Optimal. Leaf size=24 \[ -\frac {\tanh ^{-1}\left (\frac {\sqrt {a^2-\log ^2(x)}}{a}\right )}{a} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.06, 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 = {272, 65, 212}
\begin {gather*} -\frac {\tanh ^{-1}\left (\frac {\sqrt {a^2-\log ^2(x)}}{a}\right )}{a} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 65
Rule 212
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}{\sqrt {a^2-x} 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 {\tanh ^{-1}\left (\frac {\sqrt {a^2-\log ^2(x)}}{a}\right )}{a}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.01, size = 24, normalized size = 1.00 \begin {gather*} -\frac {\tanh ^{-1}\left (\frac {\sqrt {a^2-\log ^2(x)}}{a}\right )}{a} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.02, size = 39, normalized size = 1.62
method | result | size |
derivativedivides | \(-\frac {\ln \left (\frac {2 a^{2}+2 \sqrt {a^{2}}\, \sqrt {a^{2}-\ln \left (x \right )^{2}}}{\ln \left (x \right )}\right )}{\sqrt {a^{2}}}\) | \(39\) |
default | \(-\frac {\ln \left (\frac {2 a^{2}+2 \sqrt {a^{2}}\, \sqrt {a^{2}-\ln \left (x \right )^{2}}}{\ln \left (x \right )}\right )}{\sqrt {a^{2}}}\) | \(39\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 3.41, size = 37, normalized size = 1.54 \begin {gather*} -\frac {\log \left (\frac {2 \, a^{2}}{{\left | \log \left (x\right ) \right |}} + \frac {2 \, \sqrt {a^{2} - \log \left (x\right )^{2}} a}{{\left | \log \left (x\right ) \right |}}\right )}{a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.78, size = 27, normalized size = 1.12 \begin {gather*} \frac {\log \left (-\frac {a - \sqrt {a^{2} - \log \left (x\right )^{2}}}{\log \left (x\right )}\right )}{a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 0.61, size = 22, normalized size = 0.92 \begin {gather*} -\frac {\mathrm {atanh}\left (\frac {\sqrt {a^2-{\ln \left (x\right )}^2}}{a}\right )}{a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________