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 = 0.03, antiderivative size = 16, normalized size of antiderivative = 1.00, 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*}
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Mathematica [A] time = 0.03, size = 16, normalized size = 1.00 \[ \log \left (\sin ^{-1}(x)\right )-\frac {1}{2} \log \left (1-x^2\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 14, normalized size = 0.88 \[ -\frac {1}{2} \, \log \left (x^{2} - 1\right ) + \log \left (-\arcsin \relax (x)\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.88, size = 14, normalized size = 0.88 \[ -\frac {1}{2} \, \log \left ({\left | x^{2} - 1 \right |}\right ) + \log \left ({\left | \arcsin \relax (x) \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 17, normalized size = 1.06 \[ -\frac {\ln \left (x -1\right )}{2}-\frac {\ln \left (x +1\right )}{2}+\ln \left (\arcsin \relax (x )\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.41, size = 12, normalized size = 0.75 \[ -\frac {1}{2} \, \log \left (x^{2} - 1\right ) + \log \left (\arcsin \relax (x)\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.31, size = 12, normalized size = 0.75 \[ \ln \left (\mathrm {asin}\relax (x)\right )-\frac {\ln \left (x^2-1\right )}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.17, size = 12, normalized size = 0.75 \[ - \frac {\log {\left (x^{2} - 1 \right )}}{2} + \log {\left (\operatorname {asin}{\relax (x )} \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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