Optimal. Leaf size=27 \[ a \tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right )-\frac{\cos ^{-1}(a x)}{x} \]
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Rubi [A] time = 0.0224197, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {4628, 266, 63, 208} \[ a \tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right )-\frac{\cos ^{-1}(a x)}{x} \]
Antiderivative was successfully verified.
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Rule 4628
Rule 266
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{\cos ^{-1}(a x)}{x^2} \, dx &=-\frac{\cos ^{-1}(a x)}{x}-a \int \frac{1}{x \sqrt{1-a^2 x^2}} \, dx\\ &=-\frac{\cos ^{-1}(a x)}{x}-\frac{1}{2} a \operatorname{Subst}\left (\int \frac{1}{x \sqrt{1-a^2 x}} \, dx,x,x^2\right )\\ &=-\frac{\cos ^{-1}(a x)}{x}+\frac{\operatorname{Subst}\left (\int \frac{1}{\frac{1}{a^2}-\frac{x^2}{a^2}} \, dx,x,\sqrt{1-a^2 x^2}\right )}{a}\\ &=-\frac{\cos ^{-1}(a x)}{x}+a \tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right )\\ \end{align*}
Mathematica [A] time = 0.0111537, size = 34, normalized size = 1.26 \[ a \log \left (\sqrt{1-a^2 x^2}+1\right )-a \log (x)-\frac{\cos ^{-1}(a x)}{x} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 29, normalized size = 1.1 \begin{align*} a \left ( -{\frac{\arccos \left ( ax \right ) }{ax}}+{\it Artanh} \left ({\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}} \right ) \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.45474, size = 51, normalized size = 1.89 \begin{align*} a \log \left (\frac{2 \, \sqrt{-a^{2} x^{2} + 1}}{{\left | x \right |}} + \frac{2}{{\left | x \right |}}\right ) - \frac{\arccos \left (a x\right )}{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.47838, size = 203, normalized size = 7.52 \begin{align*} \frac{a x \log \left (\sqrt{-a^{2} x^{2} + 1} + 1\right ) - a x \log \left (\sqrt{-a^{2} x^{2} + 1} - 1\right ) + 2 \,{\left (x - 1\right )} \arccos \left (a x\right ) - 2 \, x \arctan \left (\frac{\sqrt{-a^{2} x^{2} + 1} a x}{a^{2} x^{2} - 1}\right )}{2 \, x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 1.61538, size = 34, normalized size = 1.26 \begin{align*} - a \left (\begin{cases} - \operatorname{acosh}{\left (\frac{1}{a x} \right )} & \text{for}\: \frac{1}{\left |{a^{2} x^{2}}\right |} > 1 \\i \operatorname{asin}{\left (\frac{1}{a x} \right )} & \text{otherwise} \end{cases}\right ) - \frac{\operatorname{acos}{\left (a x \right )}}{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.13227, size = 65, normalized size = 2.41 \begin{align*} \frac{1}{2} \, a{\left (\log \left (\sqrt{-a^{2} x^{2} + 1} + 1\right ) - \log \left (-\sqrt{-a^{2} x^{2} + 1} + 1\right )\right )} - \frac{\arccos \left (a x\right )}{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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