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 = 0.0663763, 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 \[ \tan ^{-1}\left (\frac{\log (x)}{\sqrt{a^2-\log ^2(x)}}\right ) \]

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]]

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Rubi in Sympy [A]  time = 3.99253, size = 15, normalized size = 0.83 \[ \operatorname{atan}{\left (\frac{\log{\left (x \right )}}{\sqrt{a^{2} - \log{\left (x \right )}^{2}}} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x/(a**2-ln(x)**2)**(1/2),x)

[Out]

atan(log(x)/sqrt(a**2 - log(x)**2))

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Mathematica [A]  time = 0.00765847, 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 = 0.007, size = 17, normalized size = 0.9 \[ \arctan \left ({\ln \left ( x \right ){\frac{1}{\sqrt{{a}^{2}- \left ( \ln \left ( x \right ) \right ) ^{2}}}}} \right ) \]

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 = 1.52835, size = 12, normalized size = 0.67 \[ \arcsin \left (\frac{\log \left (x\right )}{\sqrt{a^{2}}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(a^2 - log(x)^2)*x),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.208032, size = 34, normalized size = 1.89 \[ -2 \, \arctan \left (-\frac{a - \sqrt{a^{2} - \log \left (x\right )^{2}}}{\log \left (x\right )}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(a^2 - log(x)^2)*x),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. \[ \int \frac{1}{x \sqrt{\left (a - \log{\left (x \right )}\right ) \left (a + \log{\left (x \right )}\right )}}\, dx \]

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 - log(x))*(a + log(x)))), x)

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GIAC/XCAS [A]  time = 0.224048, size = 14, normalized size = 0.78 \[ \arcsin \left (\frac{{\rm ln}\left (x\right )}{a}\right ){\rm sign}\left (a\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(a^2 - log(x)^2)*x),x, algorithm="giac")

[Out]

arcsin(ln(x)/a)*sign(a)