3.136 \(\int \frac {1}{x \sqrt {4+\log ^2(x)}} \, dx\)

Optimal. Leaf size=7 \[ \sinh ^{-1}\left (\frac {\log (x)}{2}\right ) \]

[Out]

arcsinh(1/2*ln(x))

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Rubi [A]  time = 0.03, antiderivative size = 7, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {215} \[ \sinh ^{-1}\left (\frac {\log (x)}{2}\right ) \]

Antiderivative was successfully verified.

[In]

Int[1/(x*Sqrt[4 + Log[x]^2]),x]

[Out]

ArcSinh[Log[x]/2]

Rule 215

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

Rubi steps

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

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Mathematica [A]  time = 0.02, size = 7, normalized size = 1.00 \[ \sinh ^{-1}\left (\frac {\log (x)}{2}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x*Sqrt[4 + Log[x]^2]),x]

[Out]

ArcSinh[Log[x]/2]

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fricas [B]  time = 1.08, size = 16, normalized size = 2.29 \[ -\log \left (\sqrt {\log \relax (x)^{2} + 4} - \log \relax (x)\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(4+log(x)^2)^(1/2),x, algorithm="fricas")

[Out]

-log(sqrt(log(x)^2 + 4) - log(x))

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giac [B]  time = 0.20, size = 16, normalized size = 2.29 \[ -\log \left (\sqrt {\log \relax (x)^{2} + 4} - \log \relax (x)\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(4+log(x)^2)^(1/2),x, algorithm="giac")

[Out]

-log(sqrt(log(x)^2 + 4) - log(x))

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maple [A]  time = 0.07, size = 6, normalized size = 0.86 \[ \arcsinh \left (\frac {\ln \relax (x )}{2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x/(4+ln(x)^2)^(1/2),x)

[Out]

arcsinh(1/2*ln(x))

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maxima [A]  time = 1.36, size = 5, normalized size = 0.71 \[ \operatorname {arsinh}\left (\frac {1}{2} \, \log \relax (x)\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(4+log(x)^2)^(1/2),x, algorithm="maxima")

[Out]

arcsinh(1/2*log(x))

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mupad [B]  time = 0.39, size = 5, normalized size = 0.71 \[ \mathrm {asinh}\left (\frac {\ln \relax (x)}{2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x*(log(x)^2 + 4)^(1/2)),x)

[Out]

asinh(log(x)/2)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{x \sqrt {\log {\relax (x )}^{2} + 4}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(4+ln(x)**2)**(1/2),x)

[Out]

Integral(1/(x*sqrt(log(x)**2 + 4)), x)

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