Optimal. Leaf size=49 \[ \frac{2 \sin (b x)}{3 b^3}-\frac{2 x \cos (b x)}{3 b^2}+\frac{1}{3} x^3 \text{CosIntegral}(b x)-\frac{x^2 \sin (b x)}{3 b} \]
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Rubi [A] time = 0.0478551, 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 = {6504, 12, 3296, 2637} \[ \frac{2 \sin (b x)}{3 b^3}-\frac{2 x \cos (b x)}{3 b^2}+\frac{1}{3} x^3 \text{CosIntegral}(b x)-\frac{x^2 \sin (b x)}{3 b} \]
Antiderivative was successfully verified.
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Rule 6504
Rule 12
Rule 3296
Rule 2637
Rubi steps
\begin{align*} \int x^2 \text{Ci}(b x) \, dx &=\frac{1}{3} x^3 \text{Ci}(b x)-\frac{1}{3} b \int \frac{x^2 \cos (b x)}{b} \, dx\\ &=\frac{1}{3} x^3 \text{Ci}(b x)-\frac{1}{3} \int x^2 \cos (b x) \, dx\\ &=\frac{1}{3} x^3 \text{Ci}(b x)-\frac{x^2 \sin (b x)}{3 b}+\frac{2 \int x \sin (b x) \, dx}{3 b}\\ &=-\frac{2 x \cos (b x)}{3 b^2}+\frac{1}{3} x^3 \text{Ci}(b x)-\frac{x^2 \sin (b x)}{3 b}+\frac{2 \int \cos (b x) \, dx}{3 b^2}\\ &=-\frac{2 x \cos (b x)}{3 b^2}+\frac{1}{3} x^3 \text{Ci}(b x)+\frac{2 \sin (b x)}{3 b^3}-\frac{x^2 \sin (b x)}{3 b}\\ \end{align*}
Mathematica [A] time = 0.0287376, size = 44, normalized size = 0.9 \[ -\frac{\left (b^2 x^2-2\right ) \sin (b x)}{3 b^3}-\frac{2 x \cos (b x)}{3 b^2}+\frac{1}{3} x^3 \text{CosIntegral}(b x) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.05, size = 44, normalized size = 0.9 \begin{align*}{\frac{1}{{b}^{3}} \left ({\frac{{b}^{3}{x}^{3}{\it Ci} \left ( bx \right ) }{3}}-{\frac{{b}^{2}{x}^{2}\sin \left ( bx \right ) }{3}}+{\frac{2\,\sin \left ( bx \right ) }{3}}-{\frac{2\,bx\cos \left ( bx \right ) }{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 Ci}\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{Ci}\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 = 2.43547, size = 70, normalized size = 1.43 \begin{align*} - \frac{x^{3} \log{\left (b x \right )}}{3} + \frac{x^{3} \log{\left (b^{2} x^{2} \right )}}{6} + \frac{x^{3} \operatorname{Ci}{\left (b x \right )}}{3} - \frac{x^{2} \sin{\left (b x \right )}}{3 b} - \frac{2 x \cos{\left (b x \right )}}{3 b^{2}} + \frac{2 \sin{\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 = 1.17142, size = 51, normalized size = 1.04 \begin{align*} \frac{1}{3} \, x^{3} \operatorname{Ci}\left (b x\right ) - \frac{2 \, x \cos \left (b x\right )}{3 \, b^{2}} - \frac{{\left (b^{2} x^{2} - 2\right )} \sin \left (b x\right )}{3 \, b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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