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 = 0.0249832, antiderivative size = 30, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, 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 3380
Rule 3296
Rule 2638
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*}
Mathematica [A] time = 0.047531, 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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Maple [A] time = 0.032, size = 42, normalized size = 1.4 \begin{align*} -{\frac{1}{{b}^{2}} \left ( \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 ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [C] time = 1.29148, size = 69, normalized size = 2.3 \begin{align*} -\frac{{\left (\Gamma \left (2, \frac{i \, b}{x}\right ) + \Gamma \left (2, -\frac{i \, b}{x}\right )\right )} \cos \left (a\right ) -{\left (i \, \Gamma \left (2, \frac{i \, b}{x}\right ) - i \, \Gamma \left (2, -\frac{i \, b}{x}\right )\right )} \sin \left (a\right )}{2 \, b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.57816, size = 70, normalized size = 2.33 \begin{align*} -\frac{x \cos \left (\frac{a x + b}{x}\right ) + b \sin \left (\frac{a x + b}{x}\right )}{b^{2} x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 3.43995, size = 31, normalized size = 1.03 \begin{align*} \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{\left (a \right )}}{2 x^{2}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\cos \left (a + \frac{b}{x}\right )}{x^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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