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 = 0.098703, 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, Rules used = {266, 63, 203} \[ \frac{\tan ^{-1}\left (\frac{\sqrt{\log ^2(x)-a^2}}{a}\right )}{a} \]
Antiderivative was successfully verified.
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Rule 266
Rule 63
Rule 203
Rubi steps
\begin{align*} \int \frac{1}{x \log (x) \sqrt{-a^2+\log ^2(x)}} \, dx &=\operatorname{Subst}\left (\int \frac{1}{x \sqrt{-a^2+x^2}} \, dx,x,\log (x)\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{x \sqrt{-a^2+x}} \, dx,x,\log ^2(x)\right )\\ &=\operatorname{Subst}\left (\int \frac{1}{a^2+x^2} \, dx,x,\sqrt{-a^2+\log ^2(x)}\right )\\ &=\frac{\tan ^{-1}\left (\frac{\sqrt{-a^2+\log ^2(x)}}{a}\right )}{a}\\ \end{align*}
Mathematica [A] time = 0.0156958, size = 23, normalized size = 1. \[ \frac{\tan ^{-1}\left (\frac{\sqrt{\log ^2(x)-a^2}}{a}\right )}{a} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 43, normalized size = 1.9 \begin{align*} -{\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}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.33496, size = 65, normalized size = 2.83 \begin{align*} \frac{2 \, \arctan \left (\frac{\sqrt{-a^{2} + \log \left (x\right )^{2}} - \log \left (x\right )}{a}\right )}{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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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 )} \log{\left (x \right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.06522, size = 28, normalized size = 1.22 \begin{align*} \frac{\arctan \left (\frac{\sqrt{-a^{2} + \log \left (x\right )^{2}}}{a}\right )}{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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