Optimal. Leaf size=30 \[ \frac{\sqrt{1-\frac{a^2}{x^2}}}{a}-\frac{\sec ^{-1}\left (\frac{x}{a}\right )}{x} \]
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Rubi [A] time = 0.0218937, antiderivative size = 30, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {4833, 5220, 261} \[ \frac{\sqrt{1-\frac{a^2}{x^2}}}{a}-\frac{\sec ^{-1}\left (\frac{x}{a}\right )}{x} \]
Antiderivative was successfully verified.
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Rule 4833
Rule 5220
Rule 261
Rubi steps
\begin{align*} \int \frac{\cos ^{-1}\left (\frac{a}{x}\right )}{x^2} \, dx &=\int \frac{\sec ^{-1}\left (\frac{x}{a}\right )}{x^2} \, dx\\ &=-\frac{\sec ^{-1}\left (\frac{x}{a}\right )}{x}+a \int \frac{1}{\sqrt{1-\frac{a^2}{x^2}} x^3} \, dx\\ &=\frac{\sqrt{1-\frac{a^2}{x^2}}}{a}-\frac{\sec ^{-1}\left (\frac{x}{a}\right )}{x}\\ \end{align*}
Mathematica [A] time = 0.0185694, size = 30, normalized size = 1. \[ \frac{\sqrt{1-\frac{a^2}{x^2}}}{a}-\frac{\cos ^{-1}\left (\frac{a}{x}\right )}{x} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.001, size = 32, normalized size = 1.1 \begin{align*} -{\frac{1}{a} \left ({\frac{a}{x}\arccos \left ({\frac{a}{x}} \right ) }-\sqrt{1-{\frac{{a}^{2}}{{x}^{2}}}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.43795, size = 42, normalized size = 1.4 \begin{align*} -\frac{\frac{a \arccos \left (\frac{a}{x}\right )}{x} - \sqrt{-\frac{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 = 2.39408, size = 69, normalized size = 2.3 \begin{align*} -\frac{a \arccos \left (\frac{a}{x}\right ) - x \sqrt{-\frac{a^{2} - x^{2}}{x^{2}}}}{a x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 2.34016, size = 26, normalized size = 0.87 \begin{align*} \begin{cases} - \frac{\operatorname{acos}{\left (\frac{a}{x} \right )}}{x} + \frac{\sqrt{- \frac{a^{2}}{x^{2}} + 1}}{a} & \text{for}\: a \neq 0 \\- \frac{\pi }{2 x} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.2283, size = 42, normalized size = 1.4 \begin{align*} -\frac{\frac{a \arccos \left (\frac{a}{x}\right )}{x} - \sqrt{-\frac{a^{2}}{x^{2}} + 1}}{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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