### 3.2254 $$\int \frac{x}{4+4 x+x^2} \, dx$$

Optimal. Leaf size=12 $\frac{2}{x+2}+\log (x+2)$

[Out]

2/(2 + x) + Log[2 + x]

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Rubi [A]  time = 0.0064843, antiderivative size = 12, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 12, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.167, Rules used = {27, 43} $\frac{2}{x+2}+\log (x+2)$

Antiderivative was successfully veriﬁed.

[In]

Int[x/(4 + 4*x + x^2),x]

[Out]

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

Mathematica [A]  time = 0.0033978, size = 12, normalized size = 1. $\frac{2}{x+2}+\log (x+2)$

Antiderivative was successfully veriﬁed.

[In]

Integrate[x/(4 + 4*x + x^2),x]

[Out]

2/(2 + x) + Log[2 + x]

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Maple [A]  time = 0.042, size = 13, normalized size = 1.1 \begin{align*} 2\, \left ( 2+x \right ) ^{-1}+\ln \left ( 2+x \right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/(2+x)+ln(2+x)

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Maxima [A]  time = 0.956167, size = 16, normalized size = 1.33 \begin{align*} \frac{2}{x + 2} + \log \left (x + 2\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/(x + 2) + log(x + 2)

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Fricas [A]  time = 2.34795, size = 46, normalized size = 3.83 \begin{align*} \frac{{\left (x + 2\right )} \log \left (x + 2\right ) + 2}{x + 2} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((x + 2)*log(x + 2) + 2)/(x + 2)

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Sympy [A]  time = 0.080823, size = 8, normalized size = 0.67 \begin{align*} \log{\left (x + 2 \right )} + \frac{2}{x + 2} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(x**2+4*x+4),x)

[Out]

log(x + 2) + 2/(x + 2)

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Giac [A]  time = 1.1104, size = 18, normalized size = 1.5 \begin{align*} \frac{2}{x + 2} + \log \left ({\left | x + 2 \right |}\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/(x + 2) + log(abs(x + 2))