3.2845 \(\int \frac{2+x}{(1+(2+x)^2)^2} \, dx\)

Optimal. Leaf size=13 \[ -\frac{1}{2 \left ((x+2)^2+1\right )} \]

[Out]

-1/(2*(1 + (2 + x)^2))

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Rubi [A]  time = 0.0051081, antiderivative size = 13, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {372, 261} \[ -\frac{1}{2 \left ((x+2)^2+1\right )} \]

Antiderivative was successfully verified.

[In]

Int[(2 + x)/(1 + (2 + x)^2)^2,x]

[Out]

-1/(2*(1 + (2 + x)^2))

Rule 372

Int[(u_)^(m_.)*((a_) + (b_.)*(v_)^(n_))^(p_.), x_Symbol] :> Dist[u^m/(Coefficient[v, x, 1]*v^m), Subst[Int[x^m
*(a + b*x^n)^p, x], x, v], x] /; FreeQ[{a, b, m, n, p}, x] && LinearPairQ[u, v, x]

Rule 261

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a + b*x^n)^(p + 1)/(b*n*(p + 1)), x] /; FreeQ
[{a, b, m, n, p}, x] && EqQ[m, n - 1] && NeQ[p, -1]

Rubi steps

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

Mathematica [A]  time = 0.0026113, size = 13, normalized size = 1. \[ -\frac{1}{2 \left ((x+2)^2+1\right )} \]

Antiderivative was successfully verified.

[In]

Integrate[(2 + x)/(1 + (2 + x)^2)^2,x]

[Out]

-1/(2*(1 + (2 + x)^2))

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Maple [A]  time = 0.001, size = 13, normalized size = 1. \begin{align*} -{\frac{1}{2\,{x}^{2}+8\,x+10}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2+x)/(1+(2+x)^2)^2,x)

[Out]

-1/2/(x^2+4*x+5)

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Maxima [A]  time = 1.02156, size = 15, normalized size = 1.15 \begin{align*} -\frac{1}{2 \,{\left ({\left (x + 2\right )}^{2} + 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+x)/(1+(2+x)^2)^2,x, algorithm="maxima")

[Out]

-1/2/((x + 2)^2 + 1)

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Fricas [A]  time = 1.462, size = 30, normalized size = 2.31 \begin{align*} -\frac{1}{2 \,{\left (x^{2} + 4 \, x + 5\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+x)/(1+(2+x)^2)^2,x, algorithm="fricas")

[Out]

-1/2/(x^2 + 4*x + 5)

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Sympy [A]  time = 0.098278, size = 12, normalized size = 0.92 \begin{align*} - \frac{1}{2 x^{2} + 8 x + 10} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+x)/(1+(2+x)**2)**2,x)

[Out]

-1/(2*x**2 + 8*x + 10)

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Giac [A]  time = 1.10775, size = 16, normalized size = 1.23 \begin{align*} -\frac{1}{2 \,{\left (x^{2} + 4 \, x + 5\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+x)/(1+(2+x)^2)^2,x, algorithm="giac")

[Out]

-1/2/(x^2 + 4*x + 5)