3.2.48 \(\int \frac {1}{(2+2 x+x^2)^2} \, dx\) [148]

Optimal. Leaf size=26 \[ \frac {1+x}{2 \left (2+2 x+x^2\right )}+\frac {1}{2} \tan ^{-1}(1+x) \]

[Out]

1/2*(1+x)/(x^2+2*x+2)+1/2*arctan(1+x)

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Rubi [A]
time = 0.00, antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {628, 631, 210} \begin {gather*} \frac {1}{2} \text {ArcTan}(x+1)+\frac {x+1}{2 \left (x^2+2 x+2\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(2 + 2*x + x^2)^(-2),x]

[Out]

(1 + x)/(2*(2 + 2*x + x^2)) + ArcTan[1 + x]/2

Rule 210

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^(-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])
], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 628

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(b + 2*c*x)*((a + b*x + c*x^2)^(p + 1)/((p + 1
)*(b^2 - 4*a*c))), x] - Dist[2*c*((2*p + 3)/((p + 1)*(b^2 - 4*a*c))), Int[(a + b*x + c*x^2)^(p + 1), x], x] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1] && NeQ[p, -3/2] && IntegerQ[4*p]

Rule 631

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[a*(c/b^2)]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b)], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rubi steps

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

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Mathematica [A]
time = 0.01, size = 23, normalized size = 0.88 \begin {gather*} \frac {1}{2} \left (\frac {1+x}{2+2 x+x^2}+\tan ^{-1}(1+x)\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(2 + 2*x + x^2)^(-2),x]

[Out]

((1 + x)/(2 + 2*x + x^2) + ArcTan[1 + x])/2

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Maple [A]
time = 0.07, size = 25, normalized size = 0.96

method	result	size
risch	\(\frac {\frac {1}{2}+\frac {x}{2}}{x^{2}+2 x +2}+\frac {\arctan \left (1+x \right )}{2}\)	\(24\)
default	\(\frac {2+2 x}{4 x^{2}+8 x +8}+\frac {\arctan \left (1+x \right )}{2}\)	\(25\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^2+2*x+2)^2,x,method=_RETURNVERBOSE)

[Out]

1/4/(x^2+2*x+2)*(2+2*x)+1/2*arctan(1+x)

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Maxima [A]
time = 3.30, size = 22, normalized size = 0.85 \begin {gather*} \frac {x + 1}{2 \, {\left (x^{2} + 2 \, x + 2\right )}} + \frac {1}{2} \, \arctan \left (x + 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(x + 1)/(x^2 + 2*x + 2) + 1/2*arctan(x + 1)

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Fricas [A]
time = 0.64, size = 28, normalized size = 1.08 \begin {gather*} \frac {{\left (x^{2} + 2 \, x + 2\right )} \arctan \left (x + 1\right ) + x + 1}{2 \, {\left (x^{2} + 2 \, x + 2\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*((x^2 + 2*x + 2)*arctan(x + 1) + x + 1)/(x^2 + 2*x + 2)

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Sympy [A]
time = 0.04, size = 19, normalized size = 0.73 \begin {gather*} \frac {x + 1}{2 x^{2} + 4 x + 4} + \frac {\operatorname {atan}{\left (x + 1 \right )}}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(x + 1)/(2*x**2 + 4*x + 4) + atan(x + 1)/2

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Giac [A]
time = 0.50, size = 22, normalized size = 0.85 \begin {gather*} \frac {x + 1}{2 \, {\left (x^{2} + 2 \, x + 2\right )}} + \frac {1}{2} \, \arctan \left (x + 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(x + 1)/(x^2 + 2*x + 2) + 1/2*arctan(x + 1)

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Mupad [B]
time = 0.19, size = 23, normalized size = 0.88 \begin {gather*} \frac {\mathrm {atan}\left (x+1\right )}{2}+\frac {\frac {x}{2}+\frac {1}{2}}{x^2+2\,x+2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(2*x + x^2 + 2)^2,x)

[Out]

atan(x + 1)/2 + (x/2 + 1/2)/(2*x + x^2 + 2)

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