Optimal. Leaf size=24 \[ -\frac {\tanh ^{-1}\left (\frac {\sqrt {a^2-\log ^2(x)}}{a}\right )}{a} \]
________________________________________________________________________________________
Rubi [A] time = 0.10, antiderivative size = 24, 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 = {266, 63, 206} \[ -\frac {\tanh ^{-1}\left (\frac {\sqrt {a^2-\log ^2(x)}}{a}\right )}{a} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 63
Rule 206
Rule 266
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}{\sqrt {a^2-x} 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 {\tanh ^{-1}\left (\frac {\sqrt {a^2-\log ^2(x)}}{a}\right )}{a}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.02, size = 24, normalized size = 1.00 \[ -\frac {\tanh ^{-1}\left (\frac {\sqrt {a^2-\log ^2(x)}}{a}\right )}{a} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{x \log (x) \sqrt {a^2-\log ^2(x)}} \, dx \]
Verification is Not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.77, size = 27, normalized size = 1.12 \[ \frac {\log \left (-\frac {a - \sqrt {a^{2} - \log \relax (x)^{2}}}{\log \relax (x)}\right )}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.02, size = 39, normalized size = 1.62
method | result | size |
derivativedivides | \(-\frac {\ln \left (\frac {2 a^{2}+2 \sqrt {a^{2}}\, \sqrt {a^{2}-\ln \relax (x )^{2}}}{\ln \relax (x )}\right )}{\sqrt {a^{2}}}\) | \(39\) |
default | \(-\frac {\ln \left (\frac {2 a^{2}+2 \sqrt {a^{2}}\, \sqrt {a^{2}-\ln \relax (x )^{2}}}{\ln \relax (x )}\right )}{\sqrt {a^{2}}}\) | \(39\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.43, size = 37, normalized size = 1.54 \[ -\frac {\log \left (\frac {2 \, a^{2}}{{\left | \log \relax (x) \right |}} + \frac {2 \, \sqrt {a^{2} - \log \relax (x)^{2}} a}{{\left | \log \relax (x) \right |}}\right )}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.61, size = 22, normalized size = 0.92 \[ -\frac {\mathrm {atanh}\left (\frac {\sqrt {a^2-{\ln \relax (x)}^2}}{a}\right )}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{x \sqrt {\left (a - \log {\relax (x )}\right ) \left (a + \log {\relax (x )}\right )} \log {\relax (x )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________