Optimal. Leaf size=31 \[ x \text{CosIntegral}(b x)^2-\frac{2 \text{CosIntegral}(b x) \sin (b x)}{b}+\frac{\text{Si}(2 b x)}{b} \]
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Rubi [A] time = 0.0537311, antiderivative size = 31, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 6, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.833, Rules used = {6506, 6512, 12, 4406, 3299} \[ x \text{CosIntegral}(b x)^2-\frac{2 \text{CosIntegral}(b x) \sin (b x)}{b}+\frac{\text{Si}(2 b x)}{b} \]
Antiderivative was successfully verified.
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Rule 6506
Rule 6512
Rule 12
Rule 4406
Rule 3299
Rubi steps
\begin{align*} \int \text{Ci}(b x)^2 \, dx &=x \text{Ci}(b x)^2-2 \int \cos (b x) \text{Ci}(b x) \, dx\\ &=x \text{Ci}(b x)^2-\frac{2 \text{Ci}(b x) \sin (b x)}{b}+2 \int \frac{\cos (b x) \sin (b x)}{b x} \, dx\\ &=x \text{Ci}(b x)^2-\frac{2 \text{Ci}(b x) \sin (b x)}{b}+\frac{2 \int \frac{\cos (b x) \sin (b x)}{x} \, dx}{b}\\ &=x \text{Ci}(b x)^2-\frac{2 \text{Ci}(b x) \sin (b x)}{b}+\frac{2 \int \frac{\sin (2 b x)}{2 x} \, dx}{b}\\ &=x \text{Ci}(b x)^2-\frac{2 \text{Ci}(b x) \sin (b x)}{b}+\frac{\int \frac{\sin (2 b x)}{x} \, dx}{b}\\ &=x \text{Ci}(b x)^2-\frac{2 \text{Ci}(b x) \sin (b x)}{b}+\frac{\text{Si}(2 b x)}{b}\\ \end{align*}
Mathematica [A] time = 0.0076127, size = 31, normalized size = 1. \[ x \text{CosIntegral}(b x)^2-\frac{2 \text{CosIntegral}(b x) \sin (b x)}{b}+\frac{\text{Si}(2 b x)}{b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.047, size = 30, normalized size = 1. \begin{align*}{\frac{bx \left ({\it Ci} \left ( bx \right ) \right ) ^{2}-2\,{\it Ci} \left ( bx \right ) \sin \left ( bx \right ) +{\it Si} \left ( 2\,bx \right ) }{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\right )^{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 (\operatorname{Ci}\left (b x\right )^{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 \operatorname{Ci}^{2}{\left (b x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [C] time = 1.28276, size = 66, normalized size = 2.13 \begin{align*} x \operatorname{Ci}\left (b x\right )^{2} - \frac{2 \, \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 )}{2 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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