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

Optimal. Leaf size=18 \[ \frac{\tan ^{-1}\left (\frac{3 x+2}{\sqrt{2}}\right )}{\sqrt{2}} \]

[Out]

ArcTan[(2 + 3*x)/Sqrt[2]]/Sqrt[2]

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Rubi [A]  time = 0.0112172, antiderivative size = 18, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {618, 204} \[ \frac{\tan ^{-1}\left (\frac{3 x+2}{\sqrt{2}}\right )}{\sqrt{2}} \]

Antiderivative was successfully verified.

[In]

Int[(2 + 4*x + 3*x^2)^(-1),x]

[Out]

ArcTan[(2 + 3*x)/Sqrt[2]]/Sqrt[2]

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

Mathematica [A]  time = 0.0049882, size = 18, normalized size = 1. \[ \frac{\tan ^{-1}\left (\frac{3 x+2}{\sqrt{2}}\right )}{\sqrt{2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(2 + 4*x + 3*x^2)^(-1),x]

[Out]

ArcTan[(2 + 3*x)/Sqrt[2]]/Sqrt[2]

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Maple [A]  time = 0.044, size = 17, normalized size = 0.9 \begin{align*}{\frac{\sqrt{2}}{2}\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),x)

[Out]

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

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Maxima [A]  time = 1.78149, size = 22, normalized size = 1.22 \begin{align*} \frac{1}{2} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (3 \, x + 2\right )}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*sqrt(2)*arctan(1/2*sqrt(2)*(3*x + 2))

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Fricas [A]  time = 2.13459, size = 58, normalized size = 3.22 \begin{align*} \frac{1}{2} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (3 \, x + 2\right )}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*sqrt(2)*arctan(1/2*sqrt(2)*(3*x + 2))

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Sympy [A]  time = 0.10425, size = 22, normalized size = 1.22 \begin{align*} \frac{\sqrt{2} \operatorname{atan}{\left (\frac{3 \sqrt{2} x}{2} + \sqrt{2} \right )}}{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sqrt(2)*atan(3*sqrt(2)*x/2 + sqrt(2))/2

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Giac [A]  time = 1.24023, size = 22, normalized size = 1.22 \begin{align*} \frac{1}{2} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (3 \, x + 2\right )}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*sqrt(2)*arctan(1/2*sqrt(2)*(3*x + 2))