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

Optimal. Leaf size=32 \[ \frac{3}{2} \log \left (x^2-2 x+4\right )+\frac{\tan ^{-1}\left (\frac{1-x}{\sqrt{3}}\right )}{\sqrt{3}} \]

[Out]

ArcTan[(1 - x)/Sqrt[3]]/Sqrt[3] + (3*Log[4 - 2*x + x^2])/2

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Rubi [A]  time = 0.0168806, antiderivative size = 32, 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} \[ \frac{3}{2} \log \left (x^2-2 x+4\right )+\frac{\tan ^{-1}\left (\frac{1-x}{\sqrt{3}}\right )}{\sqrt{3}} \]

Antiderivative was successfully verified.

[In]

Int[(-4 + 3*x)/(4 - 2*x + x^2),x]

[Out]

ArcTan[(1 - x)/Sqrt[3]]/Sqrt[3] + (3*Log[4 - 2*x + x^2])/2

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{-4+3 x}{4-2 x+x^2} \, dx &=\frac{3}{2} \int \frac{-2+2 x}{4-2 x+x^2} \, dx-\int \frac{1}{4-2 x+x^2} \, dx\\ &=\frac{3}{2} \log \left (4-2 x+x^2\right )+2 \operatorname{Subst}\left (\int \frac{1}{-12-x^2} \, dx,x,-2+2 x\right )\\ &=\frac{\tan ^{-1}\left (\frac{1-x}{\sqrt{3}}\right )}{\sqrt{3}}+\frac{3}{2} \log \left (4-2 x+x^2\right )\\ \end{align*}

Mathematica [A]  time = 0.0085422, size = 31, normalized size = 0.97 \[ \frac{3}{2} \log \left (x^2-2 x+4\right )-\frac{\tan ^{-1}\left (\frac{x-1}{\sqrt{3}}\right )}{\sqrt{3}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(ArcTan[(-1 + x)/Sqrt[3]]/Sqrt[3]) + (3*Log[4 - 2*x + x^2])/2

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Maple [A]  time = 0.003, size = 29, normalized size = 0.9 \begin{align*}{\frac{3\,\ln \left ({x}^{2}-2\,x+4 \right ) }{2}}-{\frac{\sqrt{3}}{3}\arctan \left ({\frac{ \left ( 2\,x-2 \right ) \sqrt{3}}{6}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.45734, size = 35, normalized size = 1.09 \begin{align*} -\frac{1}{3} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (x - 1\right )}\right ) + \frac{3}{2} \, \log \left (x^{2} - 2 \, x + 4\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 0.985448, size = 90, normalized size = 2.81 \begin{align*} -\frac{1}{3} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (x - 1\right )}\right ) + \frac{3}{2} \, \log \left (x^{2} - 2 \, x + 4\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.103575, size = 36, normalized size = 1.12 \begin{align*} \frac{3 \log{\left (x^{2} - 2 x + 4 \right )}}{2} - \frac{\sqrt{3} \operatorname{atan}{\left (\frac{\sqrt{3} x}{3} - \frac{\sqrt{3}}{3} \right )}}{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3*log(x**2 - 2*x + 4)/2 - sqrt(3)*atan(sqrt(3)*x/3 - sqrt(3)/3)/3

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Giac [A]  time = 1.1138, size = 35, normalized size = 1.09 \begin{align*} -\frac{1}{3} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (x - 1\right )}\right ) + \frac{3}{2} \, \log \left (x^{2} - 2 \, x + 4\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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