Optimal. Leaf size=25 \[ b \text{CosIntegral}(b x)-\frac{\text{Si}(b x)}{x}-\frac{\sin (b x)}{x} \]
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Rubi [A] time = 0.0452787, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {6503, 12, 3297, 3302} \[ b \text{CosIntegral}(b x)-\frac{\text{Si}(b x)}{x}-\frac{\sin (b x)}{x} \]
Antiderivative was successfully verified.
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Rule 6503
Rule 12
Rule 3297
Rule 3302
Rubi steps
\begin{align*} \int \frac{\text{Si}(b x)}{x^2} \, dx &=-\frac{\text{Si}(b x)}{x}+b \int \frac{\sin (b x)}{b x^2} \, dx\\ &=-\frac{\text{Si}(b x)}{x}+\int \frac{\sin (b x)}{x^2} \, dx\\ &=-\frac{\sin (b x)}{x}-\frac{\text{Si}(b x)}{x}+b \int \frac{\cos (b x)}{x} \, dx\\ &=b \text{Ci}(b x)-\frac{\sin (b x)}{x}-\frac{\text{Si}(b x)}{x}\\ \end{align*}
Mathematica [A] time = 0.009746, size = 25, normalized size = 1. \[ b \text{CosIntegral}(b x)-\frac{\text{Si}(b x)}{x}-\frac{\sin (b x)}{x} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.054, size = 32, normalized size = 1.3 \begin{align*} b \left ( -{\frac{{\it Si} \left ( bx \right ) }{bx}}-{\frac{\sin \left ( bx \right ) }{bx}}+{\it Ci} \left ( bx \right ) \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 Si}\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{Si}\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 [A] time = 1.16882, size = 36, normalized size = 1.44 \begin{align*} - \frac{b^{3} x^{2}{{}_{3}F_{4}\left (\begin{matrix} 1, 1, \frac{3}{2} \\ 2, 2, \frac{5}{2}, \frac{5}{2} \end{matrix}\middle |{- \frac{b^{2} x^{2}}{4}} \right )}}{36} + \frac{b \log{\left (b^{2} x^{2} \right )}}{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.20313, size = 50, normalized size = 2. \begin{align*} \frac{b x \operatorname{Ci}\left (b x\right ) + b x \operatorname{Ci}\left (-b x\right ) - 2 \, \sin \left (b x\right )}{2 \, x} - \frac{\operatorname{Si}\left (b x\right )}{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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