Optimal. Leaf size=13 \[ \frac {\text {ArcSin}(a x)^4}{4 a} \]
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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 {\text {ArcSin}(a x)^4}{4 a} \end {gather*}
Antiderivative was successfully verified.
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Rule 4737
Rubi steps
\begin {align*} \int \frac {\sin ^{-1}(a x)^3}{\sqrt {1-a^2 x^2}} \, dx &=\frac {\sin ^{-1}(a x)^4}{4 a}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 13, normalized size = 1.00 \begin {gather*} \frac {\text {ArcSin}(a x)^4}{4 a} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.09, size = 12, normalized size = 0.92
method | result | size |
derivativedivides | \(\frac {\arcsin \left (a x \right )^{4}}{4 a}\) | \(12\) |
default | \(\frac {\arcsin \left (a x \right )^{4}}{4 a}\) | \(12\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.46, size = 11, normalized size = 0.85 \begin {gather*} \frac {\arcsin \left (a x\right )^{4}}{4 \, a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 7.31, size = 11, normalized size = 0.85 \begin {gather*} \frac {\arcsin \left (a x\right )^{4}}{4 \, a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.25, size = 10, normalized size = 0.77 \begin {gather*} \begin {cases} \frac {\operatorname {asin}^{4}{\left (a x \right )}}{4 a} & \text {for}\: a \neq 0 \\0 & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.46, size = 11, normalized size = 0.85 \begin {gather*} \frac {\arcsin \left (a x\right )^{4}}{4 \, a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.15, size = 11, normalized size = 0.85 \begin {gather*} \frac {{\mathrm {asin}\left (a\,x\right )}^4}{4\,a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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