3.134 \(\int \frac{1}{x \sqrt{-3+\log ^2(x)}} \, dx\)

Optimal. Leaf size=14 \[ \tanh ^{-1}\left (\frac{\log (x)}{\sqrt{\log ^2(x)-3}}\right ) \]

[Out]

ArcTanh[Log[x]/Sqrt[-3 + Log[x]^2]]

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Rubi [A]  time = 0.0323525, antiderivative size = 14, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {217, 206} \[ \tanh ^{-1}\left (\frac{\log (x)}{\sqrt{\log ^2(x)-3}}\right ) \]

Antiderivative was successfully verified.

[In]

Int[1/(x*Sqrt[-3 + Log[x]^2]),x]

[Out]

ArcTanh[Log[x]/Sqrt[-3 + Log[x]^2]]

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]

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])

Rubi steps

\begin{align*} \int \frac{1}{x \sqrt{-3+\log ^2(x)}} \, dx &=\operatorname{Subst}\left (\int \frac{1}{\sqrt{-3+x^2}} \, dx,x,\log (x)\right )\\ &=\operatorname{Subst}\left (\int \frac{1}{1-x^2} \, dx,x,\frac{\log (x)}{\sqrt{-3+\log ^2(x)}}\right )\\ &=\tanh ^{-1}\left (\frac{\log (x)}{\sqrt{-3+\log ^2(x)}}\right )\\ \end{align*}

Mathematica [B]  time = 0.0148386, size = 42, normalized size = 3. \[ \frac{1}{2} \log \left (\frac{\log (x)}{\sqrt{\log ^2(x)-3}}+1\right )-\frac{1}{2} \log \left (1-\frac{\log (x)}{\sqrt{\log ^2(x)-3}}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x*Sqrt[-3 + Log[x]^2]),x]

[Out]

-Log[1 - Log[x]/Sqrt[-3 + Log[x]^2]]/2 + Log[1 + Log[x]/Sqrt[-3 + Log[x]^2]]/2

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Maple [A]  time = 0.01, size = 13, normalized size = 0.9 \begin{align*} \ln \left ( \ln \left ( x \right ) +\sqrt{-3+ \left ( \ln \left ( x \right ) \right ) ^{2}} \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x/(-3+ln(x)^2)^(1/2),x)

[Out]

ln(ln(x)+(-3+ln(x)^2)^(1/2))

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Maxima [A]  time = 1.10158, size = 22, normalized size = 1.57 \begin{align*} \log \left (2 \, \sqrt{\log \left (x\right )^{2} - 3} + 2 \, \log \left (x\right )\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(-3+log(x)^2)^(1/2),x, algorithm="maxima")

[Out]

log(2*sqrt(log(x)^2 - 3) + 2*log(x))

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Fricas [A]  time = 1.97105, size = 47, normalized size = 3.36 \begin{align*} -\log \left (\sqrt{\log \left (x\right )^{2} - 3} - \log \left (x\right )\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(-3+log(x)^2)^(1/2),x, algorithm="fricas")

[Out]

-log(sqrt(log(x)^2 - 3) - 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} - 3}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(-3+ln(x)**2)**(1/2),x)

[Out]

Integral(1/(x*sqrt(log(x)**2 - 3)), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \mathit{sage}_{0} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(-3+log(x)^2)^(1/2),x, algorithm="giac")

[Out]

sage0*x