Optimal. Leaf size=22 \[ -\frac {\tanh ^{-1}\left (\frac {\sqrt {a^2+\log ^2(x)}}{a}\right )}{a} \]
________________________________________________________________________________________
Rubi [A] time = 0.09, antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {266, 63, 207} \[ -\frac {\tanh ^{-1}\left (\frac {\sqrt {a^2+\log ^2(x)}}{a}\right )}{a} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 63
Rule 207
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}{x \sqrt {a^2+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 = 22, 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 [B] time = 0.90, size = 44, normalized size = 2.00 \[ -\frac {\log \left (a + \sqrt {a^{2} + \log \relax (x)^{2}} - \log \relax (x)\right ) - \log \left (-a + \sqrt {a^{2} + \log \relax (x)^{2}} - \log \relax (x)\right )}{a} \]
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 = 37, normalized size = 1.68
method | result | size |
derivativedivides | \(-\frac {\ln \left (\frac {2 a^{2}+2 \sqrt {a^{2}}\, \sqrt {a^{2}+\ln \relax (x )^{2}}}{\ln \relax (x )}\right )}{\sqrt {a^{2}}}\) | \(37\) |
default | \(-\frac {\ln \left (\frac {2 a^{2}+2 \sqrt {a^{2}}\, \sqrt {a^{2}+\ln \relax (x )^{2}}}{\ln \relax (x )}\right )}{\sqrt {a^{2}}}\) | \(37\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.42, size = 13, normalized size = 0.59 \[ -\frac {\operatorname {arsinh}\left (\frac {a}{{\left | \log \relax (x) \right |}}\right )}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.58, size = 27, normalized size = 1.23 \[ \frac {\mathrm {atan}\left (\frac {\sqrt {a^2+{\ln \relax (x)}^2}}{\sqrt {-a^2}}\right )}{\sqrt {-a^2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{x \sqrt {a^{2} + \log {\relax (x )}^{2}} \log {\relax (x )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________