Optimal. Leaf size=39 \[ -\frac{\sqrt{1-x^2}}{x}-\frac{\sqrt{1-x^2} \log (x)}{x}-\sin ^{-1}(x) \]
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Rubi [A] time = 0.0483891, antiderivative size = 39, normalized size of antiderivative = 1., 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.
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Rule 2335
Rule 277
Rule 216
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 = 0.0308564, size = 25, normalized size = 0.64 \[ -\frac{\sqrt{1-x^2} (\log (x)+1)}{x}-\sin ^{-1}(x) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.033, size = 35, normalized size = 0.9 \begin{align*} -\arcsin \left ( x \right ) +{\frac{1}{x} \left ( -\ln \left ( x \right ) \sqrt{-{x}^{2}+1}-\sqrt{-{x}^{2}+1} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.43639, size = 47, normalized size = 1.21 \begin{align*} -\frac{\sqrt{-x^{2} + 1} \log \left (x\right )}{x} - \frac{\sqrt{-x^{2} + 1}}{x} - \arcsin \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.08141, size = 95, normalized size = 2.44 \begin{align*} \frac{2 \, x \arctan \left (\frac{\sqrt{-x^{2} + 1} - 1}{x}\right ) - \sqrt{-x^{2} + 1}{\left (\log \left (x\right ) + 1\right )}}{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\log{\left (x \right )}}{x^{2} \sqrt{- \left (x - 1\right ) \left (x + 1\right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.09174, size = 99, normalized size = 2.54 \begin{align*} \frac{1}{2} \,{\left (\frac{x}{\sqrt{-x^{2} + 1} - 1} - \frac{\sqrt{-x^{2} + 1} - 1}{x}\right )} \log \left (x\right ) + \frac{x}{2 \,{\left (\sqrt{-x^{2} + 1} - 1\right )}} - \frac{\sqrt{-x^{2} + 1} - 1}{2 \, x} - \arcsin \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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