3.91 \(\int \frac{1}{(2+4 x+3 x^2)^2} \, dx\)

Optimal. Leaf size=43 \[ \frac{3 x+2}{4 \left (3 x^2+4 x+2\right )}+\frac{3 \tan ^{-1}\left (\frac{3 x+2}{\sqrt{2}}\right )}{4 \sqrt{2}} \]

[Out]

(2 + 3*x)/(4*(2 + 4*x + 3*x^2)) + (3*ArcTan[(2 + 3*x)/Sqrt[2]])/(4*Sqrt[2])

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Rubi [A]  time = 0.0146884, antiderivative size = 43, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {614, 618, 204} \[ \frac{3 x+2}{4 \left (3 x^2+4 x+2\right )}+\frac{3 \tan ^{-1}\left (\frac{3 x+2}{\sqrt{2}}\right )}{4 \sqrt{2}} \]

Antiderivative was successfully verified.

[In]

Int[(2 + 4*x + 3*x^2)^(-2),x]

[Out]

(2 + 3*x)/(4*(2 + 4*x + 3*x^2)) + (3*ArcTan[(2 + 3*x)/Sqrt[2]])/(4*Sqrt[2])

Rule 614

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 618

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

Rule 204

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

Rubi steps

\begin{align*} \int \frac{1}{\left (2+4 x+3 x^2\right )^2} \, dx &=\frac{2+3 x}{4 \left (2+4 x+3 x^2\right )}+\frac{3}{4} \int \frac{1}{2+4 x+3 x^2} \, dx\\ &=\frac{2+3 x}{4 \left (2+4 x+3 x^2\right )}-\frac{3}{2} \operatorname{Subst}\left (\int \frac{1}{-8-x^2} \, dx,x,4+6 x\right )\\ &=\frac{2+3 x}{4 \left (2+4 x+3 x^2\right )}+\frac{3 \tan ^{-1}\left (\frac{2+3 x}{\sqrt{2}}\right )}{4 \sqrt{2}}\\ \end{align*}

Mathematica [A]  time = 0.0227275, size = 43, normalized size = 1. \[ \frac{3 x+2}{4 \left (3 x^2+4 x+2\right )}+\frac{3 \tan ^{-1}\left (\frac{3 x+2}{\sqrt{2}}\right )}{4 \sqrt{2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(2 + 4*x + 3*x^2)^(-2),x]

[Out]

(2 + 3*x)/(4*(2 + 4*x + 3*x^2)) + (3*ArcTan[(2 + 3*x)/Sqrt[2]])/(4*Sqrt[2])

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Maple [A]  time = 0.048, size = 37, normalized size = 0.9 \begin{align*}{\frac{4+6\,x}{24\,{x}^{2}+32\,x+16}}+{\frac{3\,\sqrt{2}}{8}\arctan \left ({\frac{ \left ( 4+6\,x \right ) \sqrt{2}}{4}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(3*x^2+4*x+2)^2,x)

[Out]

1/8*(4+6*x)/(3*x^2+4*x+2)+3/8*2^(1/2)*arctan(1/4*(4+6*x)*2^(1/2))

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Maxima [A]  time = 1.70954, size = 49, normalized size = 1.14 \begin{align*} \frac{3}{8} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (3 \, x + 2\right )}\right ) + \frac{3 \, x + 2}{4 \,{\left (3 \, x^{2} + 4 \, x + 2\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(3*x^2+4*x+2)^2,x, algorithm="maxima")

[Out]

3/8*sqrt(2)*arctan(1/2*sqrt(2)*(3*x + 2)) + 1/4*(3*x + 2)/(3*x^2 + 4*x + 2)

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Fricas [A]  time = 2.04242, size = 126, normalized size = 2.93 \begin{align*} \frac{3 \, \sqrt{2}{\left (3 \, x^{2} + 4 \, x + 2\right )} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (3 \, x + 2\right )}\right ) + 6 \, x + 4}{8 \,{\left (3 \, x^{2} + 4 \, x + 2\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(3*x^2+4*x+2)^2,x, algorithm="fricas")

[Out]

1/8*(3*sqrt(2)*(3*x^2 + 4*x + 2)*arctan(1/2*sqrt(2)*(3*x + 2)) + 6*x + 4)/(3*x^2 + 4*x + 2)

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Sympy [A]  time = 0.19527, size = 39, normalized size = 0.91 \begin{align*} \frac{3 x + 2}{12 x^{2} + 16 x + 8} + \frac{3 \sqrt{2} \operatorname{atan}{\left (\frac{3 \sqrt{2} x}{2} + \sqrt{2} \right )}}{8} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(3*x**2+4*x+2)**2,x)

[Out]

(3*x + 2)/(12*x**2 + 16*x + 8) + 3*sqrt(2)*atan(3*sqrt(2)*x/2 + sqrt(2))/8

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Giac [A]  time = 1.27949, size = 49, normalized size = 1.14 \begin{align*} \frac{3}{8} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (3 \, x + 2\right )}\right ) + \frac{3 \, x + 2}{4 \,{\left (3 \, x^{2} + 4 \, x + 2\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(3*x^2+4*x+2)^2,x, algorithm="giac")

[Out]

3/8*sqrt(2)*arctan(1/2*sqrt(2)*(3*x + 2)) + 1/4*(3*x + 2)/(3*x^2 + 4*x + 2)