Optimal. Leaf size=16 \[ \log \left (\sin ^{-1}(x)\right )-\frac {1}{2} \log \left (1-x^2\right ) \]
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Rubi [F] time = 0.16, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {\sqrt {1-x^2}+x \sin ^{-1}(x)}{\sin ^{-1}(x)-x^2 \sin ^{-1}(x)} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {\sqrt {1-x^2}+x \sin ^{-1}(x)}{\sin ^{-1}(x)-x^2 \sin ^{-1}(x)} \, dx &=\int \frac {\sqrt {1-x^2}+x \sin ^{-1}(x)}{\left (1-x^2\right ) \sin ^{-1}(x)} \, dx\\ \end {align*}
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Mathematica [A] time = 0.12, 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 = 1.33, size = 20, normalized size = 1.25 \[ -\log \relax (2) - \frac {1}{2} \, \log \left ({\left | -x^{2} + 1 \right |}\right ) + \log \left ({\left | \arcsin \relax (x) \right |}\right ) \]
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 [F] time = 0.00, size = 0, normalized size = 0.00 \[ -\int \frac {\sqrt {x + 1} \sqrt {-x + 1}}{{\left (x^{2} - 1\right )} \arctan \left (x, \sqrt {x + 1} \sqrt {-x + 1}\right )}\,{d x} - \frac {1}{2} \, \log \left (x + 1\right ) - \frac {1}{2} \, \log \left (x - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.29, 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 = 4.37, size = 14, normalized size = 0.88 \[ - \frac {\log {\left (2 x^{2} - 2 \right )}}{2} + \log {\left (\operatorname {asin}{\relax (x )} \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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