Optimal. Leaf size=16 \[ \tanh ^{-1}\left (\frac{\log (x)}{\sqrt{a^2+\log ^2(x)}}\right ) \]
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Rubi [A] time = 0.0376583, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {217, 206} \[ \tanh ^{-1}\left (\frac{\log (x)}{\sqrt{a^2+\log ^2(x)}}\right ) \]
Antiderivative was successfully verified.
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Rule 217
Rule 206
Rubi steps
\begin{align*} \int \frac{1}{x \sqrt{a^2+\log ^2(x)}} \, dx &=\operatorname{Subst}\left (\int \frac{1}{\sqrt{a^2+x^2}} \, dx,x,\log (x)\right )\\ &=\operatorname{Subst}\left (\int \frac{1}{1-x^2} \, dx,x,\frac{\log (x)}{\sqrt{a^2+\log ^2(x)}}\right )\\ &=\tanh ^{-1}\left (\frac{\log (x)}{\sqrt{a^2+\log ^2(x)}}\right )\\ \end{align*}
Mathematica [B] time = 0.021873, size = 46, normalized size = 2.88 \[ \frac{1}{2} \log \left (\frac{\log (x)}{\sqrt{a^2+\log ^2(x)}}+1\right )-\frac{1}{2} \log \left (1-\frac{\log (x)}{\sqrt{a^2+\log ^2(x)}}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 15, normalized size = 0.9 \begin{align*} \ln \left ( \ln \left ( x \right ) +\sqrt{{a}^{2}+ \left ( \ln \left ( x \right ) \right ) ^{2}} \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.948843, size = 12, normalized size = 0.75 \begin{align*} \operatorname{arsinh}\left (\frac{\log \left (x\right )}{\sqrt{a^{2}}}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.4954, size = 50, normalized size = 3.12 \begin{align*} -\log \left (\sqrt{a^{2} + \log \left (x\right )^{2}} - \log \left (x\right )\right ) \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{a^{2} + \log{\left (x \right )}^{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.11912, size = 24, normalized size = 1.5 \begin{align*} -\log \left (\sqrt{a^{2} + \log \left (x\right )^{2}} - \log \left (x\right )\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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