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

Optimal. Leaf size=24 \[ -\frac{\tanh ^{-1}\left (\frac{\sqrt{a^2-\log ^2(x)}}{a}\right )}{a} \]

[Out]

-(ArcTanh[Sqrt[a^2 - Log[x]^2]/a]/a)

_______________________________________________________________________________________

Rubi [A]  time = 0.174372, antiderivative size = 24, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136 \[ -\frac{\tanh ^{-1}\left (\frac{\sqrt{a^2-\log ^2(x)}}{a}\right )}{a} \]

Antiderivative was successfully verified.

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

[Out]

-(ArcTanh[Sqrt[a^2 - Log[x]^2]/a]/a)

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 10.9284, size = 17, normalized size = 0.71 \[ - \frac{\operatorname{atanh}{\left (\frac{\sqrt{a^{2} - \log{\left (x \right )}^{2}}}{a} \right )}}{a} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-atanh(sqrt(a**2 - log(x)**2)/a)/a

_______________________________________________________________________________________

Mathematica [A]  time = 0.0174592, size = 34, normalized size = 1.42 \[ \frac{\log (\log (x))}{a}-\frac{\log \left (a \sqrt{a^2-\log ^2(x)}+a^2\right )}{a} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Maple [A]  time = 0.007, size = 39, normalized size = 1.6 \[ -{1\ln \left ({\frac{1}{\ln \left ( x \right ) } \left ( 2\,{a}^{2}+2\,\sqrt{{a}^{2}}\sqrt{{a}^{2}- \left ( \ln \left ( x \right ) \right ) ^{2}} \right ) } \right ){\frac{1}{\sqrt{{a}^{2}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x/ln(x)/(a^2-ln(x)^2)^(1/2),x)

[Out]

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

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.199954, size = 36, normalized size = 1.5 \[ \frac{\log \left (-\frac{a - \sqrt{a^{2} - \log \left (x\right )^{2}}}{\log \left (x\right )}\right )}{a} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 20.3169, size = 4, normalized size = 0.17 \[ \mathit{sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sage0*x