Optimal. Leaf size=21 \[ \sin (a) \left (-\text{CosIntegral}\left (\frac{b}{x}\right )\right )-\cos (a) \text{Si}\left (\frac{b}{x}\right ) \]
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Rubi [A] time = 0.0282507, antiderivative size = 21, 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 = {3377, 3376, 3375} \[ \sin (a) \left (-\text{CosIntegral}\left (\frac{b}{x}\right )\right )-\cos (a) \text{Si}\left (\frac{b}{x}\right ) \]
Antiderivative was successfully verified.
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Rule 3377
Rule 3376
Rule 3375
Rubi steps
\begin{align*} \int \frac{\sin \left (a+\frac{b}{x}\right )}{x} \, dx &=\cos (a) \int \frac{\sin \left (\frac{b}{x}\right )}{x} \, dx+\sin (a) \int \frac{\cos \left (\frac{b}{x}\right )}{x} \, dx\\ &=-\text{Ci}\left (\frac{b}{x}\right ) \sin (a)-\cos (a) \text{Si}\left (\frac{b}{x}\right )\\ \end{align*}
Mathematica [A] time = 0.0469815, size = 21, normalized size = 1. \[ \sin (a) \left (-\text{CosIntegral}\left (\frac{b}{x}\right )\right )-\cos (a) \text{Si}\left (\frac{b}{x}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.009, size = 22, normalized size = 1.1 \begin{align*} -\cos \left ( a \right ){\it Si} \left ({\frac{b}{x}} \right ) -{\it Ci} \left ({\frac{b}{x}} \right ) \sin \left ( a \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [C] time = 1.14411, size = 58, normalized size = 2.76 \begin{align*} \frac{1}{2} \,{\left (i \,{\rm Ei}\left (\frac{i \, b}{x}\right ) - i \,{\rm Ei}\left (-\frac{i \, b}{x}\right )\right )} \cos \left (a\right ) - \frac{1}{2} \,{\left ({\rm Ei}\left (\frac{i \, b}{x}\right ) +{\rm Ei}\left (-\frac{i \, b}{x}\right )\right )} \sin \left (a\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.92223, size = 109, normalized size = 5.19 \begin{align*} -\frac{1}{2} \,{\left (\operatorname{Ci}\left (\frac{b}{x}\right ) + \operatorname{Ci}\left (-\frac{b}{x}\right )\right )} \sin \left (a\right ) - \cos \left (a\right ) \operatorname{Si}\left (\frac{b}{x}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.33868, size = 17, normalized size = 0.81 \begin{align*} - \sin{\left (a \right )} \operatorname{Ci}{\left (\frac{b}{x} \right )} - \cos{\left (a \right )} \operatorname{Si}{\left (\frac{b}{x} \right )} \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{\sin \left (a + \frac{b}{x}\right )}{x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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