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

Optimal. Leaf size=38 \[ \frac{1}{8} \log \left (4 x^2-4 x+3\right )+x+\frac{\tan ^{-1}\left (\frac{1-2 x}{\sqrt{2}}\right )}{4 \sqrt{2}} \]

[Out]

x + ArcTan[(1 - 2*x)/Sqrt[2]]/(4*Sqrt[2]) + Log[3 - 4*x + 4*x^2]/8

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Rubi [A]  time = 0.037902, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.217, Rules used = {1657, 634, 618, 204, 628} \[ \frac{1}{8} \log \left (4 x^2-4 x+3\right )+x+\frac{\tan ^{-1}\left (\frac{1-2 x}{\sqrt{2}}\right )}{4 \sqrt{2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

x + ArcTan[(1 - 2*x)/Sqrt[2]]/(4*Sqrt[2]) + Log[3 - 4*x + 4*x^2]/8

Rule 1657

Int[(Pq_)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[Pq*(a + b*x + c*x^2)^p, x
], x] /; FreeQ[{a, b, c}, x] && PolyQ[Pq, x] && IGtQ[p, -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{2-3 x+4 x^2}{3-4 x+4 x^2} \, dx &=\int \left (1-\frac{1-x}{3-4 x+4 x^2}\right ) \, dx\\ &=x-\int \frac{1-x}{3-4 x+4 x^2} \, dx\\ &=x+\frac{1}{8} \int \frac{-4+8 x}{3-4 x+4 x^2} \, dx-\frac{1}{2} \int \frac{1}{3-4 x+4 x^2} \, dx\\ &=x+\frac{1}{8} \log \left (3-4 x+4 x^2\right )+\operatorname{Subst}\left (\int \frac{1}{-32-x^2} \, dx,x,-4+8 x\right )\\ &=x-\frac{\tan ^{-1}\left (\frac{-1+2 x}{\sqrt{2}}\right )}{4 \sqrt{2}}+\frac{1}{8} \log \left (3-4 x+4 x^2\right )\\ \end{align*}

Mathematica [A]  time = 0.0096372, size = 38, normalized size = 1. \[ \frac{1}{8} \log \left (4 x^2-4 x+3\right )+x-\frac{\tan ^{-1}\left (\frac{2 x-1}{\sqrt{2}}\right )}{4 \sqrt{2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

x - ArcTan[(-1 + 2*x)/Sqrt[2]]/(4*Sqrt[2]) + Log[3 - 4*x + 4*x^2]/8

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Maple [A]  time = 0.004, size = 32, normalized size = 0.8 \begin{align*} x+{\frac{\ln \left ( 4\,{x}^{2}-4\,x+3 \right ) }{8}}-{\frac{\sqrt{2}}{8}\arctan \left ({\frac{ \left ( 8\,x-4 \right ) \sqrt{2}}{8}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x+1/8*ln(4*x^2-4*x+3)-1/8*2^(1/2)*arctan(1/8*(8*x-4)*2^(1/2))

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Maxima [A]  time = 1.40462, size = 42, normalized size = 1.11 \begin{align*} -\frac{1}{8} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (2 \, x - 1\right )}\right ) + x + \frac{1}{8} \, \log \left (4 \, x^{2} - 4 \, x + 3\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/8*sqrt(2)*arctan(1/2*sqrt(2)*(2*x - 1)) + x + 1/8*log(4*x^2 - 4*x + 3)

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Fricas [A]  time = 2.15747, size = 101, normalized size = 2.66 \begin{align*} -\frac{1}{8} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (2 \, x - 1\right )}\right ) + x + \frac{1}{8} \, \log \left (4 \, x^{2} - 4 \, x + 3\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/8*sqrt(2)*arctan(1/2*sqrt(2)*(2*x - 1)) + x + 1/8*log(4*x^2 - 4*x + 3)

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Sympy [A]  time = 0.10557, size = 34, normalized size = 0.89 \begin{align*} x + \frac{\log{\left (x^{2} - x + \frac{3}{4} \right )}}{8} - \frac{\sqrt{2} \operatorname{atan}{\left (\sqrt{2} x - \frac{\sqrt{2}}{2} \right )}}{8} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x + log(x**2 - x + 3/4)/8 - sqrt(2)*atan(sqrt(2)*x - sqrt(2)/2)/8

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Giac [A]  time = 1.05783, size = 42, normalized size = 1.11 \begin{align*} -\frac{1}{8} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (2 \, x - 1\right )}\right ) + x + \frac{1}{8} \, \log \left (4 \, x^{2} - 4 \, x + 3\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/8*sqrt(2)*arctan(1/2*sqrt(2)*(2*x - 1)) + x + 1/8*log(4*x^2 - 4*x + 3)