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 = 0.0801758, antiderivative size = 60, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 6, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, 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 4619
Rule 4677
Rule 261
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*}
Mathematica [A] time = 0.0112293, size = 60, normalized size = 1. \[ -\frac{6 \sqrt{1-a^2 x^2}}{a}+\frac{3 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^2}{a}+x \sin ^{-1}(a x)^3-6 x \sin ^{-1}(a x) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.024, size = 57, normalized size = 1. \begin{align*}{\frac{1}{a} \left ( ax \left ( \arcsin \left ( ax \right ) \right ) ^{3}+3\, \left ( \arcsin \left ( ax \right ) \right ) ^{2}\sqrt{-{a}^{2}{x}^{2}+1}-6\,\sqrt{-{a}^{2}{x}^{2}+1}-6\,ax\arcsin \left ( ax \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.68067, size = 77, normalized size = 1.28 \begin{align*} 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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.04743, size = 116, normalized size = 1.93 \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.832359, size = 54, normalized size = 0.9 \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.36035, size = 76, normalized size = 1.27 \begin{align*} 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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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