Optimal. Leaf size=24 \[ -\frac{\tanh ^{-1}\left (\frac{\sqrt{a^2-\log ^2(x)}}{a}\right )}{a} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0981423, antiderivative size = 24, normalized size of antiderivative = 1., 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 266
Rule 63
Rule 206
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 = 0.0156425, size = 24, normalized size = 1. \[ -\frac{\tanh ^{-1}\left (\frac{\sqrt{a^2-\log ^2(x)}}{a}\right )}{a} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.007, size = 39, normalized size = 1.6 \begin{align*} -{\ln \left ({\frac{1}{\ln \left ( x \right ) } \left ( 2\,{a}^{2}+2\,\sqrt{{a}^{2}}\sqrt{{a}^{2}- \left ( \ln \left ( x \right ) \right ) ^{2}} \right ) } \right ){\frac{1}{\sqrt{{a}^{2}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 2.37175, size = 58, normalized size = 2.42 \begin{align*} \frac{\log \left (-\frac{a - \sqrt{a^{2} - \log \left (x\right )^{2}}}{\log \left (x\right )}\right )}{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \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{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \mathit{sage}_{0} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]