Optimal. Leaf size=18 \[ \tanh ^{-1}\left (\frac {\log (x)}{\sqrt {-a^2+\log ^2(x)}}\right ) \]
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Rubi [A]
time = 0.03, antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {223, 212}
\begin {gather*} \tanh ^{-1}\left (\frac {\log (x)}{\sqrt {\log ^2(x)-a^2}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 223
Rubi steps
\begin {align*} \int \frac {1}{x \sqrt {-a^2+\log ^2(x)}} \, dx &=\text {Subst}\left (\int \frac {1}{\sqrt {-a^2+x^2}} \, dx,x,\log (x)\right )\\ &=\text {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*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(50\) vs. \(2(18)=36\).
time = 0.02, size = 50, normalized size = 2.78 \begin {gather*} -\frac {1}{2} \log \left (1-\frac {\log (x)}{\sqrt {-a^2+\log ^2(x)}}\right )+\frac {1}{2} \log \left (1+\frac {\log (x)}{\sqrt {-a^2+\log ^2(x)}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.02, size = 17, normalized size = 0.94
method | result | size |
derivativedivides | \(\ln \left (\ln \left (x \right )+\sqrt {-a^{2}+\ln \left (x \right )^{2}}\right )\) | \(17\) |
default | \(\ln \left (\ln \left (x \right )+\sqrt {-a^{2}+\ln \left (x \right )^{2}}\right )\) | \(17\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 1.22, size = 20, normalized size = 1.11 \begin {gather*} \log \left (2 \, \sqrt {-a^{2} + \log \left (x\right )^{2}} + 2 \, \log \left (x\right )\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.56, size = 20, normalized size = 1.11 \begin {gather*} -\log \left (\sqrt {-a^{2} + \log \left (x\right )^{2}} - \log \left (x\right )\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{x \sqrt {- \left (a - \log {\left (x \right )}\right ) \left (a + \log {\left (x \right )}\right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.39, size = 16, normalized size = 0.89 \begin {gather*} \ln \left (\ln \left (x\right )+\sqrt {{\ln \left (x\right )}^2-a^2}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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