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[Log[x]/2]

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Rubi [A]  time = 0.0291678, antiderivative size = 7, normalized size of antiderivative = 1., 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*}

Mathematica [A]  time = 0.0153227, size = 7, normalized size = 1. \[ \sinh ^{-1}\left (\frac{\log (x)}{2}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x*Sqrt[4 + Log[x]^2]),x]

[Out]

ArcSinh[Log[x]/2]

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Maple [A]  time = 0.01, size = 6, normalized size = 0.9 \begin{align*}{\it Arcsinh} \left ({\frac{\ln \left ( x \right ) }{2}} \right ) \end{align*}

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 = 1.61202, size = 7, normalized size = 1. \begin{align*} \operatorname{arsinh}\left (\frac{1}{2} \, \log \left (x\right )\right ) \end{align*}

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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Fricas [B]  time = 2.06776, size = 47, normalized size = 6.71 \begin{align*} -\log \left (\sqrt{\log \left (x\right )^{2} + 4} - \log \left (x\right )\right ) \end{align*}

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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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{x \sqrt{\log{\left (x \right )}^{2} + 4}}\, dx \end{align*}

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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Giac [B]  time = 1.23523, size = 22, normalized size = 3.14 \begin{align*} -\log \left (\sqrt{\log \left (x\right )^{2} + 4} - \log \left (x\right )\right ) \end{align*}

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))