Optimal. Leaf size=13 \[ -\frac {1}{2 a \text {ArcSin}(a x)^2} \]
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Rubi [A]
time = 0.02, 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 = {4737}
\begin {gather*} -\frac {1}{2 a \text {ArcSin}(a x)^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 4737
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 = 0.01, size = 13, normalized size = 1.00 \begin {gather*} -\frac {1}{2 a \text {ArcSin}(a x)^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.10, size = 12, normalized size = 0.92
method | result | size |
derivativedivides | \(-\frac {1}{2 a \arcsin \left (a x \right )^{2}}\) | \(12\) |
default | \(-\frac {1}{2 a \arcsin \left (a x \right )^{2}}\) | \(12\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.47, size = 11, normalized size = 0.85 \begin {gather*} -\frac {1}{2 \, a \arcsin \left (a x\right )^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.65, size = 11, normalized size = 0.85 \begin {gather*} -\frac {1}{2 \, a \arcsin \left (a x\right )^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.40, size = 12, normalized size = 0.92 \begin {gather*} - \frac {1}{2 a \operatorname {asin}^{2}{\left (a x \right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.45, size = 11, normalized size = 0.85 \begin {gather*} -\frac {1}{2 \, a \arcsin \left (a x\right )^{2}} \end {gather*}
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 \begin {gather*} -\frac {1}{2\,a\,{\mathrm {asin}\left (a\,x\right )}^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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