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]

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

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Rubi [A]  time = 0.04, antiderivative size = 11, normalized size of antiderivative = 1.00, 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*}

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Mathematica [A]  time = 0.02, size = 11, normalized size = 1.00 \[ \frac {1}{3} \sin ^{-1}\left (\frac {3 \log (x)}{2}\right ) \]

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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fricas [B]  time = 1.51, size = 21, normalized size = 1.91 \[ -\frac {2}{3} \, \arctan \left (\frac {\sqrt {-9 \, \log \relax (x)^{2} + 4} - 2}{3 \, \log \relax (x)}\right ) \]

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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giac [A]  time = 0.23, size = 7, normalized size = 0.64 \[ \frac {1}{3} \, \arcsin \left (\frac {3}{2} \, \log \relax (x)\right ) \]

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))

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maple [A]  time = 0.07, size = 8, normalized size = 0.73 \[ \frac {\arcsin \left (\frac {3 \ln \relax (x )}{2}\right )}{3} \]

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 = 1.22, size = 7, normalized size = 0.64 \[ \frac {1}{3} \, \arcsin \left (\frac {3}{2} \, \log \relax (x)\right ) \]

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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mupad [B]  time = 0.40, size = 7, normalized size = 0.64 \[ \frac {\mathrm {asin}\left (\frac {3\,\ln \relax (x)}{2}\right )}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x*(4 - 9*log(x)^2)^(1/2)),x)

[Out]

asin((3*log(x))/2)/3

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{x \sqrt {- \left (3 \log {\relax (x )} - 2\right ) \left (3 \log {\relax (x )} + 2\right )}}\, dx \]

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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