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

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

[Out]

ArcTan[Log[x]/Sqrt[a^2 - Log[x]^2]]

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Rubi [A]  time = 0.041188, 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, 203} \[ \tan ^{-1}\left (\frac{\log (x)}{\sqrt{a^2-\log ^2(x)}}\right ) \]

Antiderivative was successfully verified.

[In]

Int[1/(x*Sqrt[a^2 - Log[x]^2]),x]

[Out]

ArcTan[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 203

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTan[(Rt[b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[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 )\\ &=\tan ^{-1}\left (\frac{\log (x)}{\sqrt{a^2-\log ^2(x)}}\right )\\ \end{align*}

Mathematica [A]  time = 0.0241556, size = 18, normalized size = 1. \[ \tan ^{-1}\left (\frac{\log (x)}{\sqrt{a^2-\log ^2(x)}}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

ArcTan[Log[x]/Sqrt[a^2 - Log[x]^2]]

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Maple [A]  time = 0.006, size = 17, normalized size = 0.9 \begin{align*} \arctan \left ({\ln \left ( x \right ){\frac{1}{\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]

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

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Maxima [A]  time = 1.4427, size = 12, normalized size = 0.67 \begin{align*} \arcsin \left (\frac{\log \left (x\right )}{\sqrt{a^{2}}}\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]

arcsin(log(x)/sqrt(a^2))

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Fricas [A]  time = 2.47214, size = 63, normalized size = 3.5 \begin{align*} -2 \, \arctan \left (-\frac{a - \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]

-2*arctan(-(a - 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 [A]  time = 1.38456, size = 14, normalized size = 0.78 \begin{align*} \arcsin \left (\frac{\log \left (x\right )}{a}\right ) \mathrm{sgn}\left (a\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="giac")

[Out]

arcsin(log(x)/a)*sgn(a)