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 = 0.0052333, 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, Rules used = {27, 43} \[ \frac{4}{2-x}+\log (2-x) \]

Antiderivative was successfully verified.

[In]

Int[(2 + x)/(4 - 4*x + x^2),x]

[Out]

4/(2 - x) + Log[2 - x]

Rule 27

Int[(u_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[u*Cancel[(b/2 + c*x)^(2*p)/c^p], x] /; Fr
eeQ[{a, b, c}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int \frac{2+x}{4-4 x+x^2} \, dx &=\int \frac{2+x}{(-2+x)^2} \, dx\\ &=\int \left (\frac{4}{(-2+x)^2}+\frac{1}{-2+x}\right ) \, dx\\ &=\frac{4}{2-x}+\log (2-x)\\ \end{align*}

Mathematica [A]  time = 0.002953, 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 = 0.003, size = 13, normalized size = 0.8 \begin{align*} -4\, \left ( -2+x \right ) ^{-1}+\ln \left ( -2+x \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.961557, size = 16, normalized size = 1. \begin{align*} -\frac{4}{x - 2} + \log \left (x - 2\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-4/(x - 2) + log(x - 2)

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Fricas [A]  time = 1.14659, size = 46, normalized size = 2.88 \begin{align*} \frac{{\left (x - 2\right )} \log \left (x - 2\right ) - 4}{x - 2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((x - 2)*log(x - 2) - 4)/(x - 2)

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Sympy [A]  time = 0.075429, size = 8, normalized size = 0.5 \begin{align*} \log{\left (x - 2 \right )} - \frac{4}{x - 2} \end{align*}

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 [A]  time = 1.07706, size = 18, normalized size = 1.12 \begin{align*} -\frac{4}{x - 2} + \log \left ({\left | x - 2 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-4/(x - 2) + log(abs(x - 2))