3.2.35 \(\int \frac {1}{x \sqrt {4-9 \log ^2(x)}} \, dx\) [135]

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 = 0.02, 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 = {222} \begin {gather*} \frac {1}{3} \text {ArcSin}\left (\frac {3 \log (x)}{2}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(x*Sqrt[4 - 9*Log[x]^2]),x]

[Out]

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

Rule 222

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 &=\text {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 [C] Result contains complex when optimal does not.
time = 0.01, size = 26, normalized size = 2.36 \begin {gather*} \frac {1}{3} i \log \left (-3 i \log (x)+\sqrt {4-9 \log ^2(x)}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]
time = 0.01, size = 8, normalized size = 0.73

method	result	size
derivativedivides	\(\frac {\arcsin \left (\frac {3 \ln \left (x \right )}{2}\right )}{3}\)	\(8\)
default	\(\frac {\arcsin \left (\frac {3 \ln \left (x \right )}{2}\right )}{3}\)	\(8\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.49, size = 7, normalized size = 0.64 \begin {gather*} \frac {1}{3} \, \arcsin \left (\frac {3}{2} \, \log \left (x\right )\right ) \end {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. 21 vs. \(2 (7) = 14\).
time = 0.38, size = 21, normalized size = 1.91 \begin {gather*} -\frac {2}{3} \, \arctan \left (\frac {\sqrt {-9 \, \log \left (x\right )^{2} + 4} - 2}{3 \, \log \left (x\right )}\right ) \end {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{x \sqrt {- \left (3 \log {\left (x \right )} - 2\right ) \left (3 \log {\left (x \right )} + 2\right )}}\, dx \end {gather*}

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 = 4.55, size = 7, normalized size = 0.64 \begin {gather*} \frac {1}{3} \, \arcsin \left (\frac {3}{2} \, \log \left (x\right )\right ) \end {gather*}

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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Mupad [B]
time = 0.40, size = 7, normalized size = 0.64 \begin {gather*} \frac {\mathrm {asin}\left (\frac {3\,\ln \left (x\right )}{2}\right )}{3} \end {gather*}

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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