Optimal. Leaf size=82 \[ \frac{2 x^2 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)}{9 a}+\frac{4 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)}{9 a^3}-\frac{4 x}{9 a^2}+\frac{1}{3} x^3 \sin ^{-1}(a x)^2-\frac{2 x^3}{27} \]
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Rubi [A] time = 0.121434, antiderivative size = 82, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {4627, 4707, 4677, 8, 30} \[ \frac{2 x^2 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)}{9 a}+\frac{4 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)}{9 a^3}-\frac{4 x}{9 a^2}+\frac{1}{3} x^3 \sin ^{-1}(a x)^2-\frac{2 x^3}{27} \]
Antiderivative was successfully verified.
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Rule 4627
Rule 4707
Rule 4677
Rule 8
Rule 30
Rubi steps
\begin{align*} \int x^2 \sin ^{-1}(a x)^2 \, dx &=\frac{1}{3} x^3 \sin ^{-1}(a x)^2-\frac{1}{3} (2 a) \int \frac{x^3 \sin ^{-1}(a x)}{\sqrt{1-a^2 x^2}} \, dx\\ &=\frac{2 x^2 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)}{9 a}+\frac{1}{3} x^3 \sin ^{-1}(a x)^2-\frac{2 \int x^2 \, dx}{9}-\frac{4 \int \frac{x \sin ^{-1}(a x)}{\sqrt{1-a^2 x^2}} \, dx}{9 a}\\ &=-\frac{2 x^3}{27}+\frac{4 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)}{9 a^3}+\frac{2 x^2 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)}{9 a}+\frac{1}{3} x^3 \sin ^{-1}(a x)^2-\frac{4 \int 1 \, dx}{9 a^2}\\ &=-\frac{4 x}{9 a^2}-\frac{2 x^3}{27}+\frac{4 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)}{9 a^3}+\frac{2 x^2 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)}{9 a}+\frac{1}{3} x^3 \sin ^{-1}(a x)^2\\ \end{align*}
Mathematica [A] time = 0.0280522, size = 64, normalized size = 0.78 \[ \frac{-2 a x \left (a^2 x^2+6\right )+9 a^3 x^3 \sin ^{-1}(a x)^2+6 \sqrt{1-a^2 x^2} \left (a^2 x^2+2\right ) \sin ^{-1}(a x)}{27 a^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.059, size = 59, normalized size = 0.7 \begin{align*}{\frac{1}{{a}^{3}} \left ({\frac{{a}^{3}{x}^{3} \left ( \arcsin \left ( ax \right ) \right ) ^{2}}{3}}+{\frac{2\,\arcsin \left ( ax \right ) \left ({a}^{2}{x}^{2}+2 \right ) }{9}\sqrt{-{a}^{2}{x}^{2}+1}}-{\frac{2\,{a}^{3}{x}^{3}}{27}}-{\frac{4\,ax}{9}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.73538, size = 97, normalized size = 1.18 \begin{align*} \frac{1}{3} \, x^{3} \arcsin \left (a x\right )^{2} + \frac{2}{9} \, a{\left (\frac{\sqrt{-a^{2} x^{2} + 1} x^{2}}{a^{2}} + \frac{2 \, \sqrt{-a^{2} x^{2} + 1}}{a^{4}}\right )} \arcsin \left (a x\right ) - \frac{2 \,{\left (a^{2} x^{3} + 6 \, x\right )}}{27 \, a^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.41462, size = 143, normalized size = 1.74 \begin{align*} \frac{9 \, a^{3} x^{3} \arcsin \left (a x\right )^{2} - 2 \, a^{3} x^{3} + 6 \,{\left (a^{2} x^{2} + 2\right )} \sqrt{-a^{2} x^{2} + 1} \arcsin \left (a x\right ) - 12 \, a x}{27 \, a^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 2.59436, size = 76, normalized size = 0.93 \begin{align*} \begin{cases} \frac{x^{3} \operatorname{asin}^{2}{\left (a x \right )}}{3} - \frac{2 x^{3}}{27} + \frac{2 x^{2} \sqrt{- a^{2} x^{2} + 1} \operatorname{asin}{\left (a x \right )}}{9 a} - \frac{4 x}{9 a^{2}} + \frac{4 \sqrt{- a^{2} x^{2} + 1} \operatorname{asin}{\left (a x \right )}}{9 a^{3}} & \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 = 1.31157, size = 131, normalized size = 1.6 \begin{align*} \frac{{\left (a^{2} x^{2} - 1\right )} x \arcsin \left (a x\right )^{2}}{3 \, a^{2}} + \frac{x \arcsin \left (a x\right )^{2}}{3 \, a^{2}} - \frac{2 \,{\left (a^{2} x^{2} - 1\right )} x}{27 \, a^{2}} - \frac{2 \,{\left (-a^{2} x^{2} + 1\right )}^{\frac{3}{2}} \arcsin \left (a x\right )}{9 \, a^{3}} - \frac{14 \, x}{27 \, a^{2}} + \frac{2 \, \sqrt{-a^{2} x^{2} + 1} \arcsin \left (a x\right )}{3 \, a^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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