Optimal. Leaf size=55 \[ \frac{2 \sqrt{1-a^2 x^2}}{a^2}-\frac{\sqrt{1-a^2 x^2} \sin ^{-1}(a x)^2}{a^2}+\frac{2 x \sin ^{-1}(a x)}{a} \]
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Rubi [A] time = 0.0719107, antiderivative size = 55, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136, Rules used = {4677, 4619, 261} \[ \frac{2 \sqrt{1-a^2 x^2}}{a^2}-\frac{\sqrt{1-a^2 x^2} \sin ^{-1}(a x)^2}{a^2}+\frac{2 x \sin ^{-1}(a x)}{a} \]
Antiderivative was successfully verified.
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Rule 4677
Rule 4619
Rule 261
Rubi steps
\begin{align*} \int \frac{x \sin ^{-1}(a x)^2}{\sqrt{1-a^2 x^2}} \, dx &=-\frac{\sqrt{1-a^2 x^2} \sin ^{-1}(a x)^2}{a^2}+\frac{2 \int \sin ^{-1}(a x) \, dx}{a}\\ &=\frac{2 x \sin ^{-1}(a x)}{a}-\frac{\sqrt{1-a^2 x^2} \sin ^{-1}(a x)^2}{a^2}-2 \int \frac{x}{\sqrt{1-a^2 x^2}} \, dx\\ &=\frac{2 \sqrt{1-a^2 x^2}}{a^2}+\frac{2 x \sin ^{-1}(a x)}{a}-\frac{\sqrt{1-a^2 x^2} \sin ^{-1}(a x)^2}{a^2}\\ \end{align*}
Mathematica [A] time = 0.0127663, size = 51, normalized size = 0.93 \[ \frac{2 \sqrt{1-a^2 x^2}-\sqrt{1-a^2 x^2} \sin ^{-1}(a x)^2+2 a x \sin ^{-1}(a x)}{a^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.044, size = 80, normalized size = 1.5 \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 ) ^{2}{x}^{2}{a}^{2}- \left ( \arcsin \left ( ax \right ) \right ) ^{2}+2\,\arcsin \left ( ax \right ) \sqrt{-{a}^{2}{x}^{2}+1}xa-2\,{a}^{2}{x}^{2}+2 \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.49674, size = 66, normalized size = 1.2 \begin{align*} -\frac{\sqrt{-a^{2} x^{2} + 1} \arcsin \left (a x\right )^{2}}{a^{2}} + \frac{2 \,{\left (a x \arcsin \left (a x\right ) + \sqrt{-a^{2} x^{2} + 1}\right )}}{a^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.6864, size = 89, normalized size = 1.62 \begin{align*} \frac{2 \, a x \arcsin \left (a x\right ) - \sqrt{-a^{2} x^{2} + 1}{\left (\arcsin \left (a x\right )^{2} - 2\right )}}{a^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.87482, size = 49, normalized size = 0.89 \begin{align*} \begin{cases} \frac{2 x \operatorname{asin}{\left (a x \right )}}{a} - \frac{\sqrt{- a^{2} x^{2} + 1} \operatorname{asin}^{2}{\left (a x \right )}}{a^{2}} + \frac{2 \sqrt{- a^{2} x^{2} + 1}}{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 = 1.27632, size = 66, normalized size = 1.2 \begin{align*} -\frac{\sqrt{-a^{2} x^{2} + 1} \arcsin \left (a x\right )^{2}}{a^{2}} + \frac{2 \,{\left (a x \arcsin \left (a x\right ) + \sqrt{-a^{2} x^{2} + 1}\right )}}{a^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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