Optimal. Leaf size=55 \[ \frac{2 \sqrt{\pi } \text{FresnelC}\left (\frac{2 \sqrt{\sin ^{-1}(a x)}}{\sqrt{\pi }}\right )}{a^2}-\frac{2 x \sqrt{1-a^2 x^2}}{a \sqrt{\sin ^{-1}(a x)}} \]
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Rubi [A] time = 0.031562, antiderivative size = 55, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {4631, 3304, 3352} \[ \frac{2 \sqrt{\pi } \text{FresnelC}\left (\frac{2 \sqrt{\sin ^{-1}(a x)}}{\sqrt{\pi }}\right )}{a^2}-\frac{2 x \sqrt{1-a^2 x^2}}{a \sqrt{\sin ^{-1}(a x)}} \]
Antiderivative was successfully verified.
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Rule 4631
Rule 3304
Rule 3352
Rubi steps
\begin{align*} \int \frac{x}{\sin ^{-1}(a x)^{3/2}} \, dx &=-\frac{2 x \sqrt{1-a^2 x^2}}{a \sqrt{\sin ^{-1}(a x)}}+\frac{2 \operatorname{Subst}\left (\int \frac{\cos (2 x)}{\sqrt{x}} \, dx,x,\sin ^{-1}(a x)\right )}{a^2}\\ &=-\frac{2 x \sqrt{1-a^2 x^2}}{a \sqrt{\sin ^{-1}(a x)}}+\frac{4 \operatorname{Subst}\left (\int \cos \left (2 x^2\right ) \, dx,x,\sqrt{\sin ^{-1}(a x)}\right )}{a^2}\\ &=-\frac{2 x \sqrt{1-a^2 x^2}}{a \sqrt{\sin ^{-1}(a x)}}+\frac{2 \sqrt{\pi } C\left (\frac{2 \sqrt{\sin ^{-1}(a x)}}{\sqrt{\pi }}\right )}{a^2}\\ \end{align*}
Mathematica [C] time = 0.0326046, size = 91, normalized size = 1.65 \[ -\frac{i \sqrt{2} \sqrt{-i \sin ^{-1}(a x)} \text{Gamma}\left (\frac{1}{2},-2 i \sin ^{-1}(a x)\right )-i \sqrt{2} \sqrt{i \sin ^{-1}(a x)} \text{Gamma}\left (\frac{1}{2},2 i \sin ^{-1}(a x)\right )+2 \sin \left (2 \sin ^{-1}(a x)\right )}{2 a^2 \sqrt{\sin ^{-1}(a x)}} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.031, size = 43, normalized size = 0.8 \begin{align*} -{\frac{1}{{a}^{2}} \left ( -2\,\sqrt{\arcsin \left ( ax \right ) }\sqrt{\pi }{\it FresnelC} \left ( 2\,{\frac{\sqrt{\arcsin \left ( ax \right ) }}{\sqrt{\pi }}} \right ) +\sin \left ( 2\,\arcsin \left ( ax \right ) \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 \frac{x}{\operatorname{asin}^{\frac{3}{2}}{\left (a x \right )}}\, dx \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{x}{\arcsin \left (a x\right )^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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