3.2419 \(\int \frac{1}{x \sqrt{2+4 x+3 x^2}} \, dx\)

Optimal. Leaf size=31 \[ -\frac{\tanh ^{-1}\left (\frac{\sqrt{2} (x+1)}{\sqrt{3 x^2+4 x+2}}\right )}{\sqrt{2}} \]

[Out]

-(ArcTanh[(Sqrt[2]*(1 + x))/Sqrt[2 + 4*x + 3*x^2]]/Sqrt[2])

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Rubi [A]  time = 0.0373123, antiderivative size = 31, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111 \[ -\frac{\tanh ^{-1}\left (\frac{\sqrt{2} (x+1)}{\sqrt{3 x^2+4 x+2}}\right )}{\sqrt{2}} \]

Antiderivative was successfully verified.

[In]  Int[1/(x*Sqrt[2 + 4*x + 3*x^2]),x]

[Out]

-(ArcTanh[(Sqrt[2]*(1 + x))/Sqrt[2 + 4*x + 3*x^2]]/Sqrt[2])

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Rubi in Sympy [A]  time = 5.02901, size = 34, normalized size = 1.1 \[ - \frac{\sqrt{2} \operatorname{atanh}{\left (\frac{\sqrt{2} \left (4 x + 4\right )}{4 \sqrt{3 x^{2} + 4 x + 2}} \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(2)*atanh(sqrt(2)*(4*x + 4)/(4*sqrt(3*x**2 + 4*x + 2)))/2

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Mathematica [A]  time = 0.030727, size = 31, normalized size = 1. \[ \frac{\log (x)-\log \left (\sqrt{6 x^2+8 x+4}+2 x+2\right )}{\sqrt{2}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x*Sqrt[2 + 4*x + 3*x^2]),x]

[Out]

(Log[x] - Log[2 + 2*x + Sqrt[4 + 8*x + 6*x^2]])/Sqrt[2]

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Maple [A]  time = 0.005, size = 29, normalized size = 0.9 \[ -{\frac{\sqrt{2}}{2}{\it Artanh} \left ({\frac{ \left ( 4+4\,x \right ) \sqrt{2}}{4}{\frac{1}{\sqrt{3\,{x}^{2}+4\,x+2}}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x/(3*x^2+4*x+2)^(1/2),x)

[Out]

-1/2*2^(1/2)*arctanh(1/4*(4+4*x)*2^(1/2)/(3*x^2+4*x+2)^(1/2))

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Maxima [A]  time = 0.751645, size = 32, normalized size = 1.03 \[ -\frac{1}{2} \, \sqrt{2} \operatorname{arsinh}\left (\frac{\sqrt{2} x}{{\left | x \right |}} + \frac{\sqrt{2}}{{\left | x \right |}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*sqrt(2)*arcsinh(sqrt(2)*x/abs(x) + sqrt(2)/abs(x))

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Fricas [A]  time = 0.253449, size = 54, normalized size = 1.74 \[ \frac{1}{4} \, \sqrt{2} \log \left (\frac{2 \, \sqrt{2} \sqrt{3 \, x^{2} + 4 \, x + 2}{\left (x + 1\right )} - 5 \, x^{2} - 8 \, x - 4}{x^{2}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4*sqrt(2)*log((2*sqrt(2)*sqrt(3*x^2 + 4*x + 2)*(x + 1) - 5*x^2 - 8*x - 4)/x^2)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{x \sqrt{3 x^{2} + 4 x + 2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(1/(x*sqrt(3*x**2 + 4*x + 2)), x)

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GIAC/XCAS [A]  time = 0.224902, size = 81, normalized size = 2.61 \[ -\frac{1}{2} \, \sqrt{2}{\rm ln}\left (-\sqrt{3} x + \sqrt{2} + \sqrt{3 \, x^{2} + 4 \, x + 2}\right ) + \frac{1}{2} \, \sqrt{2}{\rm ln}\left ({\left | -\sqrt{3} x - \sqrt{2} + \sqrt{3 \, x^{2} + 4 \, x + 2} \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*sqrt(2)*ln(-sqrt(3)*x + sqrt(2) + sqrt(3*x^2 + 4*x + 2)) + 1/2*sqrt(2)*ln(a
bs(-sqrt(3)*x - sqrt(2) + sqrt(3*x^2 + 4*x + 2)))