Optimal. Leaf size=42 \[ -\frac{2 \sqrt{1-a^2 x^2}}{a \sqrt{c-a^2 c x^2} \sqrt{\sin ^{-1}(a x)}} \]
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Rubi [A] time = 0.0695152, antiderivative size = 42, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {4643, 4641} \[ -\frac{2 \sqrt{1-a^2 x^2}}{a \sqrt{c-a^2 c x^2} \sqrt{\sin ^{-1}(a x)}} \]
Antiderivative was successfully verified.
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Rule 4643
Rule 4641
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{c-a^2 c x^2} \sin ^{-1}(a x)^{3/2}} \, dx &=\frac{\sqrt{1-a^2 x^2} \int \frac{1}{\sqrt{1-a^2 x^2} \sin ^{-1}(a x)^{3/2}} \, dx}{\sqrt{c-a^2 c x^2}}\\ &=-\frac{2 \sqrt{1-a^2 x^2}}{a \sqrt{c-a^2 c x^2} \sqrt{\sin ^{-1}(a x)}}\\ \end{align*}
Mathematica [A] time = 0.0440858, size = 42, normalized size = 1. \[ -\frac{2 \sqrt{1-a^2 x^2}}{a \sqrt{c-a^2 c x^2} \sqrt{\sin ^{-1}(a x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.036, size = 38, normalized size = 0.9 \begin{align*} -2\,{\frac{\sqrt{-{a}^{2}{x}^{2}+1}}{\sqrt{\arcsin \left ( ax \right ) }a\sqrt{-c \left ({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 [A] time = 2.15171, size = 109, normalized size = 2.6 \begin{align*} \frac{2 \, \sqrt{-a^{2} c x^{2} + c} \sqrt{-a^{2} x^{2} + 1}}{{\left (a^{3} c x^{2} - a c\right )} \sqrt{\arcsin \left (a x\right )}} \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}{\sqrt{- c \left (a x - 1\right ) \left (a x + 1\right )} \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}{\sqrt{-a^{2} c x^{2} + c} \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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