Integrand size = 8, antiderivative size = 47 \[ \int x \sec ^{-1}\left (\frac {a}{x}\right ) \, dx=-\frac {1}{4} a x \sqrt {1-\frac {x^2}{a^2}}+\frac {1}{2} x^2 \arccos \left (\frac {x}{a}\right )+\frac {1}{4} a^2 \arcsin \left (\frac {x}{a}\right ) \]
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Time = 0.02 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5372, 4724, 327, 222} \[ \int x \sec ^{-1}\left (\frac {a}{x}\right ) \, dx=\frac {1}{4} a^2 \arcsin \left (\frac {x}{a}\right )-\frac {1}{4} a x \sqrt {1-\frac {x^2}{a^2}}+\frac {1}{2} x^2 \arccos \left (\frac {x}{a}\right ) \]
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Rule 222
Rule 327
Rule 4724
Rule 5372
Rubi steps \begin{align*} \text {integral}& = \int x \arccos \left (\frac {x}{a}\right ) \, dx \\ & = \frac {1}{2} x^2 \arccos \left (\frac {x}{a}\right )+\frac {\int \frac {x^2}{\sqrt {1-\frac {x^2}{a^2}}} \, dx}{2 a} \\ & = -\frac {1}{4} a x \sqrt {1-\frac {x^2}{a^2}}+\frac {1}{2} x^2 \arccos \left (\frac {x}{a}\right )+\frac {1}{4} a \int \frac {1}{\sqrt {1-\frac {x^2}{a^2}}} \, dx \\ & = -\frac {1}{4} a x \sqrt {1-\frac {x^2}{a^2}}+\frac {1}{2} x^2 \arccos \left (\frac {x}{a}\right )+\frac {1}{4} a^2 \arcsin \left (\frac {x}{a}\right ) \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.94 \[ \int x \sec ^{-1}\left (\frac {a}{x}\right ) \, dx=\frac {1}{4} \left (-a x \sqrt {1-\frac {x^2}{a^2}}+2 x^2 \sec ^{-1}\left (\frac {a}{x}\right )+a^2 \arcsin \left (\frac {x}{a}\right )\right ) \]
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Time = 1.37 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.43
method | result | size |
parts | \(\frac {x^{2} \operatorname {arcsec}\left (\frac {a}{x}\right )}{2}+\frac {-\frac {x \,a^{2} \sqrt {1-\frac {x^{2}}{a^{2}}}}{2}+\frac {a^{2} \arctan \left (\frac {\sqrt {\frac {1}{a^{2}}}\, x}{\sqrt {1-\frac {x^{2}}{a^{2}}}}\right )}{2 \sqrt {\frac {1}{a^{2}}}}}{2 a}\) | \(67\) |
derivativedivides | \(-a^{2} \left (-\frac {x^{2} \operatorname {arcsec}\left (\frac {a}{x}\right )}{2 a^{2}}-\frac {\sqrt {\frac {a^{2}}{x^{2}}-1}\, \left (\frac {\arctan \left (\frac {1}{\sqrt {\frac {a^{2}}{x^{2}}-1}}\right ) a^{2}}{x^{2}}-\sqrt {\frac {a^{2}}{x^{2}}-1}\right ) x^{3}}{4 \sqrt {\frac {\left (\frac {a^{2}}{x^{2}}-1\right ) x^{2}}{a^{2}}}\, a^{3}}\right )\) | \(91\) |
default | \(-a^{2} \left (-\frac {x^{2} \operatorname {arcsec}\left (\frac {a}{x}\right )}{2 a^{2}}-\frac {\sqrt {\frac {a^{2}}{x^{2}}-1}\, \left (\frac {\arctan \left (\frac {1}{\sqrt {\frac {a^{2}}{x^{2}}-1}}\right ) a^{2}}{x^{2}}-\sqrt {\frac {a^{2}}{x^{2}}-1}\right ) x^{3}}{4 \sqrt {\frac {\left (\frac {a^{2}}{x^{2}}-1\right ) x^{2}}{a^{2}}}\, a^{3}}\right )\) | \(91\) |
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Time = 0.26 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.81 \[ \int x \sec ^{-1}\left (\frac {a}{x}\right ) \, dx=-\frac {1}{4} \, x^{2} \sqrt {\frac {a^{2} - x^{2}}{x^{2}}} - \frac {1}{4} \, {\left (a^{2} - 2 \, x^{2}\right )} \operatorname {arcsec}\left (\frac {a}{x}\right ) \]
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Time = 0.14 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.87 \[ \int x \sec ^{-1}\left (\frac {a}{x}\right ) \, dx=\begin {cases} - \frac {a^{2} \operatorname {asec}{\left (\frac {a}{x} \right )}}{4} - \frac {a x \sqrt {1 - \frac {x^{2}}{a^{2}}}}{4} + \frac {x^{2} \operatorname {asec}{\left (\frac {a}{x} \right )}}{2} & \text {for}\: a \neq 0 \\\tilde {\infty } x^{2} & \text {otherwise} \end {cases} \]
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Time = 0.26 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.98 \[ \int x \sec ^{-1}\left (\frac {a}{x}\right ) \, dx=\frac {1}{2} \, x^{2} \operatorname {arcsec}\left (\frac {a}{x}\right ) + \frac {a^{3} \arcsin \left (\frac {x}{a}\right ) - a^{2} x \sqrt {-\frac {x^{2}}{a^{2}} + 1}}{4 \, a} \]
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Time = 0.26 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.83 \[ \int x \sec ^{-1}\left (\frac {a}{x}\right ) \, dx=-\frac {1}{4} \, a^{2} \arccos \left (\frac {x}{a}\right ) + \frac {1}{2} \, x^{2} \arccos \left (\frac {x}{a}\right ) - \frac {1}{4} \, a x \sqrt {-\frac {x^{2}}{a^{2}} + 1} \]
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Time = 0.73 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.81 \[ \int x \sec ^{-1}\left (\frac {a}{x}\right ) \, dx=\frac {a^2\,\mathrm {acos}\left (\frac {x}{a}\right )\,\left (\frac {2\,x^2}{a^2}-1\right )}{4}-\frac {a\,x\,\sqrt {1-\frac {x^2}{a^2}}}{4} \]
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