Optimal. Leaf size=34 \[ \frac{a \sqrt{1-a^2 x^2}}{2 x}-\frac{\cos ^{-1}(a x)}{2 x^2} \]
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Rubi [A] time = 0.015502, 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 = {4628, 264} \[ \frac{a \sqrt{1-a^2 x^2}}{2 x}-\frac{\cos ^{-1}(a x)}{2 x^2} \]
Antiderivative was successfully verified.
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Rule 4628
Rule 264
Rubi steps
\begin{align*} \int \frac{\cos ^{-1}(a x)}{x^3} \, dx &=-\frac{\cos ^{-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{\cos ^{-1}(a x)}{2 x^2}\\ \end{align*}
Mathematica [A] time = 0.0136793, size = 31, normalized size = 0.91 \[ \frac{a x \sqrt{1-a^2 x^2}-\cos ^{-1}(a x)}{2 x^2} \]
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{\arccos \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 = 1.46339, size = 38, normalized size = 1.12 \begin{align*} \frac{\sqrt{-a^{2} x^{2} + 1} a}{2 \, x} - \frac{\arccos \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 = 2.33323, size = 65, normalized size = 1.91 \begin{align*} \frac{\sqrt{-a^{2} x^{2} + 1} a x - \arccos \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 = 1.3881, size = 53, normalized size = 1.56 \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{acos}{\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 = 1.11412, 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{\arccos \left (a x\right )}{2 \, x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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