Optimal. Leaf size=34 \[ -\frac{a \sqrt{1-a^2 x^2}}{2 x}-\frac{\sin ^{-1}(a x)}{2 x^2} \]
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Rubi [A] time = 0.01436, antiderivative size = 34, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {4627, 264} \[ -\frac{a \sqrt{1-a^2 x^2}}{2 x}-\frac{\sin ^{-1}(a x)}{2 x^2} \]
Antiderivative was successfully verified.
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Rule 4627
Rule 264
Rubi steps
\begin{align*} \int \frac{\sin ^{-1}(a x)}{x^3} \, dx &=-\frac{\sin ^{-1}(a x)}{2 x^2}+\frac{1}{2} a \int \frac{1}{x^2 \sqrt{1-a^2 x^2}} \, dx\\ &=-\frac{a \sqrt{1-a^2 x^2}}{2 x}-\frac{\sin ^{-1}(a x)}{2 x^2}\\ \end{align*}
Mathematica [A] time = 0.0071846, size = 29, normalized size = 0.85 \[ -\frac{a x \sqrt{1-a^2 x^2}+\sin ^{-1}(a x)}{2 x^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 38, normalized size = 1.1 \begin{align*}{a}^{2} \left ( -{\frac{\arcsin \left ( ax \right ) }{2\,{a}^{2}{x}^{2}}}-{\frac{1}{2\,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 = 1.65813, size = 38, normalized size = 1.12 \begin{align*} -\frac{\sqrt{-a^{2} x^{2} + 1} a}{2 \, x} - \frac{\arcsin \left (a x\right )}{2 \, x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.1906, size = 66, normalized size = 1.94 \begin{align*} -\frac{\sqrt{-a^{2} x^{2} + 1} a x + \arcsin \left (a x\right )}{2 \, x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 2.11402, size = 51, normalized size = 1.5 \begin{align*} \frac{a \left (\begin{cases} - \frac{i \sqrt{a^{2} x^{2} - 1}}{x} & \text{for}\: \left |{a^{2} x^{2}}\right | > 1 \\- \frac{\sqrt{- a^{2} x^{2} + 1}}{x} & \text{otherwise} \end{cases}\right )}{2} - \frac{\operatorname{asin}{\left (a x \right )}}{2 x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.28193, size = 92, normalized size = 2.71 \begin{align*} \frac{1}{4} \,{\left (\frac{a^{4} x}{{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}{\left | a \right |}} - \frac{\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a}{x{\left | a \right |}}\right )} a - \frac{\arcsin \left (a x\right )}{2 \, x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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