Optimal. Leaf size=61 \[ -\frac{\text{Si}(2 b x)}{2 b^2}+\frac{\text{Si}(b x) \cos (b x)}{b^2}+\frac{\sin (b x) \cos (b x)}{2 b^2}+\frac{x \text{Si}(b x) \sin (b x)}{b}-\frac{x}{2 b} \]
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Rubi [A] time = 0.0659373, antiderivative size = 61, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 7, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.7, Rules used = {6519, 12, 2635, 8, 6511, 4406, 3299} \[ -\frac{\text{Si}(2 b x)}{2 b^2}+\frac{\text{Si}(b x) \cos (b x)}{b^2}+\frac{\sin (b x) \cos (b x)}{2 b^2}+\frac{x \text{Si}(b x) \sin (b x)}{b}-\frac{x}{2 b} \]
Antiderivative was successfully verified.
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Rule 6519
Rule 12
Rule 2635
Rule 8
Rule 6511
Rule 4406
Rule 3299
Rubi steps
\begin{align*} \int x \cos (b x) \text{Si}(b x) \, dx &=\frac{x \sin (b x) \text{Si}(b x)}{b}-\frac{\int \sin (b x) \text{Si}(b x) \, dx}{b}-\int \frac{\sin ^2(b x)}{b} \, dx\\ &=\frac{\cos (b x) \text{Si}(b x)}{b^2}+\frac{x \sin (b x) \text{Si}(b x)}{b}-\frac{\int \frac{\cos (b x) \sin (b x)}{b x} \, dx}{b}-\frac{\int \sin ^2(b x) \, dx}{b}\\ &=\frac{\cos (b x) \sin (b x)}{2 b^2}+\frac{\cos (b x) \text{Si}(b x)}{b^2}+\frac{x \sin (b x) \text{Si}(b x)}{b}-\frac{\int \frac{\cos (b x) \sin (b x)}{x} \, dx}{b^2}-\frac{\int 1 \, dx}{2 b}\\ &=-\frac{x}{2 b}+\frac{\cos (b x) \sin (b x)}{2 b^2}+\frac{\cos (b x) \text{Si}(b x)}{b^2}+\frac{x \sin (b x) \text{Si}(b x)}{b}-\frac{\int \frac{\sin (2 b x)}{2 x} \, dx}{b^2}\\ &=-\frac{x}{2 b}+\frac{\cos (b x) \sin (b x)}{2 b^2}+\frac{\cos (b x) \text{Si}(b x)}{b^2}+\frac{x \sin (b x) \text{Si}(b x)}{b}-\frac{\int \frac{\sin (2 b x)}{x} \, dx}{2 b^2}\\ &=-\frac{x}{2 b}+\frac{\cos (b x) \sin (b x)}{2 b^2}+\frac{\cos (b x) \text{Si}(b x)}{b^2}+\frac{x \sin (b x) \text{Si}(b x)}{b}-\frac{\text{Si}(2 b x)}{2 b^2}\\ \end{align*}
Mathematica [A] time = 0.0492865, size = 42, normalized size = 0.69 \[ \frac{-2 \text{Si}(2 b x)+4 \text{Si}(b x) (b x \sin (b x)+\cos (b x))-2 b x+\sin (2 b x)}{4 b^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.056, size = 44, normalized size = 0.7 \begin{align*}{\frac{1}{{b}^{2}} \left ({\it Si} \left ( bx \right ) \left ( \sin \left ( bx \right ) bx+\cos \left ( bx \right ) \right ) +{\frac{\sin \left ( bx \right ) \cos \left ( bx \right ) }{2}}-{\frac{bx}{2}}-{\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 x{\rm Si}\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 (x \cos \left (b x\right ) \operatorname{Si}\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 x \cos{\left (b x \right )} \operatorname{Si}{\left (b x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [C] time = 1.1768, size = 74, normalized size = 1.21 \begin{align*}{\left (\frac{x \sin \left (b x\right )}{b} + \frac{\cos \left (b x\right )}{b^{2}}\right )} \operatorname{Si}\left (b x\right ) - \frac{2 \, b x + \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^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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