Optimal. Leaf size=45 \[ \frac{x \sqrt{1-a^2 x^2}}{4 a}-\frac{\sin ^{-1}(a x)}{4 a^2}+\frac{1}{2} x^2 \sin ^{-1}(a x) \]
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Rubi [A] time = 0.0158949, antiderivative size = 45, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 6, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {4627, 321, 216} \[ \frac{x \sqrt{1-a^2 x^2}}{4 a}-\frac{\sin ^{-1}(a x)}{4 a^2}+\frac{1}{2} x^2 \sin ^{-1}(a x) \]
Antiderivative was successfully verified.
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Rule 4627
Rule 321
Rule 216
Rubi steps
\begin{align*} \int x \sin ^{-1}(a x) \, dx &=\frac{1}{2} x^2 \sin ^{-1}(a x)-\frac{1}{2} a \int \frac{x^2}{\sqrt{1-a^2 x^2}} \, dx\\ &=\frac{x \sqrt{1-a^2 x^2}}{4 a}+\frac{1}{2} x^2 \sin ^{-1}(a x)-\frac{\int \frac{1}{\sqrt{1-a^2 x^2}} \, dx}{4 a}\\ &=\frac{x \sqrt{1-a^2 x^2}}{4 a}-\frac{\sin ^{-1}(a x)}{4 a^2}+\frac{1}{2} x^2 \sin ^{-1}(a x)\\ \end{align*}
Mathematica [A] time = 0.0112639, size = 40, normalized size = 0.89 \[ \frac{a x \sqrt{1-a^2 x^2}+\left (2 a^2 x^2-1\right ) \sin ^{-1}(a x)}{4 a^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 40, normalized size = 0.9 \begin{align*}{\frac{1}{{a}^{2}} \left ({\frac{{a}^{2}{x}^{2}\arcsin \left ( ax \right ) }{2}}+{\frac{ax}{4}\sqrt{-{a}^{2}{x}^{2}+1}}-{\frac{\arcsin \left ( ax \right ) }{4}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.67149, size = 70, normalized size = 1.56 \begin{align*} \frac{1}{2} \, x^{2} \arcsin \left (a x\right ) + \frac{1}{4} \, a{\left (\frac{\sqrt{-a^{2} x^{2} + 1} x}{a^{2}} - \frac{\arcsin \left (\frac{a^{2} x}{\sqrt{a^{2}}}\right )}{\sqrt{a^{2}} a^{2}}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.20046, size = 86, normalized size = 1.91 \begin{align*} \frac{\sqrt{-a^{2} x^{2} + 1} a x +{\left (2 \, a^{2} x^{2} - 1\right )} \arcsin \left (a x\right )}{4 \, a^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.345889, size = 37, normalized size = 0.82 \begin{align*} \begin{cases} \frac{x^{2} \operatorname{asin}{\left (a x \right )}}{2} + \frac{x \sqrt{- a^{2} x^{2} + 1}}{4 a} - \frac{\operatorname{asin}{\left (a x \right )}}{4 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 = 1.23908, size = 62, normalized size = 1.38 \begin{align*} \frac{\sqrt{-a^{2} x^{2} + 1} x}{4 \, a} + \frac{{\left (a^{2} x^{2} - 1\right )} \arcsin \left (a x\right )}{2 \, a^{2}} + \frac{\arcsin \left (a x\right )}{4 \, a^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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