Optimal. Leaf size=39 \[ -\frac {\sqrt {1-x^2}}{x}-\frac {\sqrt {1-x^2} \log (x)}{x}-\sin ^{-1}(x) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.05, antiderivative size = 39, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {2335, 277, 216} \[ -\frac {\sqrt {1-x^2}}{x}-\frac {\sqrt {1-x^2} \log (x)}{x}-\sin ^{-1}(x) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 216
Rule 277
Rule 2335
Rubi steps
\begin {align*} \int \frac {\log (x)}{x^2 \sqrt {1-x^2}} \, dx &=-\frac {\sqrt {1-x^2} \log (x)}{x}+\int \frac {\sqrt {1-x^2}}{x^2} \, dx\\ &=-\frac {\sqrt {1-x^2}}{x}-\frac {\sqrt {1-x^2} \log (x)}{x}-\int \frac {1}{\sqrt {1-x^2}} \, dx\\ &=-\frac {\sqrt {1-x^2}}{x}-\sin ^{-1}(x)-\frac {\sqrt {1-x^2} \log (x)}{x}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.03, size = 25, normalized size = 0.64 \[ -\frac {\sqrt {1-x^2} (\log (x)+1)}{x}-\sin ^{-1}(x) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.44, size = 39, normalized size = 1.00 \[ \frac {2 \, x \arctan \left (\frac {\sqrt {-x^{2} + 1} - 1}{x}\right ) - \sqrt {-x^{2} + 1} {\left (\log \relax (x) + 1\right )}}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [B] time = 1.08, size = 73, normalized size = 1.87 \[ \frac {1}{2} \, {\left (\frac {x}{\sqrt {-x^{2} + 1} - 1} - \frac {\sqrt {-x^{2} + 1} - 1}{x}\right )} \log \relax (x) + \frac {x}{2 \, {\left (\sqrt {-x^{2} + 1} - 1\right )}} - \frac {\sqrt {-x^{2} + 1} - 1}{2 \, x} - \arcsin \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.11, size = 35, normalized size = 0.90 \[ -\arcsin \relax (x )+\frac {-\sqrt {-x^{2}+1}\, \ln \relax (x )-\sqrt {-x^{2}+1}}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.96, size = 35, normalized size = 0.90 \[ -\frac {\sqrt {-x^{2} + 1} \log \relax (x)}{x} - \frac {\sqrt {-x^{2} + 1}}{x} - \arcsin \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.03 \[ \int \frac {\ln \relax (x)}{x^2\,\sqrt {1-x^2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\log {\relax (x )}}{x^{2} \sqrt {- \left (x - 1\right ) \left (x + 1\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________