Optimal. Leaf size=60 \[ -\frac {6 \sqrt {1-a^2 x^2}}{a}+\frac {3 \sqrt {1-a^2 x^2} \sin ^{-1}(a x)^2}{a}+x \sin ^{-1}(a x)^3-6 x \sin ^{-1}(a x) \]
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Rubi [A] time = 0.08, antiderivative size = 60, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {4619, 4677, 261} \[ -\frac {6 \sqrt {1-a^2 x^2}}{a}+\frac {3 \sqrt {1-a^2 x^2} \sin ^{-1}(a x)^2}{a}+x \sin ^{-1}(a x)^3-6 x \sin ^{-1}(a x) \]
Antiderivative was successfully verified.
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Rule 261
Rule 4619
Rule 4677
Rubi steps
\begin {align*} \int \sin ^{-1}(a x)^3 \, dx &=x \sin ^{-1}(a x)^3-(3 a) \int \frac {x \sin ^{-1}(a x)^2}{\sqrt {1-a^2 x^2}} \, dx\\ &=\frac {3 \sqrt {1-a^2 x^2} \sin ^{-1}(a x)^2}{a}+x \sin ^{-1}(a x)^3-6 \int \sin ^{-1}(a x) \, dx\\ &=-6 x \sin ^{-1}(a x)+\frac {3 \sqrt {1-a^2 x^2} \sin ^{-1}(a x)^2}{a}+x \sin ^{-1}(a x)^3+(6 a) \int \frac {x}{\sqrt {1-a^2 x^2}} \, dx\\ &=-\frac {6 \sqrt {1-a^2 x^2}}{a}-6 x \sin ^{-1}(a x)+\frac {3 \sqrt {1-a^2 x^2} \sin ^{-1}(a x)^2}{a}+x \sin ^{-1}(a x)^3\\ \end {align*}
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Mathematica [A] time = 0.01, size = 60, normalized size = 1.00 \[ -\frac {6 \sqrt {1-a^2 x^2}}{a}+\frac {3 \sqrt {1-a^2 x^2} \sin ^{-1}(a x)^2}{a}+x \sin ^{-1}(a x)^3-6 x \sin ^{-1}(a x) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.65, size = 44, normalized size = 0.73 \[ \frac {a x \arcsin \left (a x\right )^{3} - 6 \, a x \arcsin \left (a x\right ) + 3 \, \sqrt {-a^{2} x^{2} + 1} {\left (\arcsin \left (a x\right )^{2} - 2\right )}}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.14, size = 56, normalized size = 0.93 \[ x \arcsin \left (a x\right )^{3} - 6 \, x \arcsin \left (a x\right ) + \frac {3 \, \sqrt {-a^{2} x^{2} + 1} \arcsin \left (a x\right )^{2}}{a} - \frac {6 \, \sqrt {-a^{2} x^{2} + 1}}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 57, normalized size = 0.95 \[ \frac {a x \arcsin \left (a x \right )^{3}+3 \arcsin \left (a x \right )^{2} \sqrt {-a^{2} x^{2}+1}-6 \sqrt {-a^{2} x^{2}+1}-6 a x \arcsin \left (a x \right )}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.54, size = 57, normalized size = 0.95 \[ x \arcsin \left (a x\right )^{3} + \frac {3 \, \sqrt {-a^{2} x^{2} + 1} \arcsin \left (a x\right )^{2}}{a} - \frac {6 \, {\left (a x \arcsin \left (a x\right ) + \sqrt {-a^{2} x^{2} + 1}\right )}}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.14, size = 40, normalized size = 0.67 \[ \frac {3\,\sqrt {1-a^2\,x^2}\,\left ({\mathrm {asin}\left (a\,x\right )}^2-2\right )}{a}+x\,\mathrm {asin}\left (a\,x\right )\,\left ({\mathrm {asin}\left (a\,x\right )}^2-6\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.45, size = 54, normalized size = 0.90 \[ \begin {cases} x \operatorname {asin}^{3}{\left (a x \right )} - 6 x \operatorname {asin}{\left (a x \right )} + \frac {3 \sqrt {- a^{2} x^{2} + 1} \operatorname {asin}^{2}{\left (a x \right )}}{a} - \frac {6 \sqrt {- a^{2} x^{2} + 1}}{a} & \text {for}\: a \neq 0 \\0 & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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