Optimal. Leaf size=67 \[ -\frac{\sqrt{1-a^2 x^2} \sin ^{-1}(a x)^3}{a^2}+\frac{6 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)}{a^2}-\frac{6 x}{a}+\frac{3 x \sin ^{-1}(a x)^2}{a} \]
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Rubi [A] time = 0.105192, antiderivative size = 67, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136, Rules used = {4677, 4619, 8} \[ -\frac{\sqrt{1-a^2 x^2} \sin ^{-1}(a x)^3}{a^2}+\frac{6 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)}{a^2}-\frac{6 x}{a}+\frac{3 x \sin ^{-1}(a x)^2}{a} \]
Antiderivative was successfully verified.
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Rule 4677
Rule 4619
Rule 8
Rubi steps
\begin{align*} \int \frac{x \sin ^{-1}(a x)^3}{\sqrt{1-a^2 x^2}} \, dx &=-\frac{\sqrt{1-a^2 x^2} \sin ^{-1}(a x)^3}{a^2}+\frac{3 \int \sin ^{-1}(a x)^2 \, dx}{a}\\ &=\frac{3 x \sin ^{-1}(a x)^2}{a}-\frac{\sqrt{1-a^2 x^2} \sin ^{-1}(a x)^3}{a^2}-6 \int \frac{x \sin ^{-1}(a x)}{\sqrt{1-a^2 x^2}} \, dx\\ &=\frac{6 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)}{a^2}+\frac{3 x \sin ^{-1}(a x)^2}{a}-\frac{\sqrt{1-a^2 x^2} \sin ^{-1}(a x)^3}{a^2}-\frac{6 \int 1 \, dx}{a}\\ &=-\frac{6 x}{a}+\frac{6 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)}{a^2}+\frac{3 x \sin ^{-1}(a x)^2}{a}-\frac{\sqrt{1-a^2 x^2} \sin ^{-1}(a x)^3}{a^2}\\ \end{align*}
Mathematica [A] time = 0.0167272, size = 61, normalized size = 0.91 \[ \frac{-\sqrt{1-a^2 x^2} \sin ^{-1}(a x)^3+6 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)-6 a x+3 a x \sin ^{-1}(a x)^2}{a^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.046, size = 107, normalized size = 1.6 \begin{align*} -{\frac{1}{{a}^{2} \left ({a}^{2}{x}^{2}-1 \right ) }\sqrt{-{a}^{2}{x}^{2}+1} \left ( \left ( \arcsin \left ( ax \right ) \right ) ^{3}{x}^{2}{a}^{2}- \left ( \arcsin \left ( ax \right ) \right ) ^{3}+3\, \left ( \arcsin \left ( ax \right ) \right ) ^{2}\sqrt{-{a}^{2}{x}^{2}+1}xa-6\,{a}^{2}{x}^{2}\arcsin \left ( ax \right ) +6\,\arcsin \left ( ax \right ) -6\,ax\sqrt{-{a}^{2}{x}^{2}+1} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.51214, size = 86, normalized size = 1.28 \begin{align*} \frac{3 \, x \arcsin \left (a x\right )^{2}}{a} - \frac{\sqrt{-a^{2} x^{2} + 1} \arcsin \left (a x\right )^{3}}{a^{2}} - \frac{6 \,{\left (x - \frac{\sqrt{-a^{2} x^{2} + 1} \arcsin \left (a x\right )}{a}\right )}}{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.9285, size = 119, normalized size = 1.78 \begin{align*} \frac{3 \, a x \arcsin \left (a x\right )^{2} - 6 \, a x - \sqrt{-a^{2} x^{2} + 1}{\left (\arcsin \left (a x\right )^{3} - 6 \, \arcsin \left (a x\right )\right )}}{a^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.41598, size = 61, normalized size = 0.91 \begin{align*} \begin{cases} \frac{3 x \operatorname{asin}^{2}{\left (a x \right )}}{a} - \frac{6 x}{a} - \frac{\sqrt{- a^{2} x^{2} + 1} \operatorname{asin}^{3}{\left (a x \right )}}{a^{2}} + \frac{6 \sqrt{- a^{2} x^{2} + 1} \operatorname{asin}{\left (a x \right )}}{a^{2}} & \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 = 1.41462, size = 84, normalized size = 1.25 \begin{align*} -\frac{\sqrt{-a^{2} x^{2} + 1} \arcsin \left (a x\right )^{3}}{a^{2}} + \frac{3 \,{\left (x \arcsin \left (a x\right )^{2} - 2 \, x + \frac{2 \, \sqrt{-a^{2} x^{2} + 1} \arcsin \left (a x\right )}{a}\right )}}{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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