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 = 0.02, antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, 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 264
Rule 4628
Rule 5264
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*}
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Mathematica [A] time = 0.02, 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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fricas [A] time = 0.65, size = 39, normalized size = 1.03 \[ -\frac {a^{2} \operatorname {arcsec}\left (\frac {a}{x}\right ) - x^{2} \sqrt {\frac {a^{2} - x^{2}}{x^{2}}}}{2 \, a^{2} x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.18, size = 61, normalized size = 1.61 \[ \frac {a {\left (\frac {a + \sqrt {a^{2} - x^{2}}}{a^{2} x} - \frac {x}{{\left (a + \sqrt {a^{2} - x^{2}}\right )} a^{2}}\right )}}{4 \, {\left | a \right |}} - \frac {\arccos \left (\frac {x}{a}\right )}{2 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 54, normalized size = 1.42 \[ -\frac {\frac {a^{2} \mathrm {arcsec}\left (\frac {a}{x}\right )}{2 x^{2}}-\frac {x \left (-1+\frac {a^{2}}{x^{2}}\right )}{2 \sqrt {\frac {\left (-1+\frac {a^{2}}{x^{2}}\right ) x^{2}}{a^{2}}}\, a}}{a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.42, size = 32, normalized size = 0.84 \[ -\frac {\operatorname {arcsec}\left (\frac {a}{x}\right )}{2 \, x^{2}} + \frac {\sqrt {-\frac {x^{2}}{a^{2}} + 1}}{2 \, a x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.03 \[ \int \frac {\mathrm {acos}\left (\frac {x}{a}\right )}{x^3} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 1.25, size = 53, normalized size = 1.39 \[ - \frac {\operatorname {asec}{\left (\frac {a}{x} \right )}}{2 x^{2}} - \frac {\begin {cases} - \frac {\sqrt {\frac {a^{2}}{x^{2}} - 1}}{a} & \text {for}\: \left |{\frac {a^{2}}{x^{2}}}\right | > 1 \\- \frac {i \sqrt {- \frac {a^{2}}{x^{2}} + 1}}{a} & \text {otherwise} \end {cases}}{2 a} \]
Verification of antiderivative is not currently implemented for this CAS.
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