3.272 \(\int \frac{x}{2-2 x+x^2} \, dx\)

Optimal. Leaf size=22 \[ \frac{1}{2} \log \left (x^2-2 x+2\right )-\tan ^{-1}(1-x) \]

[Out]

-ArcTan[1 - x] + Log[2 - 2*x + x^2]/2

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Rubi [A]  time = 0.0075908, antiderivative size = 22, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {634, 617, 204, 628} \[ \frac{1}{2} \log \left (x^2-2 x+2\right )-\tan ^{-1}(1-x) \]

Antiderivative was successfully verified.

[In]

Int[x/(2 - 2*x + x^2),x]

[Out]

-ArcTan[1 - x] + Log[2 - 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 617

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[(a*c)/b^2]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + (2*c*x)/b], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /;
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{x}{2-2 x+x^2} \, dx &=\frac{1}{2} \int \frac{-2+2 x}{2-2 x+x^2} \, dx+\int \frac{1}{2-2 x+x^2} \, dx\\ &=\frac{1}{2} \log \left (2-2 x+x^2\right )+\operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1-x\right )\\ &=-\tan ^{-1}(1-x)+\frac{1}{2} \log \left (2-2 x+x^2\right )\\ \end{align*}

Mathematica [A]  time = 0.0032798, size = 22, normalized size = 1. \[ \frac{1}{2} \log \left (x^2-2 x+2\right )-\tan ^{-1}(1-x) \]

Antiderivative was successfully verified.

[In]

Integrate[x/(2 - 2*x + x^2),x]

[Out]

-ArcTan[1 - x] + Log[2 - 2*x + x^2]/2

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Maple [A]  time = 0.003, size = 17, normalized size = 0.8 \begin{align*} \arctan \left ( -1+x \right ) +{\frac{\ln \left ({x}^{2}-2\,x+2 \right ) }{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.41361, size = 22, normalized size = 1. \begin{align*} \arctan \left (x - 1\right ) + \frac{1}{2} \, \log \left (x^{2} - 2 \, x + 2\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.76874, size = 54, normalized size = 2.45 \begin{align*} \arctan \left (x - 1\right ) + \frac{1}{2} \, \log \left (x^{2} - 2 \, x + 2\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.091334, size = 15, normalized size = 0.68 \begin{align*} \frac{\log{\left (x^{2} - 2 x + 2 \right )}}{2} + \operatorname{atan}{\left (x - 1 \right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(x**2 - 2*x + 2)/2 + atan(x - 1)

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Giac [A]  time = 1.07189, size = 22, normalized size = 1. \begin{align*} \arctan \left (x - 1\right ) + \frac{1}{2} \, \log \left (x^{2} - 2 \, x + 2\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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