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

Optimal. Leaf size=23 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{\log ^2(x)-a^2}}{a}\right )}{a} \]

[Out]

ArcTan[Sqrt[-a^2 + Log[x]^2]/a]/a

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Rubi [A]  time = 0.175648, antiderivative size = 23, 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 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{\log ^2(x)-a^2}}{a}\right )}{a} \]

Antiderivative was successfully verified.

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

[Out]

ArcTan[Sqrt[-a^2 + Log[x]^2]/a]/a

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Rubi in Sympy [A]  time = 10.7661, size = 15, normalized size = 0.65 \[ \frac{\operatorname{atan}{\left (\frac{\sqrt{- a^{2} + \log{\left (x \right )}^{2}}}{a} \right )}}{a} \]

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]

atan(sqrt(-a**2 + log(x)**2)/a)/a

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Mathematica [C]  time = 0.0211195, size = 38, normalized size = 1.65 \[ -\frac{i \log \left (\frac{2 \sqrt{\log ^2(x)-a^2}}{\log (x)}-\frac{2 i a}{\log (x)}\right )}{a} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.006, size = 43, normalized size = 1.9 \[ -{1\ln \left ({\frac{1}{\ln \left ( x \right ) } \left ( -2\,{a}^{2}+2\,\sqrt{-{a}^{2}}\sqrt{-{a}^{2}+ \left ( \ln \left ( x \right ) \right ) ^{2}} \right ) } \right ){\frac{1}{\sqrt{-{a}^{2}}}}} \]

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))

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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

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Fricas [A]  time = 0.21002, size = 36, normalized size = 1.57 \[ \frac{2 \, \arctan \left (\frac{\sqrt{-a^{2} + \log \left (x\right )^{2}} - \log \left (x\right )}{a}\right )}{a} \]

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]

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

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

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)

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GIAC/XCAS [A]  time = 0.208225, size = 28, normalized size = 1.22 \[ \frac{\arctan \left (\frac{\sqrt{-a^{2} +{\rm ln}\left (x\right )^{2}}}{a}\right )}{a} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

arctan(sqrt(-a^2 + ln(x)^2)/a)/a