Optimal. Leaf size=13 \[ -\frac {1}{2 a \sin ^{-1}(a x)^2} \]
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Rubi [A] time = 0.03, antiderivative size = 13, normalized size of antiderivative = 1.00, 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} \[ -\frac {1}{2 a \sin ^{-1}(a x)^2} \]
Antiderivative was successfully verified.
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Rule 4641
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {1-a^2 x^2} \sin ^{-1}(a x)^3} \, dx &=-\frac {1}{2 a \sin ^{-1}(a x)^2}\\ \end {align*}
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Mathematica [A] time = 0.00, size = 13, normalized size = 1.00 \[ -\frac {1}{2 a \sin ^{-1}(a x)^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.51, size = 11, normalized size = 0.85 \[ -\frac {1}{2 \, a \arcsin \left (a x\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.64, size = 11, normalized size = 0.85 \[ -\frac {1}{2 \, a \arcsin \left (a x\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 12, normalized size = 0.92 \[ -\frac {1}{2 a \arcsin \left (a x \right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.43, size = 11, normalized size = 0.85 \[ -\frac {1}{2 \, a \arcsin \left (a x\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.12, size = 11, normalized size = 0.85 \[ -\frac {1}{2\,a\,{\mathrm {asin}\left (a\,x\right )}^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.85, size = 12, normalized size = 0.92 \[ - \frac {1}{2 a \operatorname {asin}^{2}{\left (a x \right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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