Optimal. Leaf size=89 \[ \frac{x \sqrt{1-a^2 x^2}}{4 a^2}-\frac{x \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^2}{2 a^2}+\frac{\sin ^{-1}(a x)^3}{6 a^3}-\frac{\sin ^{-1}(a x)}{4 a^3}+\frac{x^2 \sin ^{-1}(a x)}{2 a} \]
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Rubi [A] time = 0.137485, antiderivative size = 89, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208, Rules used = {4707, 4641, 4627, 321, 216} \[ \frac{x \sqrt{1-a^2 x^2}}{4 a^2}-\frac{x \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^2}{2 a^2}+\frac{\sin ^{-1}(a x)^3}{6 a^3}-\frac{\sin ^{-1}(a x)}{4 a^3}+\frac{x^2 \sin ^{-1}(a x)}{2 a} \]
Antiderivative was successfully verified.
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Rule 4707
Rule 4641
Rule 4627
Rule 321
Rule 216
Rubi steps
\begin{align*} \int \frac{x^2 \sin ^{-1}(a x)^2}{\sqrt{1-a^2 x^2}} \, dx &=-\frac{x \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^2}{2 a^2}+\frac{\int \frac{\sin ^{-1}(a x)^2}{\sqrt{1-a^2 x^2}} \, dx}{2 a^2}+\frac{\int x \sin ^{-1}(a x) \, dx}{a}\\ &=\frac{x^2 \sin ^{-1}(a x)}{2 a}-\frac{x \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^2}{2 a^2}+\frac{\sin ^{-1}(a x)^3}{6 a^3}-\frac{1}{2} \int \frac{x^2}{\sqrt{1-a^2 x^2}} \, dx\\ &=\frac{x \sqrt{1-a^2 x^2}}{4 a^2}+\frac{x^2 \sin ^{-1}(a x)}{2 a}-\frac{x \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^2}{2 a^2}+\frac{\sin ^{-1}(a x)^3}{6 a^3}-\frac{\int \frac{1}{\sqrt{1-a^2 x^2}} \, dx}{4 a^2}\\ &=\frac{x \sqrt{1-a^2 x^2}}{4 a^2}-\frac{\sin ^{-1}(a x)}{4 a^3}+\frac{x^2 \sin ^{-1}(a x)}{2 a}-\frac{x \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^2}{2 a^2}+\frac{\sin ^{-1}(a x)^3}{6 a^3}\\ \end{align*}
Mathematica [A] time = 0.024274, size = 73, normalized size = 0.82 \[ \frac{3 a x \sqrt{1-a^2 x^2}-6 a x \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^2+\left (6 a^2 x^2-3\right ) \sin ^{-1}(a x)+2 \sin ^{-1}(a x)^3}{12 a^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.062, size = 71, normalized size = 0.8 \begin{align*}{\frac{1}{12\,{a}^{3}} \left ( -6\, \left ( \arcsin \left ( ax \right ) \right ) ^{2}\sqrt{-{a}^{2}{x}^{2}+1}xa+6\,{a}^{2}{x}^{2}\arcsin \left ( ax \right ) +2\, \left ( \arcsin \left ( ax \right ) \right ) ^{3}+3\,ax\sqrt{-{a}^{2}{x}^{2}+1}-3\,\arcsin \left ( ax \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{2} \arcsin \left (a x\right )^{2}}{\sqrt{-a^{2} x^{2} + 1}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.73193, size = 150, normalized size = 1.69 \begin{align*} \frac{2 \, \arcsin \left (a x\right )^{3} + 3 \,{\left (2 \, a^{2} x^{2} - 1\right )} \arcsin \left (a x\right ) - 3 \, \sqrt{-a^{2} x^{2} + 1}{\left (2 \, a x \arcsin \left (a x\right )^{2} - a x\right )}}{12 \, a^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.50168, size = 78, normalized size = 0.88 \begin{align*} \begin{cases} \frac{x^{2} \operatorname{asin}{\left (a x \right )}}{2 a} - \frac{x \sqrt{- a^{2} x^{2} + 1} \operatorname{asin}^{2}{\left (a x \right )}}{2 a^{2}} + \frac{x \sqrt{- a^{2} x^{2} + 1}}{4 a^{2}} + \frac{\operatorname{asin}^{3}{\left (a x \right )}}{6 a^{3}} - \frac{\operatorname{asin}{\left (a x \right )}}{4 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 = 1.24809, size = 109, normalized size = 1.22 \begin{align*} -\frac{\sqrt{-a^{2} x^{2} + 1} x \arcsin \left (a x\right )^{2}}{2 \, a^{2}} + \frac{\arcsin \left (a x\right )^{3}}{6 \, a^{3}} + \frac{\sqrt{-a^{2} x^{2} + 1} x}{4 \, a^{2}} + \frac{{\left (a^{2} x^{2} - 1\right )} \arcsin \left (a x\right )}{2 \, a^{3}} + \frac{\arcsin \left (a x\right )}{4 \, a^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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