Optimal. Leaf size=59 \[ -\frac{2 \sqrt{1-a^2 x^2}}{a \sqrt{\sin ^{-1}(a x)}}-\frac{2 \sqrt{2 \pi } S\left (\sqrt{\frac{2}{\pi }} \sqrt{\sin ^{-1}(a x)}\right )}{a} \]
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Rubi [A] time = 0.0904898, antiderivative size = 59, 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 = {4621, 4723, 3305, 3351} \[ -\frac{2 \sqrt{1-a^2 x^2}}{a \sqrt{\sin ^{-1}(a x)}}-\frac{2 \sqrt{2 \pi } S\left (\sqrt{\frac{2}{\pi }} \sqrt{\sin ^{-1}(a x)}\right )}{a} \]
Antiderivative was successfully verified.
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Rule 4621
Rule 4723
Rule 3305
Rule 3351
Rubi steps
\begin{align*} \int \frac{1}{\sin ^{-1}(a x)^{3/2}} \, dx &=-\frac{2 \sqrt{1-a^2 x^2}}{a \sqrt{\sin ^{-1}(a x)}}-(2 a) \int \frac{x}{\sqrt{1-a^2 x^2} \sqrt{\sin ^{-1}(a x)}} \, dx\\ &=-\frac{2 \sqrt{1-a^2 x^2}}{a \sqrt{\sin ^{-1}(a x)}}-\frac{2 \operatorname{Subst}\left (\int \frac{\sin (x)}{\sqrt{x}} \, dx,x,\sin ^{-1}(a x)\right )}{a}\\ &=-\frac{2 \sqrt{1-a^2 x^2}}{a \sqrt{\sin ^{-1}(a x)}}-\frac{4 \operatorname{Subst}\left (\int \sin \left (x^2\right ) \, dx,x,\sqrt{\sin ^{-1}(a x)}\right )}{a}\\ &=-\frac{2 \sqrt{1-a^2 x^2}}{a \sqrt{\sin ^{-1}(a x)}}-\frac{2 \sqrt{2 \pi } S\left (\sqrt{\frac{2}{\pi }} \sqrt{\sin ^{-1}(a x)}\right )}{a}\\ \end{align*}
Mathematica [C] time = 0.0951994, size = 87, normalized size = 1.47 \[ \frac{\sqrt{-i \sin ^{-1}(a x)} \text{Gamma}\left (\frac{1}{2},-i \sin ^{-1}(a x)\right )+\sqrt{i \sin ^{-1}(a x)} \text{Gamma}\left (\frac{1}{2},i \sin ^{-1}(a x)\right )-e^{-i \sin ^{-1}(a x)} \left (1+e^{2 i \sin ^{-1}(a x)}\right )}{a \sqrt{\sin ^{-1}(a x)}} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.035, size = 65, normalized size = 1.1 \begin{align*} -{\frac{\sqrt{2}}{a\sqrt{\pi }\arcsin \left ( ax \right ) } \left ( 2\,\arcsin \left ( ax \right ) \pi \,{\it FresnelS} \left ({\frac{\sqrt{2}\sqrt{\arcsin \left ( ax \right ) }}{\sqrt{\pi }}} \right ) +\sqrt{2}\sqrt{\arcsin \left ( ax \right ) }\sqrt{\pi }\sqrt{-{a}^{2}{x}^{2}+1} \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{1}{\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{1}{\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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