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

Optimal. Leaf size=19 \[ -\frac{\tanh ^{-1}\left (\frac{2-3 x}{\sqrt{10}}\right )}{\sqrt{10}} \]

[Out]

-(ArcTanh[(2 - 3*x)/Sqrt[10]]/Sqrt[10])

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Rubi [A]  time = 0.0358445, antiderivative size = 19, 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 \[ -\frac{\tanh ^{-1}\left (\frac{2-3 x}{\sqrt{10}}\right )}{\sqrt{10}} \]

Antiderivative was successfully verified.

[In]  Int[(2 + 4*x - 3*x^2)^(-1),x]

[Out]

-(ArcTanh[(2 - 3*x)/Sqrt[10]]/Sqrt[10])

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Rubi in Sympy [A]  time = 1.43728, size = 22, normalized size = 1.16 \[ - \frac{\sqrt{10} \operatorname{atanh}{\left (\sqrt{10} \left (- \frac{3 x}{10} + \frac{1}{5}\right ) \right )}}{10} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(10)*atanh(sqrt(10)*(-3*x/10 + 1/5))/10

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Mathematica [A]  time = 0.0352976, size = 34, normalized size = 1.79 \[ \frac{\log \left (3 x+\sqrt{10}-2\right )-\log \left (-3 x+\sqrt{10}+2\right )}{2 \sqrt{10}} \]

Antiderivative was successfully verified.

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

[Out]

(-Log[2 + Sqrt[10] - 3*x] + Log[-2 + Sqrt[10] + 3*x])/(2*Sqrt[10])

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Maple [A]  time = 0.002, size = 17, normalized size = 0.9 \[{\frac{\sqrt{10}}{10}{\it Artanh} \left ({\frac{ \left ( 6\,x-4 \right ) \sqrt{10}}{20}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(-3*x^2+4*x+2),x)

[Out]

1/10*10^(1/2)*arctanh(1/20*(6*x-4)*10^(1/2))

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Maxima [A]  time = 0.819524, size = 36, normalized size = 1.89 \[ -\frac{1}{20} \, \sqrt{10} \log \left (\frac{3 \, x - \sqrt{10} - 2}{3 \, x + \sqrt{10} - 2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/20*sqrt(10)*log((3*x - sqrt(10) - 2)/(3*x + sqrt(10) - 2))

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Fricas [A]  time = 0.219648, size = 51, normalized size = 2.68 \[ \frac{1}{20} \, \sqrt{10} \log \left (\frac{\sqrt{10}{\left (9 \, x^{2} - 12 \, x + 14\right )} + 60 \, x - 40}{3 \, x^{2} - 4 \, x - 2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/20*sqrt(10)*log((sqrt(10)*(9*x^2 - 12*x + 14) + 60*x - 40)/(3*x^2 - 4*x - 2))

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Sympy [A]  time = 0.220102, size = 39, normalized size = 2.05 \[ \frac{\sqrt{10} \log{\left (x - \frac{2}{3} + \frac{\sqrt{10}}{3} \right )}}{20} - \frac{\sqrt{10} \log{\left (x - \frac{\sqrt{10}}{3} - \frac{2}{3} \right )}}{20} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(-3*x**2+4*x+2),x)

[Out]

sqrt(10)*log(x - 2/3 + sqrt(10)/3)/20 - sqrt(10)*log(x - sqrt(10)/3 - 2/3)/20

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GIAC/XCAS [A]  time = 0.210915, size = 42, normalized size = 2.21 \[ -\frac{1}{20} \, \sqrt{10}{\rm ln}\left (\frac{{\left | 6 \, x - 2 \, \sqrt{10} - 4 \right |}}{{\left | 6 \, x + 2 \, \sqrt{10} - 4 \right |}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/20*sqrt(10)*ln(abs(6*x - 2*sqrt(10) - 4)/abs(6*x + 2*sqrt(10) - 4))