3.2424 \(\int \frac{1}{x \sqrt{-2+4 x-3 x^2}} \, dx\)

Optimal. Leaf size=33 \[ -\frac{\tan ^{-1}\left (\frac{\sqrt{2} (1-x)}{\sqrt{-3 x^2+4 x-2}}\right )}{\sqrt{2}} \]

[Out]

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

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Rubi [A]  time = 0.0376262, antiderivative size = 33, 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 \[ -\frac{\tan ^{-1}\left (\frac{\sqrt{2} (1-x)}{\sqrt{-3 x^2+4 x-2}}\right )}{\sqrt{2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 5.30233, size = 32, normalized size = 0.97 \[ \frac{\sqrt{2} \operatorname{atan}{\left (\frac{\sqrt{2} \left (4 x - 4\right )}{4 \sqrt{- 3 x^{2} + 4 x - 2}} \right )}}{2} \]

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

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Mathematica [A]  time = 0.0270354, size = 38, normalized size = 1.15 \[ -\frac{\tan ^{-1}\left (\frac{(x-1) \sqrt{-6 x^2+8 x-4}}{3 x^2-4 x+2}\right )}{\sqrt{2}} \]

Antiderivative was successfully verified.

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

[Out]

-(ArcTan[((-1 + x)*Sqrt[-4 + 8*x - 6*x^2])/(2 - 4*x + 3*x^2)]/Sqrt[2])

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Maple [A]  time = 0.008, size = 29, normalized size = 0.9 \[{\frac{\sqrt{2}}{2}\arctan \left ({\frac{ \left ( -4+4\,x \right ) \sqrt{2}}{4}{\frac{1}{\sqrt{-3\,{x}^{2}+4\,x-2}}}} \right ) } \]

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

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Maxima [A]  time = 0.751286, size = 34, normalized size = 1.03 \[ \frac{1}{2} i \, \sqrt{2} \operatorname{arsinh}\left (\frac{\sqrt{2} x}{{\left | x \right |}} - \frac{\sqrt{2}}{{\left | x \right |}}\right ) \]

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*I*sqrt(2)*arcsinh(sqrt(2)*x/abs(x) - sqrt(2)/abs(x))

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Fricas [A]  time = 0.220003, size = 86, normalized size = 2.61 \[ \frac{1}{4} \, \sqrt{-2} \log \left (\frac{\sqrt{-2} \sqrt{-3 \, x^{2} + 4 \, x - 2} + 2 \, x - 2}{x}\right ) - \frac{1}{4} \, \sqrt{-2} \log \left (-\frac{\sqrt{-2} \sqrt{-3 \, x^{2} + 4 \, x - 2} - 2 \, x + 2}{x}\right ) \]

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((sqrt(-2)*sqrt(-3*x^2 + 4*x - 2) + 2*x - 2)/x) - 1/4*sqrt(-2)*l
og(-(sqrt(-2)*sqrt(-3*x^2 + 4*x - 2) - 2*x + 2)/x)

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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 = 0.257964, size = 163, normalized size = 4.94 \[ -\frac{2 \, \sqrt{3} i{\rm ln}\left (192 \, i{\left (\sqrt{6} + \sqrt{2}\right )} + \frac{384 \,{\left (\sqrt{2} i + \sqrt{3} \sqrt{-3 \, x^{2} + 4 \, x - 2}\right )}}{3 \, x - 2}\right )}{\sqrt{6} + \sqrt{2}} + \frac{2 \, \sqrt{3} i{\rm ln}\left (-192 \, i{\left (\sqrt{6} - \sqrt{2}\right )} + \frac{384 \,{\left (\sqrt{2} i + \sqrt{3} \sqrt{-3 \, x^{2} + 4 \, x - 2}\right )}}{3 \, x - 2}\right )}{\sqrt{6} - \sqrt{2}} \]

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]

-2*sqrt(3)*i*ln(192*i*(sqrt(6) + sqrt(2)) + 384*(sqrt(2)*i + sqrt(3)*sqrt(-3*x^2
+ 4*x - 2))/(3*x - 2))/(sqrt(6) + sqrt(2)) + 2*sqrt(3)*i*ln(-192*i*(sqrt(6) - s
qrt(2)) + 384*(sqrt(2)*i + sqrt(3)*sqrt(-3*x^2 + 4*x - 2))/(3*x - 2))/(sqrt(6) -
sqrt(2))