\(\int \frac {1}{x \sqrt {4+\log ^2(x)}} \, dx\) [136]

Optimal result

Integrand size = 14, antiderivative size = 7 \[ \int \frac {1}{x \sqrt {4+\log ^2(x)}} \, dx=\text {arcsinh}\left (\frac {\log (x)}{2}\right ) \]

[Out]

arcsinh(1/2*ln(x))

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 7, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {221} \[ \int \frac {1}{x \sqrt {4+\log ^2(x)}} \, dx=\text {arcsinh}\left (\frac {\log (x)}{2}\right ) \]

[In]

Int[1/(x*Sqrt[4 + Log[x]^2]),x]

[Out]

ArcSinh[Log[x]/2]

Rule 221

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*} \text {integral}& = \text {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 [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(18\) vs. \(2(7)=14\).

Time = 0.03 (sec) , antiderivative size = 18, normalized size of antiderivative = 2.57 \[ \int \frac {1}{x \sqrt {4+\log ^2(x)}} \, dx=-\log \left (-\log (x)+\sqrt {4+\log ^2(x)}\right ) \]

[In]

Integrate[1/(x*Sqrt[4 + Log[x]^2]),x]

[Out]

-Log[-Log[x] + Sqrt[4 + Log[x]^2]]

Maple [A] (verified)

Time = 0.17 (sec) , antiderivative size = 6, normalized size of antiderivative = 0.86

method	result	size
derivativedivides	\(\operatorname {arcsinh}\left (\frac {\ln \left (x \right )}{2}\right )\)	\(6\)
default	\(\operatorname {arcsinh}\left (\frac {\ln \left (x \right )}{2}\right )\)	\(6\)

[In]

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

[Out]

arcsinh(1/2*ln(x))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 16 vs. \(2 (5) = 10\).

Time = 0.30 (sec) , antiderivative size = 16, normalized size of antiderivative = 2.29 \[ \int \frac {1}{x \sqrt {4+\log ^2(x)}} \, dx=-\log \left (\sqrt {\log \left (x\right )^{2} + 4} - \log \left (x\right )\right ) \]

[In]

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

[Out]

-log(sqrt(log(x)^2 + 4) - log(x))

Sympy [F]

\[ \int \frac {1}{x \sqrt {4+\log ^2(x)}} \, dx=\int \frac {1}{x \sqrt {\log {\left (x \right )}^{2} + 4}}\, dx \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 5, normalized size of antiderivative = 0.71 \[ \int \frac {1}{x \sqrt {4+\log ^2(x)}} \, dx=\operatorname {arsinh}\left (\frac {1}{2} \, \log \left (x\right )\right ) \]

[In]

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

[Out]

arcsinh(1/2*log(x))

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 16 vs. \(2 (5) = 10\).

Time = 0.35 (sec) , antiderivative size = 16, normalized size of antiderivative = 2.29 \[ \int \frac {1}{x \sqrt {4+\log ^2(x)}} \, dx=-\log \left (\sqrt {\log \left (x\right )^{2} + 4} - \log \left (x\right )\right ) \]

[In]

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

[Out]

-log(sqrt(log(x)^2 + 4) - log(x))

Mupad [B] (verification not implemented)

Time = 1.53 (sec) , antiderivative size = 5, normalized size of antiderivative = 0.71 \[ \int \frac {1}{x \sqrt {4+\log ^2(x)}} \, dx=\mathrm {asinh}\left (\frac {\ln \left (x\right )}{2}\right ) \]

[In]

int(1/(x*(log(x)^2 + 4)^(1/2)),x)

[Out]

asinh(log(x)/2)