3.5.17 \(\int \frac {-4+3 x}{4-2 x+x^2} \, dx\) [417]

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

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 32, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {648, 632, 210, 642} \begin {gather*} \frac {\text {ArcTan}\left (\frac {1-x}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {3}{2} \log \left (x^2-2 x+4\right ) \end {gather*}

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 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]

Rule 642

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]

Rule 648

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]

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 \text {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*}

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Mathematica [A]
time = 0.01, size = 31, normalized size = 0.97 \begin {gather*} -\frac {\tan ^{-1}\left (\frac {-1+x}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {3}{2} \log \left (4-2 x+x^2\right ) \end {gather*}

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 = 0.35, size = 29, normalized size = 0.91

method	result	size
risch	\(\frac {3 \ln \left (x^{2}-2 x +4\right )}{2}-\frac {\sqrt {3}\, \arctan \left (\frac {\left (-1+x \right ) \sqrt {3}}{3}\right )}{3}\)	\(27\)
default	\(\frac {3 \ln \left (x^{2}-2 x +4\right )}{2}-\frac {\sqrt {3}\, \arctan \left (\frac {\left (2 x -2\right ) \sqrt {3}}{6}\right )}{3}\)	\(29\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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 = 0.51, size = 26, normalized size = 0.81 \begin {gather*} -\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 {gather*}

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 = 0.40, size = 26, normalized size = 0.81 \begin {gather*} -\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 {gather*}

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 = 0.03, size = 36, normalized size = 1.12 \begin {gather*} \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 {gather*}

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 = 4.64, size = 26, normalized size = 0.81 \begin {gather*} -\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 {gather*}

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)

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Mupad [B]
time = 0.04, size = 30, normalized size = 0.94 \begin {gather*} \frac {3\,\ln \left (x^2-2\,x+4\right )}{2}-\frac {\sqrt {3}\,\mathrm {atan}\left (\frac {\sqrt {3}\,x}{3}-\frac {\sqrt {3}}{3}\right )}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((3*x - 4)/(x^2 - 2*x + 4),x)

[Out]

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

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