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

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 = 0.0373894, 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 \[ \frac{3 x+2}{4 \left (3 x^2+4 x+2\right )}+\frac{3 \tan ^{-1}\left (\frac{3 x+2}{\sqrt{2}}\right )}{4 \sqrt{2}} \]

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])

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Rubi in Sympy [A]  time = 1.86408, size = 36, normalized size = 0.84 \[ \frac{6 x + 4}{8 \left (3 x^{2} + 4 x + 2\right )} + \frac{3 \sqrt{2} \operatorname{atan}{\left (\sqrt{2} \left (\frac{3 x}{2} + 1\right ) \right )}}{8} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(3*x**2+4*x+2)**2,x)

[Out]

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

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Mathematica [A]  time = 0.0398516, 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 = 0.005, size = 37, normalized size = 0.9 \[{\frac{6\,x+4}{24\,{x}^{2}+32\,x+16}}+{\frac{3\,\sqrt{2}}{8}\arctan \left ({\frac{ \left ( 6\,x+4 \right ) \sqrt{2}}{4}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(3*x^2+4*x+2)^2,x)

[Out]

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

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Maxima [A]  time = 0.790684, size = 49, normalized size = 1.14 \[ \frac{3}{8} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (3 \, x + 2\right )}\right ) + \frac{3 \, x + 2}{4 \,{\left (3 \, x^{2} + 4 \, x + 2\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((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 = 0.21222, size = 68, normalized size = 1.58 \[ \frac{\sqrt{2}{\left (3 \,{\left (3 \, x^{2} + 4 \, x + 2\right )} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (3 \, x + 2\right )}\right ) + \sqrt{2}{\left (3 \, x + 2\right )}\right )}}{8 \,{\left (3 \, x^{2} + 4 \, x + 2\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((3*x^2 + 4*x + 2)^(-2),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.335114, size = 39, normalized size = 0.91 \[ \frac{3 x + 2}{12 x^{2} + 16 x + 8} + \frac{3 \sqrt{2} \operatorname{atan}{\left (\frac{3 \sqrt{2} x}{2} + \sqrt{2} \right )}}{8} \]

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/XCAS [A]  time = 0.207236, size = 49, normalized size = 1.14 \[ \frac{3}{8} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (3 \, x + 2\right )}\right ) + \frac{3 \, x + 2}{4 \,{\left (3 \, x^{2} + 4 \, x + 2\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((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)