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

Optimal. Leaf size=16 \[ \tanh ^{-1}\left (\frac {\log (x)}{\sqrt {a^2+\log ^2(x)}}\right ) \]

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Rubi [A]  time = 0.04, antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {217, 206} \[ \tanh ^{-1}\left (\frac {\log (x)}{\sqrt {a^2+\log ^2(x)}}\right ) \]

Antiderivative was successfully verified.

[In]

Int[1/(x*Sqrt[a^2 + Log[x]^2]),x]

[Out]

ArcTanh[Log[x]/Sqrt[a^2 + Log[x]^2]]

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 217

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

Rubi steps

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

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Mathematica [B]  time = 0.03, size = 46, normalized size = 2.88 \[ \frac {1}{2} \log \left (\frac {\log (x)}{\sqrt {a^2+\log ^2(x)}}+1\right )-\frac {1}{2} \log \left (1-\frac {\log (x)}{\sqrt {a^2+\log ^2(x)}}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x*Sqrt[a^2 + Log[x]^2]),x]

[Out]

-1/2*Log[1 - Log[x]/Sqrt[a^2 + Log[x]^2]] + Log[1 + Log[x]/Sqrt[a^2 + Log[x]^2]]/2

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

Verification is Not applicable to the result.

[In]

IntegrateAlgebraic[1/(x*Sqrt[a^2 + Log[x]^2]),x]

[Out]

Could not integrate

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fricas [A]  time = 0.80, size = 18, normalized size = 1.12 \[ -\log \left (\sqrt {a^{2} + \log \relax (x)^{2}} - \log \relax (x)\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-log(sqrt(a^2 + log(x)^2) - log(x))

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giac [A]  time = 0.60, size = 18, normalized size = 1.12 \[ -\log \left (\sqrt {a^{2} + \log \relax (x)^{2}} - \log \relax (x)\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-log(sqrt(a^2 + log(x)^2) - log(x))

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maple [A]  time = 0.02, size = 15, normalized size = 0.94

method	result	size
derivativedivides	\(\ln \left (\ln \relax (x )+\sqrt {a^{2}+\ln \relax (x )^{2}}\right )\)	\(15\)
default	\(\ln \left (\ln \relax (x )+\sqrt {a^{2}+\ln \relax (x )^{2}}\right )\)	\(15\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

ln(ln(x)+(a^2+ln(x)^2)^(1/2))

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maxima [A]  time = 0.42, size = 7, normalized size = 0.44 \[ \operatorname {arsinh}\left (\frac {\log \relax (x)}{a}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

arcsinh(log(x)/a)

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mupad [B]  time = 0.43, size = 14, normalized size = 0.88 \[ \ln \left (\ln \relax (x)+\sqrt {a^2+{\ln \relax (x)}^2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x*(log(x)^2 + a^2)^(1/2)),x)

[Out]

log(log(x) + (log(x)^2 + a^2)^(1/2))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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