Optimal. Leaf size=38 \[ \frac{\sqrt{1-\frac{x^2}{a^2}}}{2 a x}-\frac{\cos ^{-1}\left (\frac{x}{a}\right )}{2 x^2} \]
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Rubi [A] time = 0.0213249, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {5264, 4628, 264} \[ \frac{\sqrt{1-\frac{x^2}{a^2}}}{2 a x}-\frac{\cos ^{-1}\left (\frac{x}{a}\right )}{2 x^2} \]
Antiderivative was successfully verified.
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Rule 5264
Rule 4628
Rule 264
Rubi steps
\begin{align*} \int \frac{\sec ^{-1}\left (\frac{a}{x}\right )}{x^3} \, dx &=\int \frac{\cos ^{-1}\left (\frac{x}{a}\right )}{x^3} \, dx\\ &=-\frac{\cos ^{-1}\left (\frac{x}{a}\right )}{2 x^2}-\frac{\int \frac{1}{x^2 \sqrt{1-\frac{x^2}{a^2}}} \, dx}{2 a}\\ &=\frac{\sqrt{1-\frac{x^2}{a^2}}}{2 a x}-\frac{\cos ^{-1}\left (\frac{x}{a}\right )}{2 x^2}\\ \end{align*}
Mathematica [A] time = 0.0208916, size = 36, normalized size = 0.95 \[ \frac{x \sqrt{1-\frac{x^2}{a^2}}-a \sec ^{-1}\left (\frac{a}{x}\right )}{2 a x^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.172, size = 54, normalized size = 1.4 \begin{align*} -{\frac{1}{{a}^{2}} \left ({\frac{{a}^{2}}{2\,{x}^{2}}{\rm arcsec} \left ({\frac{a}{x}}\right )}-{\frac{x}{2\,a} \left ( -1+{\frac{{a}^{2}}{{x}^{2}}} \right ){\frac{1}{\sqrt{{\frac{{x}^{2}}{{a}^{2}} \left ( -1+{\frac{{a}^{2}}{{x}^{2}}} \right ) }}}}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.28492, size = 84, normalized size = 2.21 \begin{align*} -\frac{a^{2} \operatorname{arcsec}\left (\frac{a}{x}\right ) - x^{2} \sqrt{\frac{a^{2} - x^{2}}{x^{2}}}}{2 \, a^{2} x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{asec}{\left (\frac{a}{x} \right )}}{x^{3}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.11039, size = 82, normalized size = 2.16 \begin{align*} \frac{a{\left (\frac{a + \sqrt{a^{2} - x^{2}}}{a^{2} x} - \frac{x}{{\left (a + \sqrt{a^{2} - x^{2}}\right )} a^{2}}\right )}}{4 \,{\left | a \right |}} - \frac{\arccos \left (\frac{x}{a}\right )}{2 \, x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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