3.2420 \(\int \frac{1}{x \sqrt{2+4 x-3 x^2}} \, dx\)

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 = 0.0375791, 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 = 5.21531, 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 = 0.0318178, 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 = 0.006, 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 = 0.754304, size = 47, normalized size = 1.52 \[ -\frac{1}{2} \, \sqrt{2} \log \left (\frac{2 \, \sqrt{2} \sqrt{-3 \, x^{2} + 4 \, x + 2}}{{\left | x \right |}} + \frac{4}{{\left | x \right |}} + 4\right ) \]

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)*log(2*sqrt(2)*sqrt(-3*x^2 + 4*x + 2)/abs(x) + 4/abs(x) + 4)

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Fricas [A]  time = 0.22869, size = 53, normalized size = 1.71 \[ \frac{1}{4} \, \sqrt{2} \log \left (-\frac{2 \, \sqrt{2} \sqrt{-3 \, x^{2} + 4 \, x + 2}{\left (x + 1\right )} + x^{2} - 8 \, x - 4}{x^{2}}\right ) \]

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) + 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 = 0.278415, size = 132, normalized size = 4.26 \[ -\frac{1}{6} \, \sqrt{6} \sqrt{3}{\rm ln}\left (\frac{{\left | -14 \, \sqrt{10} - 14 \, \sqrt{6} + \frac{28 \,{\left (\sqrt{3} \sqrt{-3 \, x^{2} + 4 \, x + 2} - \sqrt{10}\right )}}{3 \, x - 2} \right |}}{{\left | -14 \, \sqrt{10} + 14 \, \sqrt{6} + \frac{28 \,{\left (\sqrt{3} \sqrt{-3 \, x^{2} + 4 \, x + 2} - \sqrt{10}\right )}}{3 \, x - 2} \right |}}\right ) \]

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]

-1/6*sqrt(6)*sqrt(3)*ln(abs(-14*sqrt(10) - 14*sqrt(6) + 28*(sqrt(3)*sqrt(-3*x^2
+ 4*x + 2) - sqrt(10))/(3*x - 2))/abs(-14*sqrt(10) + 14*sqrt(6) + 28*(sqrt(3)*sq
rt(-3*x^2 + 4*x + 2) - sqrt(10))/(3*x - 2)))