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]

_______________________________________________________________________________________

Rubi [A]  time = 0.0269848, 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 \[ \frac{\tan ^{-1}\left (\frac{3 x+2}{\sqrt{2}}\right )}{\sqrt{2}} \]

Antiderivative was successfully verified.

[In]  Int[(2 + 4*x + 3*x^2)^(-1),x]

[Out]

ArcTan[(2 + 3*x)/Sqrt[2]]/Sqrt[2]

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 1.41611, size = 19, normalized size = 1.06 \[ \frac{\sqrt{2} \operatorname{atan}{\left (\sqrt{2} \left (\frac{3 x}{2} + 1\right ) \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(2)*atan(sqrt(2)*(3*x/2 + 1))/2

_______________________________________________________________________________________

Mathematica [A]  time = 0.00811029, 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]

_______________________________________________________________________________________

Maple [A]  time = 0.005, size = 17, normalized size = 0.9 \[{\frac{\sqrt{2}}{2}\arctan \left ({\frac{ \left ( 6\,x+4 \right ) \sqrt{2}}{4}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(3*x^2+4*x+2),x)

[Out]

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

_______________________________________________________________________________________

Maxima [A]  time = 0.794678, size = 22, normalized size = 1.22 \[ \frac{1}{2} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (3 \, x + 2\right )}\right ) \]

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))

_______________________________________________________________________________________

Fricas [A]  time = 0.223852, size = 22, normalized size = 1.22 \[ \frac{1}{2} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (3 \, x + 2\right )}\right ) \]

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))

_______________________________________________________________________________________

Sympy [A]  time = 0.22521, size = 22, normalized size = 1.22 \[ \frac{\sqrt{2} \operatorname{atan}{\left (\frac{3 \sqrt{2} x}{2} + \sqrt{2} \right )}}{2} \]

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

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.206447, size = 22, normalized size = 1.22 \[ \frac{1}{2} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (3 \, x + 2\right )}\right ) \]

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))