Optimal. Leaf size=44 \[ x \sqrt{\sin ^{-1}(a x)}-\frac{\sqrt{\frac{\pi }{2}} S\left (\sqrt{\frac{2}{\pi }} \sqrt{\sin ^{-1}(a x)}\right )}{a} \]
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Rubi [A] time = 0.0896557, antiderivative size = 44, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {4619, 4723, 3305, 3351} \[ x \sqrt{\sin ^{-1}(a x)}-\frac{\sqrt{\frac{\pi }{2}} S\left (\sqrt{\frac{2}{\pi }} \sqrt{\sin ^{-1}(a x)}\right )}{a} \]
Antiderivative was successfully verified.
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Rule 4619
Rule 4723
Rule 3305
Rule 3351
Rubi steps
\begin{align*} \int \sqrt{\sin ^{-1}(a x)} \, dx &=x \sqrt{\sin ^{-1}(a x)}-\frac{1}{2} a \int \frac{x}{\sqrt{1-a^2 x^2} \sqrt{\sin ^{-1}(a x)}} \, dx\\ &=x \sqrt{\sin ^{-1}(a x)}-\frac{\operatorname{Subst}\left (\int \frac{\sin (x)}{\sqrt{x}} \, dx,x,\sin ^{-1}(a x)\right )}{2 a}\\ &=x \sqrt{\sin ^{-1}(a x)}-\frac{\operatorname{Subst}\left (\int \sin \left (x^2\right ) \, dx,x,\sqrt{\sin ^{-1}(a x)}\right )}{a}\\ &=x \sqrt{\sin ^{-1}(a x)}-\frac{\sqrt{\frac{\pi }{2}} S\left (\sqrt{\frac{2}{\pi }} \sqrt{\sin ^{-1}(a x)}\right )}{a}\\ \end{align*}
Mathematica [C] time = 0.0308365, size = 66, normalized size = 1.5 \[ \frac{\sqrt{-i \sin ^{-1}(a x)} \text{Gamma}\left (\frac{3}{2},-i \sin ^{-1}(a x)\right )+\sqrt{i \sin ^{-1}(a x)} \text{Gamma}\left (\frac{3}{2},i \sin ^{-1}(a x)\right )}{2 a \sqrt{\sin ^{-1}(a x)}} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.032, size = 49, normalized size = 1.1 \begin{align*}{\frac{1}{2\,a} \left ( -\sqrt{2}\sqrt{\arcsin \left ( ax \right ) }\sqrt{\pi }{\it FresnelS} \left ({\frac{\sqrt{2}}{\sqrt{\pi }}\sqrt{\arcsin \left ( ax \right ) }} \right ) +2\,ax\arcsin \left ( ax \right ) \right ){\frac{1}{\sqrt{\arcsin \left ( ax \right ) }}}} \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 \sqrt{\operatorname{asin}{\left (a x \right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [C] time = 1.3613, size = 112, normalized size = 2.55 \begin{align*} -\frac{\left (i - 1\right ) \, \sqrt{2} \sqrt{\pi } \operatorname{erf}\left (\left (\frac{1}{2} i - \frac{1}{2}\right ) \, \sqrt{2} \sqrt{\arcsin \left (a x\right )}\right )}{8 \, a} + \frac{\left (i + 1\right ) \, \sqrt{2} \sqrt{\pi } \operatorname{erf}\left (-\left (\frac{1}{2} i + \frac{1}{2}\right ) \, \sqrt{2} \sqrt{\arcsin \left (a x\right )}\right )}{8 \, a} - \frac{i \, \sqrt{\arcsin \left (a x\right )} e^{\left (i \, \arcsin \left (a x\right )\right )}}{2 \, a} + \frac{i \, \sqrt{\arcsin \left (a x\right )} e^{\left (-i \, \arcsin \left (a x\right )\right )}}{2 \, a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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