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 = 0.01, antiderivative size = 13, normalized size of antiderivative = 1.00, 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 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]

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]

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*}

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Mathematica [A]  time = 0.00, size = 13, normalized size = 1.00 \[ -\frac {1}{2 \left ((x+2)^2+1\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-1/2*1/(1 + (2 + x)^2)

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fricas [A]  time = 0.65, size = 12, normalized size = 0.92 \[ -\frac {1}{2 \, {\left (x^{2} + 4 \, x + 5\right )}} \]

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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giac [A]  time = 0.20, size = 12, normalized size = 0.92 \[ -\frac {1}{2 \, {\left (x^{2} + 4 \, x + 5\right )}} \]

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)

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maple [A]  time = 0.00, size = 13, normalized size = 1.00 \[ -\frac {1}{2 \left (x^{2}+4 x +5\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.58, size = 11, normalized size = 0.85 \[ -\frac {1}{2 \, {\left ({\left (x + 2\right )}^{2} + 1\right )}} \]

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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mupad [B]  time = 1.14, size = 13, normalized size = 1.00 \[ -\frac {1}{2\,\left ({\left (x+2\right )}^2+1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x + 2)/((x + 2)^2 + 1)^2,x)

[Out]

-1/(2*((x + 2)^2 + 1))

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sympy [A]  time = 0.12, size = 12, normalized size = 0.92 \[ - \frac {1}{2 x^{2} + 8 x + 10} \]

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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