Optimal. Leaf size=23 \[ \frac {\tan ^{-1}\left (\frac {\sqrt {-a^2+\log ^2(x)}}{a}\right )}{a} \]
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Rubi [A]
time = 0.06, antiderivative size = 23, normalized size of antiderivative = 1.00, 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 = {272, 65, 209}
\begin {gather*} \frac {\text {ArcTan}\left (\frac {\sqrt {\log ^2(x)-a^2}}{a}\right )}{a} \end {gather*}
Antiderivative was successfully verified.
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Rule 65
Rule 209
Rule 272
Rubi steps
\begin {align*} \int \frac {1}{x \log (x) \sqrt {-a^2+\log ^2(x)}} \, dx &=\text {Subst}\left (\int \frac {1}{x \sqrt {-a^2+x^2}} \, dx,x,\log (x)\right )\\ &=\frac {1}{2} \text {Subst}\left (\int \frac {1}{x \sqrt {-a^2+x}} \, dx,x,\log ^2(x)\right )\\ &=\text {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*}
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Mathematica [A]
time = 0.01, size = 23, normalized size = 1.00 \begin {gather*} \frac {\tan ^{-1}\left (\frac {\sqrt {-a^2+\log ^2(x)}}{a}\right )}{a} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.01, size = 43, normalized size = 1.87
method | result | size |
derivativedivides | \(-\frac {\ln \left (\frac {-2 a^{2}+2 \sqrt {-a^{2}}\, \sqrt {-a^{2}+\ln \left (x \right )^{2}}}{\ln \left (x \right )}\right )}{\sqrt {-a^{2}}}\) | \(43\) |
default | \(-\frac {\ln \left (\frac {-2 a^{2}+2 \sqrt {-a^{2}}\, \sqrt {-a^{2}+\ln \left (x \right )^{2}}}{\ln \left (x \right )}\right )}{\sqrt {-a^{2}}}\) | \(43\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 3.47, size = 13, normalized size = 0.57 \begin {gather*} -\frac {\arcsin \left (\frac {a}{{\left | \log \left (x\right ) \right |}}\right )}{a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.69, size = 27, normalized size = 1.17 \begin {gather*} \frac {2 \, \arctan \left (\frac {\sqrt {-a^{2} + \log \left (x\right )^{2}} - \log \left (x\right )}{a}\right )}{a} \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 )} \log {\left (x \right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 2.07, size = 21, normalized size = 0.91 \begin {gather*} \frac {\arctan \left (\frac {\sqrt {-a^{2} + \log \left (x\right )^{2}}}{a}\right )}{a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.52, size = 25, normalized size = 1.09 \begin {gather*} \frac {\mathrm {atan}\left (\frac {\sqrt {{\ln \left (x\right )}^2-a^2}}{\sqrt {a^2}}\right )}{\sqrt {a^2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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