3.116 \(\int \frac{1}{\sqrt{-8+6 x+9 x^2}} \, dx\)

Optimal. Leaf size=25 \[ \frac{1}{3} \tanh ^{-1}\left (\frac{3 x+1}{\sqrt{9 x^2+6 x-8}}\right ) \]

[Out]

ArcTanh[(1 + 3*x)/Sqrt[-8 + 6*x + 9*x^2]]/3

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Rubi [A]  time = 0.0155982, antiderivative size = 25, 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 \[ \frac{1}{3} \tanh ^{-1}\left (\frac{3 x+1}{\sqrt{9 x^2+6 x-8}}\right ) \]

Antiderivative was successfully verified.

[In]  Int[1/Sqrt[-8 + 6*x + 9*x^2],x]

[Out]

ArcTanh[(1 + 3*x)/Sqrt[-8 + 6*x + 9*x^2]]/3

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Rubi in Sympy [A]  time = 1.48554, size = 22, normalized size = 0.88 \[ \frac{\operatorname{atanh}{\left (\frac{18 x + 6}{6 \sqrt{9 x^{2} + 6 x - 8}} \right )}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(9*x**2+6*x-8)**(1/2),x)

[Out]

atanh((18*x + 6)/(6*sqrt(9*x**2 + 6*x - 8)))/3

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Mathematica [A]  time = 0.00978668, size = 24, normalized size = 0.96 \[ \frac{1}{3} \log \left (\sqrt{9 x^2+6 x-8}+3 x+1\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[1/Sqrt[-8 + 6*x + 9*x^2],x]

[Out]

Log[1 + 3*x + Sqrt[-8 + 6*x + 9*x^2]]/3

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Maple [A]  time = 0.005, size = 30, normalized size = 1.2 \[{\frac{\sqrt{9}}{9}\ln \left ({\frac{ \left ( 9\,x+3 \right ) \sqrt{9}}{9}}+\sqrt{9\,{x}^{2}+6\,x-8} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(9*x^2+6*x-8)^(1/2),x)

[Out]

1/9*ln(1/9*(9*x+3)*9^(1/2)+(9*x^2+6*x-8)^(1/2))*9^(1/2)

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Maxima [A]  time = 0.872102, size = 30, normalized size = 1.2 \[ \frac{1}{3} \, \log \left (18 \, x + 6 \, \sqrt{9 \, x^{2} + 6 \, x - 8} + 6\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/sqrt(9*x^2 + 6*x - 8),x, algorithm="maxima")

[Out]

1/3*log(18*x + 6*sqrt(9*x^2 + 6*x - 8) + 6)

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Fricas [A]  time = 0.233257, size = 27, normalized size = 1.08 \[ -\frac{1}{3} \, \log \left (-3 \, x + \sqrt{9 \, x^{2} + 6 \, x - 8} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/sqrt(9*x^2 + 6*x - 8),x, algorithm="fricas")

[Out]

-1/3*log(-3*x + sqrt(9*x^2 + 6*x - 8) - 1)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{9 x^{2} + 6 x - 8}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(9*x**2+6*x-8)**(1/2),x)

[Out]

Integral(1/sqrt(9*x**2 + 6*x - 8), x)

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GIAC/XCAS [A]  time = 0.211914, size = 28, normalized size = 1.12 \[ -\frac{1}{3} \,{\rm ln}\left ({\left | -3 \, x + \sqrt{9 \, x^{2} + 6 \, x - 8} - 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/sqrt(9*x^2 + 6*x - 8),x, algorithm="giac")

[Out]

-1/3*ln(abs(-3*x + sqrt(9*x^2 + 6*x - 8) - 1))