Integrand size = 10, antiderivative size = 30 \[ \int \frac {\arccos \left (\frac {a}{x}\right )}{x^2} \, dx=\frac {\sqrt {1-\frac {a^2}{x^2}}}{a}-\frac {\sec ^{-1}\left (\frac {x}{a}\right )}{x} \]
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Time = 0.02 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {4917, 5328, 267} \[ \int \frac {\arccos \left (\frac {a}{x}\right )}{x^2} \, dx=\frac {\sqrt {1-\frac {a^2}{x^2}}}{a}-\frac {\sec ^{-1}\left (\frac {x}{a}\right )}{x} \]
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Rule 267
Rule 4917
Rule 5328
Rubi steps \begin{align*} \text {integral}& = \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*}
Time = 0.02 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.00 \[ \int \frac {\arccos \left (\frac {a}{x}\right )}{x^2} \, dx=\frac {\sqrt {1-\frac {a^2}{x^2}}}{a}-\frac {\arccos \left (\frac {a}{x}\right )}{x} \]
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Time = 0.09 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.07
method | result | size |
derivativedivides | \(-\frac {\frac {a \arccos \left (\frac {a}{x}\right )}{x}-\sqrt {1-\frac {a^{2}}{x^{2}}}}{a}\) | \(32\) |
default | \(-\frac {\frac {a \arccos \left (\frac {a}{x}\right )}{x}-\sqrt {1-\frac {a^{2}}{x^{2}}}}{a}\) | \(32\) |
parts | \(-\frac {\arccos \left (\frac {a}{x}\right )}{x}-\frac {a^{2}-x^{2}}{a \sqrt {-\frac {a^{2}-x^{2}}{x^{2}}}\, x^{2}}\) | \(46\) |
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Time = 0.23 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.20 \[ \int \frac {\arccos \left (\frac {a}{x}\right )}{x^2} \, dx=-\frac {a \arccos \left (\frac {a}{x}\right ) - x \sqrt {-\frac {a^{2} - x^{2}}{x^{2}}}}{a x} \]
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Time = 0.39 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.87 \[ \int \frac {\arccos \left (\frac {a}{x}\right )}{x^2} \, dx=\begin {cases} - \frac {\operatorname {acos}{\left (\frac {a}{x} \right )}}{x} + \frac {\sqrt {- \frac {a^{2}}{x^{2}} + 1}}{a} & \text {for}\: a \neq 0 \\- \frac {\pi }{2 x} & \text {otherwise} \end {cases} \]
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Time = 0.28 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.03 \[ \int \frac {\arccos \left (\frac {a}{x}\right )}{x^2} \, dx=-\frac {\frac {a \arccos \left (\frac {a}{x}\right )}{x} - \sqrt {-\frac {a^{2}}{x^{2}} + 1}}{a} \]
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Time = 0.29 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.03 \[ \int \frac {\arccos \left (\frac {a}{x}\right )}{x^2} \, dx=-\frac {\frac {a \arccos \left (\frac {a}{x}\right )}{x} - \sqrt {-\frac {a^{2}}{x^{2}} + 1}}{a} \]
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Time = 0.03 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.93 \[ \int \frac {\arccos \left (\frac {a}{x}\right )}{x^2} \, dx=\frac {\sqrt {1-\frac {a^2}{x^2}}}{a}-\frac {\mathrm {acos}\left (\frac {a}{x}\right )}{x} \]
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