Optimal. Leaf size=13 \[ -\frac {\sin \left (a+\frac {b}{x}\right )}{b} \]
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Rubi [A]
time = 0.01, antiderivative size = 13, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {3461, 2717}
\begin {gather*} -\frac {\sin \left (a+\frac {b}{x}\right )}{b} \end {gather*}
Antiderivative was successfully verified.
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Rule 2717
Rule 3461
Rubi steps
\begin {align*} \int \frac {\cos \left (a+\frac {b}{x}\right )}{x^2} \, dx &=-\text {Subst}\left (\int \cos (a+b x) \, dx,x,\frac {1}{x}\right )\\ &=-\frac {\sin \left (a+\frac {b}{x}\right )}{b}\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 13, normalized size = 1.00 \begin {gather*} -\frac {\sin \left (a+\frac {b}{x}\right )}{b} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.04, size = 14, normalized size = 1.08
method | result | size |
derivativedivides | \(-\frac {\sin \left (a +\frac {b}{x}\right )}{b}\) | \(14\) |
default | \(-\frac {\sin \left (a +\frac {b}{x}\right )}{b}\) | \(14\) |
risch | \(-\frac {\sin \left (\frac {a x +b}{x}\right )}{b}\) | \(16\) |
norman | \(-\frac {2 \tan \left (\frac {a}{2}+\frac {b}{2 x}\right )}{b \left (1+\tan ^{2}\left (\frac {a}{2}+\frac {b}{2 x}\right )\right )}\) | \(34\) |
meijerg | \(-\frac {\cos \left (a \right ) \sin \left (\frac {b}{x}\right )}{b}+\frac {\sqrt {\pi }\, \sin \left (a \right ) \left (\frac {1}{\sqrt {\pi }}-\frac {\cos \left (\frac {b}{x}\right )}{\sqrt {\pi }}\right )}{b}\) | \(39\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 13, normalized size = 1.00 \begin {gather*} -\frac {\sin \left (a + \frac {b}{x}\right )}{b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.39, size = 15, normalized size = 1.15 \begin {gather*} -\frac {\sin \left (\frac {a x + b}{x}\right )}{b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.34, size = 15, normalized size = 1.15 \begin {gather*} \begin {cases} - \frac {\sin {\left (a + \frac {b}{x} \right )}}{b} & \text {for}\: b \neq 0 \\- \frac {\cos {\left (a \right )}}{x} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.41, size = 15, normalized size = 1.15 \begin {gather*} -\frac {\sin \left (\frac {a x + b}{x}\right )}{b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.26, size = 13, normalized size = 1.00 \begin {gather*} -\frac {\sin \left (a+\frac {b}{x}\right )}{b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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