Integrand size = 10, antiderivative size = 31 \[ \int \frac {\sec ^{-1}\left (\frac {a}{x}\right )}{x^2} \, dx=-\frac {\arccos \left (\frac {x}{a}\right )}{x}+\frac {\text {arctanh}\left (\sqrt {1-\frac {x^2}{a^2}}\right )}{a} \]
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Time = 0.02 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5372, 4724, 272, 65, 214} \[ \int \frac {\sec ^{-1}\left (\frac {a}{x}\right )}{x^2} \, dx=\frac {\text {arctanh}\left (\sqrt {1-\frac {x^2}{a^2}}\right )}{a}-\frac {\arccos \left (\frac {x}{a}\right )}{x} \]
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Rule 65
Rule 214
Rule 272
Rule 4724
Rule 5372
Rubi steps \begin{align*} \text {integral}& = \int \frac {\arccos \left (\frac {x}{a}\right )}{x^2} \, dx \\ & = -\frac {\arccos \left (\frac {x}{a}\right )}{x}-\frac {\int \frac {1}{x \sqrt {1-\frac {x^2}{a^2}}} \, dx}{a} \\ & = -\frac {\arccos \left (\frac {x}{a}\right )}{x}-\frac {\text {Subst}\left (\int \frac {1}{x \sqrt {1-\frac {x}{a^2}}} \, dx,x,x^2\right )}{2 a} \\ & = -\frac {\arccos \left (\frac {x}{a}\right )}{x}+a \text {Subst}\left (\int \frac {1}{a^2-a^2 x^2} \, dx,x,\sqrt {1-\frac {x^2}{a^2}}\right ) \\ & = -\frac {\arccos \left (\frac {x}{a}\right )}{x}+\frac {\text {arctanh}\left (\sqrt {1-\frac {x^2}{a^2}}\right )}{a} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(93\) vs. \(2(31)=62\).
Time = 0.09 (sec) , antiderivative size = 93, normalized size of antiderivative = 3.00 \[ \int \frac {\sec ^{-1}\left (\frac {a}{x}\right )}{x^2} \, dx=-\frac {\sec ^{-1}\left (\frac {a}{x}\right )}{x}+\frac {\sqrt {-1+\frac {a^2}{x^2}} x \left (-\log \left (1-\frac {a}{\sqrt {-1+\frac {a^2}{x^2}} x}\right )+\log \left (1+\frac {a}{\sqrt {-1+\frac {a^2}{x^2}} x}\right )\right )}{2 a^2 \sqrt {1-\frac {x^2}{a^2}}} \]
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Time = 0.06 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.97
method | result | size |
parts | \(-\frac {\operatorname {arcsec}\left (\frac {a}{x}\right )}{x}+\frac {\operatorname {arctanh}\left (\frac {1}{\sqrt {1-\frac {x^{2}}{a^{2}}}}\right )}{a}\) | \(30\) |
derivativedivides | \(-\frac {\frac {a \,\operatorname {arcsec}\left (\frac {a}{x}\right )}{x}-\ln \left (\frac {a}{x}+\frac {a \sqrt {1-\frac {x^{2}}{a^{2}}}}{x}\right )}{a}\) | \(44\) |
default | \(-\frac {\frac {a \,\operatorname {arcsec}\left (\frac {a}{x}\right )}{x}-\ln \left (\frac {a}{x}+\frac {a \sqrt {1-\frac {x^{2}}{a^{2}}}}{x}\right )}{a}\) | \(44\) |
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Leaf count of result is larger than twice the leaf count of optimal. 107 vs. \(2 (29) = 58\).
Time = 0.27 (sec) , antiderivative size = 107, normalized size of antiderivative = 3.45 \[ \int \frac {\sec ^{-1}\left (\frac {a}{x}\right )}{x^2} \, dx=-\frac {2 \, a x \arctan \left (-\frac {x^{2} \sqrt {\frac {a^{2} - x^{2}}{x^{2}}}}{a^{2} - x^{2}}\right ) - 2 \, {\left (a x - a\right )} \operatorname {arcsec}\left (\frac {a}{x}\right ) - x \log \left (x \sqrt {\frac {a^{2} - x^{2}}{x^{2}}} + a\right ) + x \log \left (x \sqrt {\frac {a^{2} - x^{2}}{x^{2}}} - a\right )}{2 \, a x} \]
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Result contains complex when optimal does not.
Time = 1.07 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.94 \[ \int \frac {\sec ^{-1}\left (\frac {a}{x}\right )}{x^2} \, dx=- \frac {\operatorname {asec}{\left (\frac {a}{x} \right )}}{x} - \frac {\begin {cases} - \operatorname {acosh}{\left (\frac {a}{x} \right )} & \text {for}\: \left |{\frac {a^{2}}{x^{2}}}\right | > 1 \\i \operatorname {asin}{\left (\frac {a}{x} \right )} & \text {otherwise} \end {cases}}{a} \]
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none
Time = 0.18 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.68 \[ \int \frac {\sec ^{-1}\left (\frac {a}{x}\right )}{x^2} \, dx=-\frac {\frac {2 \, a \operatorname {arcsec}\left (\frac {a}{x}\right )}{x} - \log \left (\sqrt {-\frac {x^{2}}{a^{2}} + 1} + 1\right ) + \log \left (-\sqrt {-\frac {x^{2}}{a^{2}} + 1} + 1\right )}{2 \, a} \]
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Leaf count of result is larger than twice the leaf count of optimal. 61 vs. \(2 (29) = 58\).
Time = 0.27 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.97 \[ \int \frac {\sec ^{-1}\left (\frac {a}{x}\right )}{x^2} \, dx=\frac {a {\left (\frac {\log \left ({\left | a + \sqrt {a^{2} - x^{2}} \right |}\right )}{a} - \frac {\log \left ({\left | -a + \sqrt {a^{2} - x^{2}} \right |}\right )}{a}\right )}}{2 \, {\left | a \right |}} - \frac {\arccos \left (\frac {x}{a}\right )}{x} \]
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Time = 0.71 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.94 \[ \int \frac {\sec ^{-1}\left (\frac {a}{x}\right )}{x^2} \, dx=\frac {\mathrm {atanh}\left (\frac {1}{\sqrt {1-\frac {x^2}{a^2}}}\right )}{a}-\frac {\mathrm {acos}\left (\frac {x}{a}\right )}{x} \]
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