Optimal. Leaf size=51 \[ \frac {\sqrt {1-\frac {a^2}{x^2}}}{4 a x}-\frac {\csc ^{-1}\left (\frac {x}{a}\right )}{4 a^2}-\frac {\sec ^{-1}\left (\frac {x}{a}\right )}{2 x^2} \]
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Rubi [A] time = 0.03, antiderivative size = 51, 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 = {4833, 5220, 335, 321, 216} \[ \frac {\sqrt {1-\frac {a^2}{x^2}}}{4 a x}-\frac {\csc ^{-1}\left (\frac {x}{a}\right )}{4 a^2}-\frac {\sec ^{-1}\left (\frac {x}{a}\right )}{2 x^2} \]
Antiderivative was successfully verified.
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Rule 216
Rule 321
Rule 335
Rule 4833
Rule 5220
Rubi steps
\begin {align*} \int \frac {\cos ^{-1}\left (\frac {a}{x}\right )}{x^3} \, dx &=\int \frac {\sec ^{-1}\left (\frac {x}{a}\right )}{x^3} \, dx\\ &=-\frac {\sec ^{-1}\left (\frac {x}{a}\right )}{2 x^2}+\frac {1}{2} a \int \frac {1}{\sqrt {1-\frac {a^2}{x^2}} x^4} \, dx\\ &=-\frac {\sec ^{-1}\left (\frac {x}{a}\right )}{2 x^2}-\frac {1}{2} a \operatorname {Subst}\left (\int \frac {x^2}{\sqrt {1-a^2 x^2}} \, dx,x,\frac {1}{x}\right )\\ &=\frac {\sqrt {1-\frac {a^2}{x^2}}}{4 a x}-\frac {\sec ^{-1}\left (\frac {x}{a}\right )}{2 x^2}-\frac {\operatorname {Subst}\left (\int \frac {1}{\sqrt {1-a^2 x^2}} \, dx,x,\frac {1}{x}\right )}{4 a}\\ &=\frac {\sqrt {1-\frac {a^2}{x^2}}}{4 a x}-\frac {\csc ^{-1}\left (\frac {x}{a}\right )}{4 a^2}-\frac {\sec ^{-1}\left (\frac {x}{a}\right )}{2 x^2}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 50, normalized size = 0.98 \[ \frac {a x \sqrt {1-\frac {a^2}{x^2}}-2 a^2 \cos ^{-1}\left (\frac {a}{x}\right )-x^2 \sin ^{-1}\left (\frac {a}{x}\right )}{4 a^2 x^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 47, normalized size = 0.92 \[ \frac {a x \sqrt {-\frac {a^{2} - x^{2}}{x^{2}}} - {\left (2 \, a^{2} - x^{2}\right )} \arccos \left (\frac {a}{x}\right )}{4 \, a^{2} x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.81, size = 44, normalized size = 0.86 \[ \frac {\frac {\arccos \left (\frac {a}{x}\right )}{a} - \frac {2 \, a \arccos \left (\frac {a}{x}\right )}{x^{2}} + \frac {\sqrt {-\frac {a^{2}}{x^{2}} + 1}}{x}}{4 \, a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 47, normalized size = 0.92 \[ -\frac {\frac {\arccos \left (\frac {a}{x}\right ) a^{2}}{2 x^{2}}-\frac {a \sqrt {1-\frac {a^{2}}{x^{2}}}}{4 x}+\frac {\arcsin \left (\frac {a}{x}\right )}{4}}{a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.41, size = 77, normalized size = 1.51 \[ -\frac {1}{4} \, a {\left (\frac {x \sqrt {-\frac {a^{2}}{x^{2}} + 1}}{a^{2} x^{2} {\left (\frac {a^{2}}{x^{2}} - 1\right )} - a^{4}} - \frac {\arctan \left (\frac {x \sqrt {-\frac {a^{2}}{x^{2}} + 1}}{a}\right )}{a^{3}}\right )} - \frac {\arccos \left (\frac {a}{x}\right )}{2 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.28, size = 42, normalized size = 0.82 \[ \frac {\sqrt {1-\frac {a^2}{x^2}}}{4\,a\,x}-\frac {\mathrm {acos}\left (\frac {a}{x}\right )\,\left (\frac {2\,a^2}{x^2}-1\right )}{4\,a^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 3.10, size = 100, normalized size = 1.96 \[ \frac {a \left (\begin {cases} \frac {i \sqrt {\frac {a^{2}}{x^{2}} - 1}}{2 a^{2} x} + \frac {i \operatorname {acosh}{\left (\frac {a}{x} \right )}}{2 a^{3}} & \text {for}\: \left |{\frac {a^{2}}{x^{2}}}\right | > 1 \\- \frac {1}{2 x^{3} \sqrt {- \frac {a^{2}}{x^{2}} + 1}} + \frac {1}{2 a^{2} x \sqrt {- \frac {a^{2}}{x^{2}} + 1}} - \frac {\operatorname {asin}{\left (\frac {a}{x} \right )}}{2 a^{3}} & \text {otherwise} \end {cases}\right )}{2} - \frac {\operatorname {acos}{\left (\frac {a}{x} \right )}}{2 x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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