Optimal. Leaf size=13 \[ \frac{\sin ^{-1}(a x)^3}{3 a} \]
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Rubi [A] time = 0.0338384, antiderivative size = 13, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.048, Rules used = {4641} \[ \frac{\sin ^{-1}(a x)^3}{3 a} \]
Antiderivative was successfully verified.
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Rule 4641
Rubi steps
\begin{align*} \int \frac{\sin ^{-1}(a x)^2}{\sqrt{1-a^2 x^2}} \, dx &=\frac{\sin ^{-1}(a x)^3}{3 a}\\ \end{align*}
Mathematica [A] time = 0.0038623, size = 13, normalized size = 1. \[ \frac{\sin ^{-1}(a x)^3}{3 a} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 12, normalized size = 0.9 \begin{align*}{\frac{ \left ( \arcsin \left ( ax \right ) \right ) ^{3}}{3\,a}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.51888, size = 15, normalized size = 1.15 \begin{align*} \frac{\arcsin \left (a x\right )^{3}}{3 \, a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.62441, size = 28, normalized size = 2.15 \begin{align*} \frac{\arcsin \left (a x\right )^{3}}{3 \, a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.474915, size = 10, normalized size = 0.77 \begin{align*} \begin{cases} \frac{\operatorname{asin}^{3}{\left (a x \right )}}{3 a} & \text{for}\: a \neq 0 \\0 & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.25424, size = 15, normalized size = 1.15 \begin{align*} \frac{\arcsin \left (a x\right )^{3}}{3 \, a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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