Optimal. Leaf size=35 \[ \frac{\text{CosIntegral}(2 b x)}{2 b}-\frac{\text{CosIntegral}(b x) \cos (b x)}{b}+\frac{\log (x)}{2 b} \]
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Rubi [A] time = 0.0612325, antiderivative size = 35, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.444, Rules used = {6518, 12, 3312, 3302} \[ \frac{\text{CosIntegral}(2 b x)}{2 b}-\frac{\text{CosIntegral}(b x) \cos (b x)}{b}+\frac{\log (x)}{2 b} \]
Antiderivative was successfully verified.
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Rule 6518
Rule 12
Rule 3312
Rule 3302
Rubi steps
\begin{align*} \int \text{Ci}(b x) \sin (b x) \, dx &=-\frac{\cos (b x) \text{Ci}(b x)}{b}+\int \frac{\cos ^2(b x)}{b x} \, dx\\ &=-\frac{\cos (b x) \text{Ci}(b x)}{b}+\frac{\int \frac{\cos ^2(b x)}{x} \, dx}{b}\\ &=-\frac{\cos (b x) \text{Ci}(b x)}{b}+\frac{\int \left (\frac{1}{2 x}+\frac{\cos (2 b x)}{2 x}\right ) \, dx}{b}\\ &=-\frac{\cos (b x) \text{Ci}(b x)}{b}+\frac{\log (x)}{2 b}+\frac{\int \frac{\cos (2 b x)}{x} \, dx}{2 b}\\ &=-\frac{\cos (b x) \text{Ci}(b x)}{b}+\frac{\text{Ci}(2 b x)}{2 b}+\frac{\log (x)}{2 b}\\ \end{align*}
Mathematica [A] time = 0.0132629, size = 37, normalized size = 1.06 \[ \frac{\text{CosIntegral}(2 b x)}{2 b}-\frac{\text{CosIntegral}(b x) \cos (b x)}{b}+\frac{\log (b x)}{2 b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.052, size = 34, normalized size = 1. \begin{align*} -{\frac{{\it Ci} \left ( bx \right ) \cos \left ( bx \right ) }{b}}+{\frac{\ln \left ( bx \right ) }{2\,b}}+{\frac{{\it Ci} \left ( 2\,bx \right ) }{2\,b}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\rm Ci}\left (b x\right ) \sin \left (b x\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\operatorname{Ci}\left (b x\right ) \sin \left (b x\right ), x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sin{\left (b x \right )} \operatorname{Ci}{\left (b x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.16776, size = 46, normalized size = 1.31 \begin{align*} -\frac{\cos \left (b x\right ) \operatorname{Ci}\left (b x\right )}{b} + \frac{\operatorname{Ci}\left (2 \, b x\right ) + \operatorname{Ci}\left (-2 \, b x\right ) + 2 \, \log \left (x\right )}{4 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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