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