3.2411 \(\int \frac{1}{x \sqrt{-4-12 x-9 x^2}} \, dx\)

Optimal. Leaf size=27 \[ -\frac{(3 x+2) \tanh ^{-1}(3 x+1)}{\sqrt{-9 x^2-12 x-4}} \]

[Out]

-(((2 + 3*x)*ArcTanh[1 + 3*x])/Sqrt[-4 - 12*x - 9*x^2])

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Rubi [B]  time = 0.0489856, antiderivative size = 55, normalized size of antiderivative = 2.04, number of steps used = 4, number of rules used = 4, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222 \[ \frac{(3 x+2) \log (x)}{2 \sqrt{-9 x^2-12 x-4}}-\frac{(3 x+2) \log (3 x+2)}{2 \sqrt{-9 x^2-12 x-4}} \]

Antiderivative was successfully verified.

[In]  Int[1/(x*Sqrt[-4 - 12*x - 9*x^2]),x]

[Out]

((2 + 3*x)*Log[x])/(2*Sqrt[-4 - 12*x - 9*x^2]) - ((2 + 3*x)*Log[2 + 3*x])/(2*Sqr
t[-4 - 12*x - 9*x^2])

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Rubi in Sympy [A]  time = 7.6756, size = 53, normalized size = 1.96 \[ - \frac{3 \sqrt{- 9 x^{2} - 12 x - 4} \log{\left (x \right )}}{2 \left (9 x + 6\right )} + \frac{3 \sqrt{- 9 x^{2} - 12 x - 4} \log{\left (3 x + 2 \right )}}{2 \left (9 x + 6\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-3*sqrt(-9*x**2 - 12*x - 4)*log(x)/(2*(9*x + 6)) + 3*sqrt(-9*x**2 - 12*x - 4)*lo
g(3*x + 2)/(2*(9*x + 6))

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Mathematica [A]  time = 0.0130054, size = 33, normalized size = 1.22 \[ \frac{(3 x+2) (\log (x)-\log (3 x+2))}{2 \sqrt{-(3 x+2)^2}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x*Sqrt[-4 - 12*x - 9*x^2]),x]

[Out]

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

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Maple [A]  time = 0.005, size = 30, normalized size = 1.1 \[{\frac{ \left ( 2+3\,x \right ) \left ( \ln \left ( x \right ) -\ln \left ( 2+3\,x \right ) \right ) }{2}{\frac{1}{\sqrt{- \left ( 2+3\,x \right ) ^{2}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x/(-(2+3*x)^2)^(1/2),x)

[Out]

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

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Maxima [A]  time = 0.753562, size = 32, normalized size = 1.19 \[ -\frac{1}{2} i \, \left (-1\right )^{12 \, x + 8} \log \left (\frac{12 \, x}{{\left | x \right |}} + \frac{8}{{\left | x \right |}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*I*(-1)^(12*x + 8)*log(12*x/abs(x) + 8/abs(x))

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Fricas [A]  time = 0.217407, size = 15, normalized size = 0.56 \[ \frac{1}{2} i \, \log \left (x + \frac{2}{3}\right ) - \frac{1}{2} i \, \log \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*I*log(x + 2/3) - 1/2*I*log(x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(1/(x*sqrt(-(3*x + 2)**2)), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \mathit{undef} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

undef