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

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

[Out]

ArcSin[(3*Log[x])/2]/3

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Rubi [A]  time = 0.0350316, antiderivative size = 11, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.062, Rules used = {216} \[ \frac{1}{3} \sin ^{-1}\left (\frac{3 \log (x)}{2}\right ) \]

Antiderivative was successfully verified.

[In]

Int[1/(x*Sqrt[4 - 9*Log[x]^2]),x]

[Out]

ArcSin[(3*Log[x])/2]/3

Rule 216

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[(Rt[-b, 2]*x)/Sqrt[a]]/Rt[-b, 2], x] /; FreeQ[{a, b}
, x] && GtQ[a, 0] && NegQ[b]

Rubi steps

\begin{align*} \int \frac{1}{x \sqrt{4-9 \log ^2(x)}} \, dx &=\operatorname{Subst}\left (\int \frac{1}{\sqrt{4-9 x^2}} \, dx,x,\log (x)\right )\\ &=\frac{1}{3} \sin ^{-1}\left (\frac{3 \log (x)}{2}\right )\\ \end{align*}

Mathematica [A]  time = 0.0188951, size = 11, normalized size = 1. \[ \frac{1}{3} \sin ^{-1}\left (\frac{3 \log (x)}{2}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

ArcSin[(3*Log[x])/2]/3

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Maple [A]  time = 0.012, size = 8, normalized size = 0.7 \begin{align*}{\frac{1}{3}\arcsin \left ({\frac{3\,\ln \left ( x \right ) }{2}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x/(4-9*ln(x)^2)^(1/2),x)

[Out]

1/3*arcsin(3/2*ln(x))

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Maxima [A]  time = 1.64437, size = 9, normalized size = 0.82 \begin{align*} \frac{1}{3} \, \arcsin \left (\frac{3}{2} \, \log \left (x\right )\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(4-9*log(x)^2)^(1/2),x, algorithm="maxima")

[Out]

1/3*arcsin(3/2*log(x))

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Fricas [B]  time = 2.03161, size = 72, normalized size = 6.55 \begin{align*} -\frac{2}{3} \, \arctan \left (\frac{\sqrt{-9 \, \log \left (x\right )^{2} + 4} - 2}{3 \, \log \left (x\right )}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(4-9*log(x)^2)^(1/2),x, algorithm="fricas")

[Out]

-2/3*arctan(1/3*(sqrt(-9*log(x)^2 + 4) - 2)/log(x))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(4-9*ln(x)**2)**(1/2),x)

[Out]

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

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Giac [A]  time = 1.19744, size = 9, normalized size = 0.82 \begin{align*} \frac{1}{3} \, \arcsin \left (\frac{3}{2} \, \log \left (x\right )\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(4-9*log(x)^2)^(1/2),x, algorithm="giac")

[Out]

1/3*arcsin(3/2*log(x))