Optimal. Leaf size=25 \[ \frac{\sqrt{1-a^2 x^2}}{a}+x \sin ^{-1}(a x) \]
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Rubi [A] time = 0.0075506, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 4, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {4619, 261} \[ \frac{\sqrt{1-a^2 x^2}}{a}+x \sin ^{-1}(a x) \]
Antiderivative was successfully verified.
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Rule 4619
Rule 261
Rubi steps
\begin{align*} \int \sin ^{-1}(a x) \, dx &=x \sin ^{-1}(a x)-a \int \frac{x}{\sqrt{1-a^2 x^2}} \, dx\\ &=\frac{\sqrt{1-a^2 x^2}}{a}+x \sin ^{-1}(a x)\\ \end{align*}
Mathematica [A] time = 0.0075927, size = 25, normalized size = 1. \[ \frac{\sqrt{1-a^2 x^2}}{a}+x \sin ^{-1}(a x) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.001, size = 25, normalized size = 1. \begin{align*}{\frac{1}{a} \left ( ax\arcsin \left ( ax \right ) +\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 = 1.68877, size = 32, normalized size = 1.28 \begin{align*} \frac{a x \arcsin \left (a x\right ) + \sqrt{-a^{2} x^{2} + 1}}{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.10614, size = 57, normalized size = 2.28 \begin{align*} \frac{a x \arcsin \left (a x\right ) + \sqrt{-a^{2} x^{2} + 1}}{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.618542, size = 20, normalized size = 0.8 \begin{align*} \begin{cases} x \operatorname{asin}{\left (a x \right )} + \frac{\sqrt{- a^{2} x^{2} + 1}}{a} & \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 = 1.33051, size = 32, normalized size = 1.28 \begin{align*} \frac{a x \arcsin \left (a x\right ) + \sqrt{-a^{2} x^{2} + 1}}{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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