Optimal. Leaf size=28 \[ -\frac{\sqrt{x^2} \sin ^{-1}\left (\sqrt{1-x^2}\right )^2}{2 x} \]
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Rubi [A] time = 0.0332714, antiderivative size = 28, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {4834, 4641} \[ -\frac{\sqrt{x^2} \sin ^{-1}\left (\sqrt{1-x^2}\right )^2}{2 x} \]
Antiderivative was successfully verified.
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Rule 4834
Rule 4641
Rubi steps
\begin{align*} \int \frac{\sin ^{-1}\left (\sqrt{1-x^2}\right )}{\sqrt{1-x^2}} \, dx &=-\frac{\sqrt{x^2} \operatorname{Subst}\left (\int \frac{\sin ^{-1}(x)}{\sqrt{1-x^2}} \, dx,x,\sqrt{1-x^2}\right )}{x}\\ &=-\frac{\sqrt{x^2} \sin ^{-1}\left (\sqrt{1-x^2}\right )^2}{2 x}\\ \end{align*}
Mathematica [A] time = 0.0166199, size = 28, normalized size = 1. \[ -\frac{\sqrt{x^2} \sin ^{-1}\left (\sqrt{1-x^2}\right )^2}{2 x} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.056, size = 0, normalized size = 0. \begin{align*} \int{\arcsin \left ( \sqrt{-{x}^{2}+1} \right ){\frac{1}{\sqrt{-{x}^{2}+1}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\arcsin \left (\sqrt{-x^{2} + 1}\right )}{\sqrt{-x^{2} + 1}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.55128, size = 42, normalized size = 1.5 \begin{align*} -\frac{1}{2} \, \arcsin \left (\sqrt{-x^{2} + 1}\right )^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 2.92055, size = 27, normalized size = 0.96 \begin{align*} \frac{x \operatorname{asin}^{2}{\left (x \right )}}{2 \sqrt{x^{2}}} + \operatorname{asin}{\left (x \right )} \operatorname{asin}{\left (\sqrt{1 - x^{2}} \right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\arcsin \left (\sqrt{-x^{2} + 1}\right )}{\sqrt{-x^{2} + 1}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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