Optimal. Leaf size=59 \[ \frac{3 a \text{Unintegrable}\left (\frac{x}{\left (1-a^2 x^2\right )^{5/2} \sin ^{-1}(a x)},x\right )}{c^2}-\frac{1}{a c^2 \left (1-a^2 x^2\right )^{3/2} \sin ^{-1}(a x)} \]
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Rubi [A] time = 0.0974797, 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 = {} \[ \int \frac{1}{\left (c-a^2 c x^2\right )^2 \sin ^{-1}(a x)^2} \, dx \]
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 )^2 \sin ^{-1}(a x)^2} \, dx &=-\frac{1}{a c^2 \left (1-a^2 x^2\right )^{3/2} \sin ^{-1}(a x)}+\frac{(3 a) \int \frac{x}{\left (1-a^2 x^2\right )^{5/2} \sin ^{-1}(a x)} \, dx}{c^2}\\ \end{align*}
Mathematica [A] time = 14.5328, size = 0, normalized size = 0. \[ \int \frac{1}{\left (c-a^2 c x^2\right )^2 \sin ^{-1}(a x)^2} \, dx \]
Verification is Not applicable to the result.
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Maple [A] time = 0.289, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{ \left ( -{a}^{2}c{x}^{2}+c \right ) ^{2} \left ( \arcsin \left ( ax \right ) \right ) ^{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [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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Fricas [A] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{1}{{\left (a^{4} c^{2} x^{4} - 2 \, a^{2} c^{2} x^{2} + c^{2}\right )} \arcsin \left (a x\right )^{2}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int \frac{1}{a^{4} x^{4} \operatorname{asin}^{2}{\left (a x \right )} - 2 a^{2} x^{2} \operatorname{asin}^{2}{\left (a x \right )} + \operatorname{asin}^{2}{\left (a x \right )}}\, dx}{c^{2}} \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 )}^{2} \arcsin \left (a x\right )^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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