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

Optimal. Leaf size=16 \[ \frac{4}{2-x}+\log (2-x) \]

[Out]

4/(2 - x) + Log[2 - x]

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Rubi [A]  time = 0.0155304, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143 \[ \frac{4}{2-x}+\log (2-x) \]

Antiderivative was successfully verified.

[In]  Int[(2 + x)/(4 - 4*x + x^2),x]

[Out]

4/(2 - x) + Log[2 - x]

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Rubi in Sympy [A]  time = 2.18911, size = 8, normalized size = 0.5 \[ \log{\left (- x + 2 \right )} + \frac{4}{- x + 2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((2+x)/(x**2-4*x+4),x)

[Out]

log(-x + 2) + 4/(-x + 2)

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Mathematica [A]  time = 0.00527172, size = 12, normalized size = 0.75 \[ \log (x-2)-\frac{4}{x-2} \]

Antiderivative was successfully verified.

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

[Out]

-4/(-2 + x) + Log[-2 + x]

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Maple [A]  time = 0.01, size = 13, normalized size = 0.8 \[ \ln \left ( -2+x \right ) -4\, \left ( -2+x \right ) ^{-1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((2+x)/(x^2-4*x+4),x)

[Out]

ln(-2+x)-4/(-2+x)

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Maxima [A]  time = 1.36311, size = 16, normalized size = 1. \[ -\frac{4}{x - 2} + \log \left (x - 2\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-4/(x - 2) + log(x - 2)

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Fricas [A]  time = 0.194168, size = 22, normalized size = 1.38 \[ \frac{{\left (x - 2\right )} \log \left (x - 2\right ) - 4}{x - 2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

((x - 2)*log(x - 2) - 4)/(x - 2)

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Sympy [A]  time = 0.07067, size = 8, normalized size = 0.5 \[ \log{\left (x - 2 \right )} - \frac{4}{x - 2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2+x)/(x**2-4*x+4),x)

[Out]

log(x - 2) - 4/(x - 2)

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GIAC/XCAS [A]  time = 0.212257, size = 18, normalized size = 1.12 \[ -\frac{4}{x - 2} +{\rm ln}\left ({\left | x - 2 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-4/(x - 2) + ln(abs(x - 2))