/2))

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Maxima [A]  time = 1.52259, size = 41, normalized size = 1.11 \begin{align*} -\frac{1}{15} \, \sqrt{5} \arctan \left (\frac{1}{5} \, \sqrt{5}{\left (3 \, x - 2\right )}\right ) + \frac{1}{6} \, \log \left (3 \, x^{2} - 4 \, x + 3\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-1+x)/(3*x^2-4*x+3),x, algorithm="maxima")

[Out]

-1/15*sqrt(5)*arctan(1/5*sqrt(5)*(3*x - 2)) + 1/6*log(3*x^2 - 4*x + 3)

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Fricas [A]  time = 1.15898, size = 97, normalized size = 2.62 \begin{align*} -\frac{1}{15} \, \sqrt{5} \arctan \left (\frac{1}{5} \, \sqrt{5}{\left (3 \, x - 2\right )}\right ) + \frac{1}{6} \, \log \left (3 \, x^{2} - 4 \, x + 3\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-1+x)/(3*x^2-4*x+3),x, algorithm="fricas")

[Out]

-1/15*sqrt(5)*arctan(1/5*sqrt(5)*(3*x - 2)) + 1/6*log(3*x^2 - 4*x + 3)

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Sympy [A]  time = 0.106238, size = 39, normalized size = 1.05 \begin{align*} \frac{\log{\left (x^{2} - \frac{4 x}{3} + 1 \right )}}{6} - \frac{\sqrt{5} \operatorname{atan}{\left (\frac{3 \sqrt{5} x}{5} - \frac{2 \sqrt{5}}{5} \right )}}{15} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-1+x)/(3*x**2-4*x+3),x)

[Out]

log(x**2 - 4*x/3 + 1)/6 - sqrt(5)*atan(3*sqrt(5)*x/5 - 2*sqrt(5)/5)/15

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Giac [A]  time = 1.08268, size = 41, normalized size = 1.11 \begin{align*} -\frac{1}{15} \, \sqrt{5} \arctan \left (\frac{1}{5} \, \sqrt{5}{\left (3 \, x - 2\right )}\right ) + \frac{1}{6} \, \log \left (3 \, x^{2} - 4 \, x + 3\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-1+x)/(3*x^2-4*x+3),x, algorithm="giac")

[Out]

-1/15*sqrt(5)*arctan(1/5*sqrt(5)*(3*x - 2)) + 1/6*log(3*x^2 - 4*x + 3)