Optimal. Leaf size=46 \[ -\frac{1}{4} b^2 \text{CosIntegral}(b x)-\frac{\text{CosIntegral}(b x)}{2 x^2}-\frac{\cos (b x)}{4 x^2}+\frac{b \sin (b x)}{4 x} \]
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Rubi [A] time = 0.0675109, antiderivative size = 46, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {6504, 12, 3297, 3302} \[ -\frac{1}{4} b^2 \text{CosIntegral}(b x)-\frac{\text{CosIntegral}(b x)}{2 x^2}-\frac{\cos (b x)}{4 x^2}+\frac{b \sin (b x)}{4 x} \]
Antiderivative was successfully verified.
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Rule 6504
Rule 12
Rule 3297
Rule 3302
Rubi steps
\begin{align*} \int \frac{\text{Ci}(b x)}{x^3} \, dx &=-\frac{\text{Ci}(b x)}{2 x^2}+\frac{1}{2} b \int \frac{\cos (b x)}{b x^3} \, dx\\ &=-\frac{\text{Ci}(b x)}{2 x^2}+\frac{1}{2} \int \frac{\cos (b x)}{x^3} \, dx\\ &=-\frac{\cos (b x)}{4 x^2}-\frac{\text{Ci}(b x)}{2 x^2}-\frac{1}{4} b \int \frac{\sin (b x)}{x^2} \, dx\\ &=-\frac{\cos (b x)}{4 x^2}-\frac{\text{Ci}(b x)}{2 x^2}+\frac{b \sin (b x)}{4 x}-\frac{1}{4} b^2 \int \frac{\cos (b x)}{x} \, dx\\ &=-\frac{\cos (b x)}{4 x^2}-\frac{1}{4} b^2 \text{Ci}(b x)-\frac{\text{Ci}(b x)}{2 x^2}+\frac{b \sin (b x)}{4 x}\\ \end{align*}
Mathematica [A] time = 0.0138352, size = 46, normalized size = 1. \[ -\frac{1}{4} b^2 \text{CosIntegral}(b x)-\frac{\text{CosIntegral}(b x)}{2 x^2}-\frac{\cos (b x)}{4 x^2}+\frac{b \sin (b x)}{4 x} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.053, size = 48, normalized size = 1. \begin{align*}{b}^{2} \left ( -{\frac{{\it Ci} \left ( bx \right ) }{2\,{b}^{2}{x}^{2}}}-{\frac{\cos \left ( bx \right ) }{4\,{b}^{2}{x}^{2}}}+{\frac{\sin \left ( bx \right ) }{4\,bx}}-{\frac{{\it Ci} \left ( bx \right ) }{4}} \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 \frac{{\rm Ci}\left (b x\right )}{x^{3}}\,{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 )}{x^{3}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 3.13406, size = 87, normalized size = 1.89 \begin{align*} \frac{b^{2} \log{\left (b x \right )}}{4} - \frac{b^{2} \log{\left (b^{2} x^{2} \right )}}{8} - \frac{b^{2} \operatorname{Ci}{\left (b x \right )}}{4} + \frac{b \sin{\left (b x \right )}}{4 x} + \frac{\log{\left (b x \right )}}{2 x^{2}} - \frac{\log{\left (b^{2} x^{2} \right )}}{4 x^{2}} - \frac{\cos{\left (b x \right )}}{4 x^{2}} - \frac{\operatorname{Ci}{\left (b x \right )}}{2 x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.16096, size = 72, normalized size = 1.57 \begin{align*} -\frac{b^{2} x^{2} \operatorname{Ci}\left (b x\right ) + b^{2} x^{2} \operatorname{Ci}\left (-b x\right ) - 2 \, b x \sin \left (b x\right ) + 2 \, \cos \left (b x\right )}{8 \, x^{2}} - \frac{\operatorname{Ci}\left (b x\right )}{2 \, x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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