Optimal. Leaf size=25 \[ \frac{\text{CosIntegral}(b x) \sin (b x)}{b}-\frac{\text{Si}(2 b x)}{2 b} \]
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Rubi [A] time = 0.043309, antiderivative size = 25, 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 = {6512, 12, 4406, 3299} \[ \frac{\text{CosIntegral}(b x) \sin (b x)}{b}-\frac{\text{Si}(2 b x)}{2 b} \]
Antiderivative was successfully verified.
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Rule 6512
Rule 12
Rule 4406
Rule 3299
Rubi steps
\begin{align*} \int \cos (b x) \text{Ci}(b x) \, dx &=\frac{\text{Ci}(b x) \sin (b x)}{b}-\int \frac{\cos (b x) \sin (b x)}{b x} \, dx\\ &=\frac{\text{Ci}(b x) \sin (b x)}{b}-\frac{\int \frac{\cos (b x) \sin (b x)}{x} \, dx}{b}\\ &=\frac{\text{Ci}(b x) \sin (b x)}{b}-\frac{\int \frac{\sin (2 b x)}{2 x} \, dx}{b}\\ &=\frac{\text{Ci}(b x) \sin (b x)}{b}-\frac{\int \frac{\sin (2 b x)}{x} \, dx}{2 b}\\ &=\frac{\text{Ci}(b x) \sin (b x)}{b}-\frac{\text{Si}(2 b x)}{2 b}\\ \end{align*}
Mathematica [A] time = 0.0115221, size = 25, normalized size = 1. \[ \frac{\text{CosIntegral}(b x) \sin (b x)}{b}-\frac{\text{Si}(2 b x)}{2 b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.049, size = 22, normalized size = 0.9 \begin{align*}{\frac{1}{b} \left ({\it Ci} \left ( bx \right ) \sin \left ( bx \right ) -{\frac{{\it Si} \left ( 2\,bx \right ) }{2}} \right ) } \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 ) \cos \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 (\cos \left (b x\right ) \operatorname{Ci}\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 \cos{\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 [C] time = 1.25457, size = 54, normalized size = 2.16 \begin{align*} \frac{\operatorname{Ci}\left (b x\right ) \sin \left (b x\right )}{b} - \frac{\Im \left ( \operatorname{Ci}\left (2 \, b x\right ) \right ) - \Im \left ( \operatorname{Ci}\left (-2 \, b x\right ) \right ) + 2 \, \operatorname{Si}\left (2 \, b x\right )}{4 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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