Optimal. Leaf size=30 \[ -\frac {\cos \left (a+\frac {b}{x}\right )}{b^2}-\frac {\sin \left (a+\frac {b}{x}\right )}{b x} \]
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Rubi [A] time = 0.02, antiderivative size = 30, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3380, 3296, 2638} \[ -\frac {\cos \left (a+\frac {b}{x}\right )}{b^2}-\frac {\sin \left (a+\frac {b}{x}\right )}{b x} \]
Antiderivative was successfully verified.
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Rule 2638
Rule 3296
Rule 3380
Rubi steps
\begin {align*} \int \frac {\cos \left (a+\frac {b}{x}\right )}{x^3} \, dx &=-\operatorname {Subst}\left (\int x \cos (a+b x) \, dx,x,\frac {1}{x}\right )\\ &=-\frac {\sin \left (a+\frac {b}{x}\right )}{b x}+\frac {\operatorname {Subst}\left (\int \sin (a+b x) \, dx,x,\frac {1}{x}\right )}{b}\\ &=-\frac {\cos \left (a+\frac {b}{x}\right )}{b^2}-\frac {\sin \left (a+\frac {b}{x}\right )}{b x}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 29, normalized size = 0.97 \[ -\frac {b \sin \left (a+\frac {b}{x}\right )+x \cos \left (a+\frac {b}{x}\right )}{b^2 x} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.75, size = 33, normalized size = 1.10 \[ -\frac {x \cos \left (\frac {a x + b}{x}\right ) + b \sin \left (\frac {a x + b}{x}\right )}{b^{2} x} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.39, size = 49, normalized size = 1.63 \[ \frac {a \sin \left (\frac {a x + b}{x}\right ) - \frac {{\left (a x + b\right )} \sin \left (\frac {a x + b}{x}\right )}{x} - \cos \left (\frac {a x + b}{x}\right )}{b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 42, normalized size = 1.40 \[ -\frac {\cos \left (a +\frac {b}{x}\right )+\left (a +\frac {b}{x}\right ) \sin \left (a +\frac {b}{x}\right )-a \sin \left (a +\frac {b}{x}\right )}{b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [C] time = 1.69, size = 51, normalized size = 1.70 \[ -\frac {{\left (\Gamma \left (2, \frac {i \, b}{x}\right ) + \Gamma \left (2, -\frac {i \, b}{x}\right )\right )} \cos \relax (a) - {\left (i \, \Gamma \left (2, \frac {i \, b}{x}\right ) - i \, \Gamma \left (2, -\frac {i \, b}{x}\right )\right )} \sin \relax (a)}{2 \, b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.28, size = 30, normalized size = 1.00 \[ -\frac {\cos \left (a+\frac {b}{x}\right )}{b^2}-\frac {\sin \left (a+\frac {b}{x}\right )}{b\,x} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.80, size = 31, normalized size = 1.03 \[ \begin {cases} - \frac {\sin {\left (a + \frac {b}{x} \right )}}{b x} - \frac {\cos {\left (a + \frac {b}{x} \right )}}{b^{2}} & \text {for}\: b \neq 0 \\- \frac {\cos {\relax (a )}}{2 x^{2}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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