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 = 0.0165922, antiderivative size = 34, normalized size of antiderivative = 1., 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 4833
Rule 5220
Rule 191
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*}
Mathematica [A] time = 0.021294, 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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Maple [A] time = 0.004, size = 39, normalized size = 1.2 \begin{align*} -{a}^{2} \left ( -{\frac{{x}^{2}}{2\,{a}^{2}}\arccos \left ({\frac{a}{x}} \right ) }+{\frac{x}{2\,a}\sqrt{1-{\frac{{a}^{2}}{{x}^{2}}}}} \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.44162, size = 38, normalized size = 1.12 \begin{align*} \frac{1}{2} \, x^{2} \arccos \left (\frac{a}{x}\right ) - \frac{1}{2} \, a x \sqrt{-\frac{a^{2}}{x^{2}} + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.56326, size = 73, normalized size = 2.15 \begin{align*} \frac{1}{2} \, x^{2} \arccos \left (\frac{a}{x}\right ) - \frac{1}{2} \, a x \sqrt{-\frac{a^{2} - x^{2}}{x^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 2.68407, size = 49, normalized size = 1.44 \begin{align*} - \frac{a \left (\begin{cases} a \sqrt{-1 + \frac{x^{2}}{a^{2}}} & \text{for}\: \frac{\left |{x^{2}}\right |}{\left |{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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.24789, size = 58, normalized size = 1.71 \begin{align*} \frac{1}{2} \, x^{2} \arccos \left (\frac{a}{x}\right ) + \frac{1}{2} \,{\left (\sqrt{-a^{2}} \mathrm{sgn}\left (x\right ) - \frac{\sqrt{-a^{2} + x^{2}}}{\mathrm{sgn}\left (x\right )}\right )} a \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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