3.55 \(\int \frac{1}{\sqrt{5-4 x+3 x^2}} \, dx\)

Optimal. Leaf size=19 \[ -\frac{\sinh ^{-1}\left (\frac{2-3 x}{\sqrt{11}}\right )}{\sqrt{3}} \]

[Out]

-(ArcSinh[(2 - 3*x)/Sqrt[11]]/Sqrt[3])

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Rubi [A]  time = 0.0278955, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143 \[ -\frac{\sinh ^{-1}\left (\frac{2-3 x}{\sqrt{11}}\right )}{\sqrt{3}} \]

Antiderivative was successfully verified.

[In]  Int[1/Sqrt[5 - 4*x + 3*x^2],x]

[Out]

-(ArcSinh[(2 - 3*x)/Sqrt[11]]/Sqrt[3])

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Rubi in Sympy [A]  time = 0.732332, size = 32, normalized size = 1.68 \[ \frac{\sqrt{3} \operatorname{atanh}{\left (\frac{\sqrt{3} \left (6 x - 4\right )}{6 \sqrt{3 x^{2} - 4 x + 5}} \right )}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(3*x**2-4*x+5)**(1/2),x)

[Out]

sqrt(3)*atanh(sqrt(3)*(6*x - 4)/(6*sqrt(3*x**2 - 4*x + 5)))/3

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Mathematica [A]  time = 0.0131011, size = 18, normalized size = 0.95 \[ \frac{\sinh ^{-1}\left (\frac{3 x-2}{\sqrt{11}}\right )}{\sqrt{3}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/Sqrt[5 - 4*x + 3*x^2],x]

[Out]

ArcSinh[(-2 + 3*x)/Sqrt[11]]/Sqrt[3]

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Maple [A]  time = 0.004, size = 15, normalized size = 0.8 \[{\frac{\sqrt{3}}{3}{\it Arcsinh} \left ({\frac{3\,\sqrt{11}}{11} \left ( x-{\frac{2}{3}} \right ) } \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(3*x^2-4*x+5)^(1/2),x)

[Out]

1/3*3^(1/2)*arcsinh(3/11*11^(1/2)*(x-2/3))

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Maxima [A]  time = 1.54136, size = 22, normalized size = 1.16 \[ \frac{1}{3} \, \sqrt{3} \operatorname{arsinh}\left (\frac{1}{11} \, \sqrt{11}{\left (3 \, x - 2\right )}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/sqrt(3*x^2 - 4*x + 5),x, algorithm="maxima")

[Out]

1/3*sqrt(3)*arcsinh(1/11*sqrt(11)*(3*x - 2))

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Fricas [A]  time = 0.226077, size = 55, normalized size = 2.89 \[ \frac{1}{6} \, \sqrt{3} \log \left (-\sqrt{3}{\left (18 \, x^{2} - 24 \, x + 19\right )} - 6 \, \sqrt{3 \, x^{2} - 4 \, x + 5}{\left (3 \, x - 2\right )}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/sqrt(3*x^2 - 4*x + 5),x, algorithm="fricas")

[Out]

1/6*sqrt(3)*log(-sqrt(3)*(18*x^2 - 24*x + 19) - 6*sqrt(3*x^2 - 4*x + 5)*(3*x - 2
))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{3 x^{2} - 4 x + 5}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(3*x**2-4*x+5)**(1/2),x)

[Out]

Integral(1/sqrt(3*x**2 - 4*x + 5), x)

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GIAC/XCAS [A]  time = 0.216172, size = 45, normalized size = 2.37 \[ -\frac{1}{3} \, \sqrt{3}{\rm ln}\left (-\sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} - 4 \, x + 5}\right )} + 2\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/sqrt(3*x^2 - 4*x + 5),x, algorithm="giac")

[Out]

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