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 = 0.03, antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, 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 63
Rule 208
Rule 266
Rule 4628
Rule 5264
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*}
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Mathematica [B] time = 0.14, size = 93, normalized size = 3.00 \[ \frac {x \sqrt {\frac {a^2}{x^2}-1} \left (\log \left (\frac {a}{x \sqrt {\frac {a^2}{x^2}-1}}+1\right )-\log \left (1-\frac {a}{x \sqrt {\frac {a^2}{x^2}-1}}\right )\right )}{2 a^2 \sqrt {1-\frac {x^2}{a^2}}}-\frac {\sec ^{-1}\left (\frac {a}{x}\right )}{x} \]
Antiderivative was successfully verified.
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fricas [B] time = 2.12, size = 107, normalized size = 3.45 \[ -\frac {2 \, a x \arctan \left (-\frac {x^{2} \sqrt {\frac {a^{2} - x^{2}}{x^{2}}}}{a^{2} - x^{2}}\right ) - 2 \, {\left (a x - a\right )} \operatorname {arcsec}\left (\frac {a}{x}\right ) - x \log \left (x \sqrt {\frac {a^{2} - x^{2}}{x^{2}}} + a\right ) + x \log \left (x \sqrt {\frac {a^{2} - x^{2}}{x^{2}}} - a\right )}{2 \, a x} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.14, size = 61, normalized size = 1.97 \[ \frac {a {\left (\frac {\log \left ({\left | a + \sqrt {a^{2} - x^{2}} \right |}\right )}{a} - \frac {\log \left ({\left | -a + \sqrt {a^{2} - x^{2}} \right |}\right )}{a}\right )}}{2 \, {\left | a \right |}} - \frac {\arccos \left (\frac {x}{a}\right )}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 41, normalized size = 1.32 \[ -\frac {\mathrm {arcsec}\left (\frac {a}{x}\right )}{x}+\frac {\ln \left (\frac {a}{x}+\frac {a \sqrt {1-\frac {x^{2}}{a^{2}}}}{x}\right )}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.34, size = 52, normalized size = 1.68 \[ -\frac {\frac {2 \, a \operatorname {arcsec}\left (\frac {a}{x}\right )}{x} - \log \left (\sqrt {-\frac {x^{2}}{a^{2}} + 1} + 1\right ) + \log \left (-\sqrt {-\frac {x^{2}}{a^{2}} + 1} + 1\right )}{2 \, a} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.60, size = 29, normalized size = 0.94 \[ \frac {\mathrm {atanh}\left (\frac {1}{\sqrt {1-\frac {x^2}{a^2}}}\right )}{a}-\frac {\mathrm {acos}\left (\frac {x}{a}\right )}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 2.17, size = 29, normalized size = 0.94 \[ - \frac {\operatorname {asec}{\left (\frac {a}{x} \right )}}{x} - \frac {\begin {cases} - \operatorname {acosh}{\left (\frac {a}{x} \right )} & \text {for}\: \left |{\frac {a^{2}}{x^{2}}}\right | > 1 \\i \operatorname {asin}{\left (\frac {a}{x} \right )} & \text {otherwise} \end {cases}}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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