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 = 0.02, antiderivative size = 30, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, 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 261
Rule 4833
Rule 5220
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*}
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Mathematica [A] time = 0.02, size = 30, normalized size = 1.00 \[ \frac {\sqrt {1-\frac {a^2}{x^2}}}{a}-\frac {\cos ^{-1}\left (\frac {a}{x}\right )}{x} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 36, normalized size = 1.20 \[ -\frac {a \arccos \left (\frac {a}{x}\right ) - x \sqrt {-\frac {a^{2} - x^{2}}{x^{2}}}}{a x} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 31, normalized size = 1.03 \[ -\frac {\frac {a \arccos \left (\frac {a}{x}\right )}{x} - \sqrt {-\frac {a^{2}}{x^{2}} + 1}}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 32, normalized size = 1.07 \[ -\frac {\frac {a \arccos \left (\frac {a}{x}\right )}{x}-\sqrt {1-\frac {a^{2}}{x^{2}}}}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.40, size = 31, normalized size = 1.03 \[ -\frac {\frac {a \arccos \left (\frac {a}{x}\right )}{x} - \sqrt {-\frac {a^{2}}{x^{2}} + 1}}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.03, size = 28, normalized size = 0.93 \[ \frac {\sqrt {1-\frac {a^2}{x^2}}}{a}-\frac {\mathrm {acos}\left (\frac {a}{x}\right )}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.72, size = 26, normalized size = 0.87 \[ \begin {cases} - \frac {\operatorname {acos}{\left (\frac {a}{x} \right )}}{x} + \frac {\sqrt {- \frac {a^{2}}{x^{2}} + 1}}{a} & \text {for}\: a \neq 0 \\- \frac {\pi }{2 x} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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