Optimal. Leaf size=49 \[ -\frac{2 x \sin (b x)}{3 b^2}-\frac{2 \cos (b x)}{3 b^3}+\frac{1}{3} x^3 \text{Si}(b x)+\frac{x^2 \cos (b x)}{3 b} \]
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Rubi [A] time = 0.0461829, antiderivative size = 49, 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 = {6503, 12, 3296, 2638} \[ -\frac{2 x \sin (b x)}{3 b^2}-\frac{2 \cos (b x)}{3 b^3}+\frac{1}{3} x^3 \text{Si}(b x)+\frac{x^2 \cos (b x)}{3 b} \]
Antiderivative was successfully verified.
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Rule 6503
Rule 12
Rule 3296
Rule 2638
Rubi steps
\begin{align*} \int x^2 \text{Si}(b x) \, dx &=\frac{1}{3} x^3 \text{Si}(b x)-\frac{1}{3} b \int \frac{x^2 \sin (b x)}{b} \, dx\\ &=\frac{1}{3} x^3 \text{Si}(b x)-\frac{1}{3} \int x^2 \sin (b x) \, dx\\ &=\frac{x^2 \cos (b x)}{3 b}+\frac{1}{3} x^3 \text{Si}(b x)-\frac{2 \int x \cos (b x) \, dx}{3 b}\\ &=\frac{x^2 \cos (b x)}{3 b}-\frac{2 x \sin (b x)}{3 b^2}+\frac{1}{3} x^3 \text{Si}(b x)+\frac{2 \int \sin (b x) \, dx}{3 b^2}\\ &=-\frac{2 \cos (b x)}{3 b^3}+\frac{x^2 \cos (b x)}{3 b}-\frac{2 x \sin (b x)}{3 b^2}+\frac{1}{3} x^3 \text{Si}(b x)\\ \end{align*}
Mathematica [A] time = 0.0271712, size = 41, normalized size = 0.84 \[ \frac{b^3 x^3 \text{Si}(b x)+\left (b^2 x^2-2\right ) \cos (b x)-2 b x \sin (b x)}{3 b^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.054, size = 44, normalized size = 0.9 \begin{align*}{\frac{1}{{b}^{3}} \left ({\frac{{b}^{3}{x}^{3}{\it Si} \left ( bx \right ) }{3}}+{\frac{{b}^{2}{x}^{2}\cos \left ( bx \right ) }{3}}-{\frac{2\,\cos \left ( bx \right ) }{3}}-{\frac{2\,\sin \left ( bx \right ) bx}{3}} \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^{2}{\rm Si}\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^{2} \operatorname{Si}\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 = 1.47372, size = 46, normalized size = 0.94 \begin{align*} \frac{x^{3} \operatorname{Si}{\left (b x \right )}}{3} + \frac{x^{2} \cos{\left (b x \right )}}{3 b} - \frac{2 x \sin{\left (b x \right )}}{3 b^{2}} - \frac{2 \cos{\left (b x \right )}}{3 b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.28744, size = 51, normalized size = 1.04 \begin{align*} \frac{1}{3} \, x^{3} \operatorname{Si}\left (b x\right ) - \frac{2 \, x \sin \left (b x\right )}{3 \, b^{2}} + \frac{{\left (b^{2} x^{2} - 2\right )} \cos \left (b x\right )}{3 \, b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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