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 = 0.02, antiderivative size = 28, normalized size of antiderivative = 1.00, 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 = {4918, 4737}
\begin {gather*} -\frac {\sqrt {x^2} \text {ArcSin}\left (\sqrt {1-x^2}\right )^2}{2 x} \end {gather*}
Antiderivative was successfully verified.
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Rule 4737
Rule 4918
Rubi steps
\begin {align*} \int \frac {\sin ^{-1}\left (\sqrt {1-x^2}\right )}{\sqrt {1-x^2}} \, dx &=-\frac {\sqrt {x^2} \text {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*}
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Mathematica [A]
time = 0.01, size = 28, normalized size = 1.00 \begin {gather*} -\frac {\sqrt {x^2} \sin ^{-1}\left (\sqrt {1-x^2}\right )^2}{2 x} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.06, size = 0, normalized size = 0.00 \[\int \frac {\arcsin \left (\sqrt {-x^{2}+1}\right )}{\sqrt {-x^{2}+1}}\, dx\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.61, size = 14, normalized size = 0.50 \begin {gather*} -\frac {1}{2} \, \arcsin \left (\sqrt {-x^{2} + 1}\right )^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.41, size = 22, normalized size = 0.79 \begin {gather*} - \frac {\sqrt {x^{2}} \operatorname {asin}^{2}{\left (\sqrt {1 - x^{2}} \right )}}{2 x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int \frac {\mathrm {asin}\left (\sqrt {1-x^2}\right )}{\sqrt {1-x^2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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