Optimal. Leaf size=47 \[ \frac{\text{CosIntegral}(2 a+2 b x)}{2 b}-\frac{\text{CosIntegral}(a+b x) \cos (a+b x)}{b}+\frac{\log (a+b x)}{2 b} \]
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Rubi [A] time = 0.076047, antiderivative size = 47, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {6518, 3312, 3302} \[ \frac{\text{CosIntegral}(2 a+2 b x)}{2 b}-\frac{\text{CosIntegral}(a+b x) \cos (a+b x)}{b}+\frac{\log (a+b x)}{2 b} \]
Antiderivative was successfully verified.
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Rule 6518
Rule 3312
Rule 3302
Rubi steps
\begin{align*} \int \text{Ci}(a+b x) \sin (a+b x) \, dx &=-\frac{\cos (a+b x) \text{Ci}(a+b x)}{b}+\int \frac{\cos ^2(a+b x)}{a+b x} \, dx\\ &=-\frac{\cos (a+b x) \text{Ci}(a+b x)}{b}+\int \left (\frac{1}{2 (a+b x)}+\frac{\cos (2 a+2 b x)}{2 (a+b x)}\right ) \, dx\\ &=-\frac{\cos (a+b x) \text{Ci}(a+b x)}{b}+\frac{\log (a+b x)}{2 b}+\frac{1}{2} \int \frac{\cos (2 a+2 b x)}{a+b x} \, dx\\ &=-\frac{\cos (a+b x) \text{Ci}(a+b x)}{b}+\frac{\text{Ci}(2 a+2 b x)}{2 b}+\frac{\log (a+b x)}{2 b}\\ \end{align*}
Mathematica [A] time = 0.0241664, size = 46, normalized size = 0.98 \[ \frac{\text{CosIntegral}(2 (a+b x))}{2 b}-\frac{\text{CosIntegral}(a+b x) \cos (a+b x)}{b}+\frac{\log (a+b x)}{2 b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.052, size = 44, normalized size = 0.9 \begin{align*}{\frac{{\it Ci} \left ( 2\,bx+2\,a \right ) }{2\,b}}-{\frac{{\it Ci} \left ( bx+a \right ) \cos \left ( bx+a \right ) }{b}}+{\frac{\ln \left ( bx+a \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 + a\right ) \sin \left (b x + a\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 + a\right ) \sin \left (b x + a\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 (a + b x \right )} \operatorname{Ci}{\left (a + b x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.19141, size = 130, normalized size = 2.77 \begin{align*} -\frac{\cos \left (b x + a\right ) \operatorname{Ci}\left (b x + a\right )}{b} + \frac{\cos \left (2 \, a\right )^{2} \operatorname{Ci}\left (2 \, b x + 2 \, a\right ) + \cos \left (2 \, a\right )^{2} \operatorname{Ci}\left (-2 \, b x - 2 \, a\right ) + \operatorname{Ci}\left (2 \, b x + 2 \, a\right ) \sin \left (2 \, a\right )^{2} + \operatorname{Ci}\left (-2 \, b x - 2 \, a\right ) \sin \left (2 \, a\right )^{2} + 2 \, \log \left (b x + a\right )}{4 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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