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 = 0.0607728, antiderivative size = 25, normalized size of antiderivative = 1., 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 4723
Rule 3305
Rule 3351
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*}
Mathematica [C] time = 0.0762037, size = 53, normalized size = 2.12 \[ -\frac{\sqrt{-i \sin ^{-1}(x)} \text{Gamma}\left (\frac{1}{2},-i \sin ^{-1}(x)\right )+\sqrt{i \sin ^{-1}(x)} \text{Gamma}\left (\frac{1}{2},i \sin ^{-1}(x)\right )}{2 \sqrt{\sin ^{-1}(x)}} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.067, size = 20, normalized size = 0.8 \begin{align*}{\it FresnelS} \left ({\frac{\sqrt{2}}{\sqrt{\pi }}\sqrt{\arcsin \left ( x \right ) }} \right ) \sqrt{2}\sqrt{\pi } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: RuntimeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x}{\sqrt{- \left (x - 1\right ) \left (x + 1\right )} \sqrt{\operatorname{asin}{\left (x \right )}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [C] time = 1.35562, size = 50, normalized size = 2. \begin{align*} \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 \left (x\right )}\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 \left (x\right )}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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