Optimal. Leaf size=34 \[ \frac {1}{2} x^2 \sec ^{-1}\left (\frac {x}{a}\right )-\frac {1}{2} a x \sqrt {1-\frac {a^2}{x^2}} \]
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Rubi [A] time = 0.02, antiderivative size = 34, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {4833, 5220, 191} \[ \frac {1}{2} x^2 \sec ^{-1}\left (\frac {x}{a}\right )-\frac {1}{2} a x \sqrt {1-\frac {a^2}{x^2}} \]
Antiderivative was successfully verified.
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Rule 191
Rule 4833
Rule 5220
Rubi steps
\begin {align*} \int x \cos ^{-1}\left (\frac {a}{x}\right ) \, dx &=\int x \sec ^{-1}\left (\frac {x}{a}\right ) \, dx\\ &=\frac {1}{2} x^2 \sec ^{-1}\left (\frac {x}{a}\right )-\frac {1}{2} a \int \frac {1}{\sqrt {1-\frac {a^2}{x^2}}} \, dx\\ &=-\frac {1}{2} a \sqrt {1-\frac {a^2}{x^2}} x+\frac {1}{2} x^2 \sec ^{-1}\left (\frac {x}{a}\right )\\ \end {align*}
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Mathematica [A] time = 0.02, size = 33, normalized size = 0.97 \[ \frac {1}{2} \left (x^2 \cos ^{-1}\left (\frac {a}{x}\right )-a x \sqrt {1-\frac {a^2}{x^2}}\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 32, normalized size = 0.94 \[ \frac {1}{2} \, x^{2} \arccos \left (\frac {a}{x}\right ) - \frac {1}{2} \, a x \sqrt {-\frac {a^{2} - x^{2}}{x^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.21, size = 64, normalized size = 1.88 \[ -\frac {a^{3} {\left (\frac {x {\left (\sqrt {-\frac {a^{2}}{x^{2}} + 1} - 1\right )}}{a} - \frac {a}{x {\left (\sqrt {-\frac {a^{2}}{x^{2}} + 1} - 1\right )}}\right )} - 2 \, a x^{2} \arccos \left (\frac {a}{x}\right )}{4 \, a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 39, normalized size = 1.15 \[ -a^{2} \left (-\frac {x^{2} \arccos \left (\frac {a}{x}\right )}{2 a^{2}}+\frac {\sqrt {1-\frac {a^{2}}{x^{2}}}\, x}{2 a}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.40, size = 28, normalized size = 0.82 \[ \frac {1}{2} \, x^{2} \arccos \left (\frac {a}{x}\right ) - \frac {1}{2} \, a x \sqrt {-\frac {a^{2}}{x^{2}} + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.24, size = 28, normalized size = 0.82 \[ \frac {x^2\,\mathrm {acos}\left (\frac {a}{x}\right )}{2}-\frac {a\,x\,\sqrt {1-\frac {a^2}{x^2}}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.54, size = 48, normalized size = 1.41 \[ - \frac {a \left (\begin {cases} a \sqrt {-1 + \frac {x^{2}}{a^{2}}} & \text {for}\: \left |{\frac {x^{2}}{a^{2}}}\right | > 1 \\i a \sqrt {1 - \frac {x^{2}}{a^{2}}} & \text {otherwise} \end {cases}\right )}{2} + \frac {x^{2} \operatorname {acos}{\left (\frac {a}{x} \right )}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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