Optimal. Leaf size=44 \[ \frac{1}{2} b \text{CosIntegral}(b x)^2+b \text{CosIntegral}(2 b x)-\frac{\text{CosIntegral}(b x) \sin (b x)}{x}-\frac{\sin (2 b x)}{2 x} \]
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Rubi [A] time = 0.102781, antiderivative size = 44, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {6522, 6686, 12, 4406, 3297, 3302} \[ \frac{1}{2} b \text{CosIntegral}(b x)^2+b \text{CosIntegral}(2 b x)-\frac{\text{CosIntegral}(b x) \sin (b x)}{x}-\frac{\sin (2 b x)}{2 x} \]
Antiderivative was successfully verified.
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Rule 6522
Rule 6686
Rule 12
Rule 4406
Rule 3297
Rule 3302
Rubi steps
\begin{align*} \int \frac{\text{Ci}(b x) \sin (b x)}{x^2} \, dx &=-\frac{\text{Ci}(b x) \sin (b x)}{x}+b \int \frac{\cos (b x) \text{Ci}(b x)}{x} \, dx+b \int \frac{\cos (b x) \sin (b x)}{b x^2} \, dx\\ &=\frac{1}{2} b \text{Ci}(b x)^2-\frac{\text{Ci}(b x) \sin (b x)}{x}+\int \frac{\cos (b x) \sin (b x)}{x^2} \, dx\\ &=\frac{1}{2} b \text{Ci}(b x)^2-\frac{\text{Ci}(b x) \sin (b x)}{x}+\int \frac{\sin (2 b x)}{2 x^2} \, dx\\ &=\frac{1}{2} b \text{Ci}(b x)^2-\frac{\text{Ci}(b x) \sin (b x)}{x}+\frac{1}{2} \int \frac{\sin (2 b x)}{x^2} \, dx\\ &=\frac{1}{2} b \text{Ci}(b x)^2-\frac{\text{Ci}(b x) \sin (b x)}{x}-\frac{\sin (2 b x)}{2 x}+b \int \frac{\cos (2 b x)}{x} \, dx\\ &=\frac{1}{2} b \text{Ci}(b x)^2+b \text{Ci}(2 b x)-\frac{\text{Ci}(b x) \sin (b x)}{x}-\frac{\sin (2 b x)}{2 x}\\ \end{align*}
Mathematica [A] time = 0.0095825, size = 44, normalized size = 1. \[ \frac{1}{2} b \text{CosIntegral}(b x)^2+b \text{CosIntegral}(2 b x)-\frac{\text{CosIntegral}(b x) \sin (b x)}{x}-\frac{\sin (2 b x)}{2 x} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.059, size = 0, normalized size = 0. \begin{align*} \int{\frac{{\it Ci} \left ( bx \right ) \sin \left ( bx \right ) }{{x}^{2}}}\, dx \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 \frac{{\rm Ci}\left (b x\right ) \sin \left (b x\right )}{x^{2}}\,{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 (\frac{\operatorname{Ci}\left (b x\right ) \sin \left (b x\right )}{x^{2}}, 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 \frac{\sin{\left (b x \right )} \operatorname{Ci}{\left (b x \right )}}{x^{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\rm Ci}\left (b x\right ) \sin \left (b x\right )}{x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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