Optimal. Leaf size=29 \[ \frac{x}{a}-\frac{\sqrt{1-a^2 x^2} \sin ^{-1}(a x)}{a^2} \]
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Rubi [A] time = 0.0405081, antiderivative size = 29, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {4677, 8} \[ \frac{x}{a}-\frac{\sqrt{1-a^2 x^2} \sin ^{-1}(a x)}{a^2} \]
Antiderivative was successfully verified.
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Rule 4677
Rule 8
Rubi steps
\begin{align*} \int \frac{x \sin ^{-1}(a x)}{\sqrt{1-a^2 x^2}} \, dx &=-\frac{\sqrt{1-a^2 x^2} \sin ^{-1}(a x)}{a^2}+\frac{\int 1 \, dx}{a}\\ &=\frac{x}{a}-\frac{\sqrt{1-a^2 x^2} \sin ^{-1}(a x)}{a^2}\\ \end{align*}
Mathematica [A] time = 0.0083922, size = 29, normalized size = 1. \[ \frac{x}{a}-\frac{\sqrt{1-a^2 x^2} \sin ^{-1}(a x)}{a^2} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.038, size = 62, normalized size = 2.1 \begin{align*} -{\frac{1}{{a}^{2} \left ({a}^{2}{x}^{2}-1 \right ) }\sqrt{-{a}^{2}{x}^{2}+1} \left ({a}^{2}{x}^{2}\arcsin \left ( ax \right ) -\arcsin \left ( ax \right ) +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 = 1.53852, size = 36, normalized size = 1.24 \begin{align*} \frac{x}{a} - \frac{\sqrt{-a^{2} x^{2} + 1} \arcsin \left (a x\right )}{a^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.13272, size = 59, normalized size = 2.03 \begin{align*} \frac{a x - \sqrt{-a^{2} x^{2} + 1} \arcsin \left (a x\right )}{a^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.521907, size = 24, normalized size = 0.83 \begin{align*} \begin{cases} \frac{x}{a} - \frac{\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 = 1.40187, size = 36, normalized size = 1.24 \begin{align*} \frac{x}{a} - \frac{\sqrt{-a^{2} x^{2} + 1} \arcsin \left (a x\right )}{a^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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