Optimal. Leaf size=25 \[ \sqrt {2 \pi } S\left (\sqrt {\frac {2}{\pi }} \sqrt {\sin ^{-1}(x)}\right ) \]
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Rubi [A] time = 0.06, antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {4723, 3305, 3351} \[ \sqrt {2 \pi } S\left (\sqrt {\frac {2}{\pi }} \sqrt {\sin ^{-1}(x)}\right ) \]
Antiderivative was successfully verified.
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Rule 3305
Rule 3351
Rule 4723
Rubi steps
\begin {align*} \int \frac {x}{\sqrt {1-x^2} \sqrt {\sin ^{-1}(x)}} \, dx &=\operatorname {Subst}\left (\int \frac {\sin (x)}{\sqrt {x}} \, dx,x,\sin ^{-1}(x)\right )\\ &=2 \operatorname {Subst}\left (\int \sin \left (x^2\right ) \, dx,x,\sqrt {\sin ^{-1}(x)}\right )\\ &=\sqrt {2 \pi } S\left (\sqrt {\frac {2}{\pi }} \sqrt {\sin ^{-1}(x)}\right )\\ \end {align*}
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Mathematica [C] time = 0.09, size = 53, normalized size = 2.12 \[ -\frac {\sqrt {-i \sin ^{-1}(x)} \Gamma \left (\frac {1}{2},-i \sin ^{-1}(x)\right )+\sqrt {i \sin ^{-1}(x)} \Gamma \left (\frac {1}{2},i \sin ^{-1}(x)\right )}{2 \sqrt {\sin ^{-1}(x)}} \]
Warning: Unable to verify antiderivative.
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fricas [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [C] time = 0.77, size = 37, normalized size = 1.48 \[ \left (\frac {1}{4} i - \frac {1}{4}\right ) \, \sqrt {2} \sqrt {\pi } \operatorname {erf}\left (\left (\frac {1}{2} i - \frac {1}{2}\right ) \, \sqrt {2} \sqrt {\arcsin \relax (x)}\right ) - \left (\frac {1}{4} i + \frac {1}{4}\right ) \, \sqrt {2} \sqrt {\pi } \operatorname {erf}\left (-\left (\frac {1}{2} i + \frac {1}{2}\right ) \, \sqrt {2} \sqrt {\arcsin \relax (x)}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.13, size = 20, normalized size = 0.80 \[ \mathrm {S}\left (\frac {\sqrt {2}\, \sqrt {\arcsin \relax (x )}}{\sqrt {\pi }}\right ) \sqrt {2}\, \sqrt {\pi } \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]
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 \[ \int \frac {x}{\sqrt {\mathrm {asin}\relax (x)}\,\sqrt {1-x^2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x}{\sqrt {- \left (x - 1\right ) \left (x + 1\right )} \sqrt {\operatorname {asin}{\relax (x )}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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