Optimal. Leaf size=16 \[ \log \left (\sin ^{-1}(x)\right )-\frac{1}{2} \log \left (1-x^2\right ) \]
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Rubi [A] time = 0.0282553, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.071, Rules used = {260, 4639} \[ \log \left (\sin ^{-1}(x)\right )-\frac{1}{2} \log \left (1-x^2\right ) \]
Antiderivative was successfully verified.
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Rule 260
Rule 4639
Rubi steps
\begin{align*} \int \left (\frac{x}{1-x^2}+\frac{1}{\sqrt{1-x^2} \sin ^{-1}(x)}\right ) \, dx &=\int \frac{x}{1-x^2} \, dx+\int \frac{1}{\sqrt{1-x^2} \sin ^{-1}(x)} \, dx\\ &=-\frac{1}{2} \log \left (1-x^2\right )+\log \left (\sin ^{-1}(x)\right )\\ \end{align*}
Mathematica [A] time = 0.0233533, size = 16, normalized size = 1. \[ \log \left (\sin ^{-1}(x)\right )-\frac{1}{2} \log \left (1-x^2\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.009, size = 17, normalized size = 1.1 \begin{align*} -{\frac{\ln \left ( x-1 \right ) }{2}}-{\frac{\ln \left ( x+1 \right ) }{2}}+\ln \left ( \arcsin \left ( x \right ) \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.45961, size = 16, normalized size = 1. \begin{align*} -\frac{1}{2} \, \log \left (x^{2} - 1\right ) + \log \left (\arcsin \left (x\right )\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.08817, size = 50, normalized size = 3.12 \begin{align*} -\frac{1}{2} \, \log \left (x^{2} - 1\right ) + \log \left (-\arcsin \left (x\right )\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.235039, size = 12, normalized size = 0.75 \begin{align*} - \frac{\log{\left (x^{2} - 1 \right )}}{2} + \log{\left (\operatorname{asin}{\left (x \right )} \right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.14248, size = 19, normalized size = 1.19 \begin{align*} -\frac{1}{2} \, \log \left ({\left | x^{2} - 1 \right |}\right ) + \log \left ({\left | \arcsin \left (x\right ) \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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