Optimal. Leaf size=35 \[ -\frac{\cos (b x)}{2 b^2}+\frac{1}{2} x^2 \text{CosIntegral}(b x)-\frac{x \sin (b x)}{2 b} \]
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Rubi [A] time = 0.0235583, antiderivative size = 35, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 6, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.667, Rules used = {6504, 12, 3296, 2638} \[ -\frac{\cos (b x)}{2 b^2}+\frac{1}{2} x^2 \text{CosIntegral}(b x)-\frac{x \sin (b x)}{2 b} \]
Antiderivative was successfully verified.
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Rule 6504
Rule 12
Rule 3296
Rule 2638
Rubi steps
\begin{align*} \int x \text{Ci}(b x) \, dx &=\frac{1}{2} x^2 \text{Ci}(b x)-\frac{1}{2} b \int \frac{x \cos (b x)}{b} \, dx\\ &=\frac{1}{2} x^2 \text{Ci}(b x)-\frac{1}{2} \int x \cos (b x) \, dx\\ &=\frac{1}{2} x^2 \text{Ci}(b x)-\frac{x \sin (b x)}{2 b}+\frac{\int \sin (b x) \, dx}{2 b}\\ &=-\frac{\cos (b x)}{2 b^2}+\frac{1}{2} x^2 \text{Ci}(b x)-\frac{x \sin (b x)}{2 b}\\ \end{align*}
Mathematica [A] time = 0.0079792, size = 35, normalized size = 1. \[ -\frac{\cos (b x)}{2 b^2}+\frac{1}{2} x^2 \text{CosIntegral}(b x)-\frac{x \sin (b x)}{2 b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.05, size = 32, normalized size = 0.9 \begin{align*}{\frac{1}{{b}^{2}} \left ({\frac{{b}^{2}{x}^{2}{\it Ci} \left ( bx \right ) }{2}}-{\frac{\sin \left ( bx \right ) bx}{2}}-{\frac{\cos \left ( 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 Ci}\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 \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 [A] time = 1.74918, size = 53, normalized size = 1.51 \begin{align*} - \frac{x^{2} \log{\left (b x \right )}}{2} + \frac{x^{2} \log{\left (b^{2} x^{2} \right )}}{4} + \frac{x^{2} \operatorname{Ci}{\left (b x \right )}}{2} - \frac{x \sin{\left (b x \right )}}{2 b} - \frac{\cos{\left (b x \right )}}{2 b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.16488, size = 39, normalized size = 1.11 \begin{align*} \frac{1}{2} \, x^{2} \operatorname{Ci}\left (b x\right ) - \frac{x \sin \left (b x\right )}{2 \, b} - \frac{\cos \left (b x\right )}{2 \, b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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