Optimal. Leaf size=102 \[ \frac{4 a \sqrt{1-a^2 x^2} \text{Unintegrable}\left (\frac{x}{\left (1-a^2 x^2\right )^2 \sqrt{\sin ^{-1}(a x)}},x\right )}{c \sqrt{c-a^2 c x^2}}-\frac{2 \sqrt{1-a^2 x^2}}{a \left (c-a^2 c x^2\right )^{3/2} \sqrt{\sin ^{-1}(a x)}} \]
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Rubi [A] time = 0.0917123, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int \frac{1}{\left (c-a^2 c x^2\right )^{3/2} \sin ^{-1}(a x)^{3/2}} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin{align*} \int \frac{1}{\left (c-a^2 c x^2\right )^{3/2} \sin ^{-1}(a x)^{3/2}} \, dx &=-\frac{2 \sqrt{1-a^2 x^2}}{a \left (c-a^2 c x^2\right )^{3/2} \sqrt{\sin ^{-1}(a x)}}+\frac{\left (4 a \sqrt{1-a^2 x^2}\right ) \int \frac{x}{\left (1-a^2 x^2\right )^2 \sqrt{\sin ^{-1}(a x)}} \, dx}{c \sqrt{c-a^2 c x^2}}\\ \end{align*}
Mathematica [A] time = 0.840648, size = 0, normalized size = 0. \[ \int \frac{1}{\left (c-a^2 c x^2\right )^{3/2} \sin ^{-1}(a x)^{3/2}} \, dx \]
Verification is Not applicable to the result.
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Maple [A] time = 0.211, size = 0, normalized size = 0. \begin{align*} \int{ \left ( -{a}^{2}c{x}^{2}+c \right ) ^{-{\frac{3}{2}}} \left ( \arcsin \left ( ax \right ) \right ) ^{-{\frac{3}{2}}}}\, dx \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(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (-a^{2} c x^{2} + c\right )}^{\frac{3}{2}} \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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