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

Optimal. Leaf size=18 \[ \tanh ^{-1}\left (\frac{\log (x)}{\sqrt{\log ^2(x)-a^2}}\right ) \]

[Out]

ArcTanh[Log[x]/Sqrt[-a^2 + Log[x]^2]]

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Rubi [A]  time = 0.042227, antiderivative size = 18, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {217, 206} \[ \tanh ^{-1}\left (\frac{\log (x)}{\sqrt{\log ^2(x)-a^2}}\right ) \]

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 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{-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*}

Mathematica [B]  time = 0.0241887, size = 50, normalized size = 2.78 \[ \frac{1}{2} \log \left (\frac{\log (x)}{\sqrt{\log ^2(x)-a^2}}+1\right )-\frac{1}{2} \log \left (1-\frac{\log (x)}{\sqrt{\log ^2(x)-a^2}}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.007, size = 17, normalized size = 0.9 \begin{align*} \ln \left ( \ln \left ( x \right ) +\sqrt{-{a}^{2}+ \left ( \ln \left ( x \right ) \right ) ^{2}} \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.935055, size = 27, normalized size = 1.5 \begin{align*} \log \left (2 \, \sqrt{-a^{2} + \log \left (x\right )^{2}} + 2 \, \log \left (x\right )\right ) \end{align*}

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]

log(2*sqrt(-a^2 + log(x)^2) + 2*log(x))

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Fricas [A]  time = 2.41528, size = 51, normalized size = 2.83 \begin{align*} -\log \left (\sqrt{-a^{2} + \log \left (x\right )^{2}} - \log \left (x\right )\right ) \end{align*}

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

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 - log(x))*(a + log(x)))), x)

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Giac [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

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]

Timed out