Optimal. Leaf size=31 \[ \frac{\tanh ^{-1}\left (\sqrt{1-\frac{x^2}{a^2}}\right )}{a}-\frac{\cos ^{-1}\left (\frac{x}{a}\right )}{x} \]
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Rubi [A] time = 0.0296163, antiderivative size = 31, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {5264, 4628, 266, 63, 208} \[ \frac{\tanh ^{-1}\left (\sqrt{1-\frac{x^2}{a^2}}\right )}{a}-\frac{\cos ^{-1}\left (\frac{x}{a}\right )}{x} \]
Antiderivative was successfully verified.
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Rule 5264
Rule 4628
Rule 266
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{\sec ^{-1}\left (\frac{a}{x}\right )}{x^2} \, dx &=\int \frac{\cos ^{-1}\left (\frac{x}{a}\right )}{x^2} \, dx\\ &=-\frac{\cos ^{-1}\left (\frac{x}{a}\right )}{x}-\frac{\int \frac{1}{x \sqrt{1-\frac{x^2}{a^2}}} \, dx}{a}\\ &=-\frac{\cos ^{-1}\left (\frac{x}{a}\right )}{x}-\frac{\operatorname{Subst}\left (\int \frac{1}{x \sqrt{1-\frac{x}{a^2}}} \, dx,x,x^2\right )}{2 a}\\ &=-\frac{\cos ^{-1}\left (\frac{x}{a}\right )}{x}+a \operatorname{Subst}\left (\int \frac{1}{a^2-a^2 x^2} \, dx,x,\sqrt{1-\frac{x^2}{a^2}}\right )\\ &=-\frac{\cos ^{-1}\left (\frac{x}{a}\right )}{x}+\frac{\tanh ^{-1}\left (\sqrt{1-\frac{x^2}{a^2}}\right )}{a}\\ \end{align*}
Mathematica [B] time = 0.13649, size = 93, normalized size = 3. \[ \frac{x \sqrt{\frac{a^2}{x^2}-1} \left (\log \left (\frac{a}{x \sqrt{\frac{a^2}{x^2}-1}}+1\right )-\log \left (1-\frac{a}{x \sqrt{\frac{a^2}{x^2}-1}}\right )\right )}{2 a^2 \sqrt{1-\frac{x^2}{a^2}}}-\frac{\sec ^{-1}\left (\frac{a}{x}\right )}{x} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.145, size = 41, normalized size = 1.3 \begin{align*} -{\frac{1}{x}{\rm arcsec} \left ({\frac{a}{x}}\right )}+{\frac{1}{a}\ln \left ({\frac{a}{x}}+{\frac{a}{x}\sqrt{1-{\frac{{x}^{2}}{{a}^{2}}}}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.961596, size = 70, normalized size = 2.26 \begin{align*} -\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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.41124, size = 225, normalized size = 7.26 \begin{align*} -\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} \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^{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.108, size = 82, normalized size = 2.65 \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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