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 = 0.0184598, 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 \[ \frac{1}{2} \log \left (x^2-2 x+2\right )-\tan ^{-1}(1-x) \]

Antiderivative was successfully verified.

[In]  Int[x/(2 - 2*x + x^2),x]

[Out]

-ArcTan[1 - x] + Log[2 - 2*x + x^2]/2

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Rubi in Sympy [A]  time = 2.01616, size = 15, normalized size = 0.68 \[ \frac{\log{\left (x^{2} - 2 x + 2 \right )}}{2} + \operatorname{atan}{\left (x - 1 \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x/(x**2-2*x+2),x)

[Out]

log(x**2 - 2*x + 2)/2 + atan(x - 1)

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Mathematica [A]  time = 0.00538467, 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 = 0.006, size = 17, normalized size = 0.8 \[ \arctan \left ( -1+x \right ) +{\frac{\ln \left ({x}^{2}-2\,x+2 \right ) }{2}} \]

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 = 1.51392, size = 22, normalized size = 1. \[ \arctan \left (x - 1\right ) + \frac{1}{2} \, \log \left (x^{2} - 2 \, x + 2\right ) \]

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 = 0.197379, size = 22, normalized size = 1. \[ \arctan \left (x - 1\right ) + \frac{1}{2} \, \log \left (x^{2} - 2 \, x + 2\right ) \]

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 = 0.093142, size = 15, normalized size = 0.68 \[ \frac{\log{\left (x^{2} - 2 x + 2 \right )}}{2} + \operatorname{atan}{\left (x - 1 \right )} \]

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/XCAS [A]  time = 0.222689, size = 22, normalized size = 1. \[ \arctan \left (x - 1\right ) + \frac{1}{2} \,{\rm ln}\left (x^{2} - 2 \, x + 2\right ) \]

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*ln(x^2 - 2*x + 2)