Optimal. Leaf size=63 \[ -\frac{3 x^2 \cos (b x)}{4 b^2}+\frac{3 x \sin (b x)}{2 b^3}+\frac{3 \cos (b x)}{2 b^4}+\frac{1}{4} x^4 \text{CosIntegral}(b x)-\frac{x^3 \sin (b x)}{4 b} \]
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Rubi [A] time = 0.0713873, antiderivative size = 63, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {6504, 12, 3296, 2638} \[ -\frac{3 x^2 \cos (b x)}{4 b^2}+\frac{3 x \sin (b x)}{2 b^3}+\frac{3 \cos (b x)}{2 b^4}+\frac{1}{4} x^4 \text{CosIntegral}(b x)-\frac{x^3 \sin (b x)}{4 b} \]
Antiderivative was successfully verified.
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Rule 6504
Rule 12
Rule 3296
Rule 2638
Rubi steps
\begin{align*} \int x^3 \text{Ci}(b x) \, dx &=\frac{1}{4} x^4 \text{Ci}(b x)-\frac{1}{4} b \int \frac{x^3 \cos (b x)}{b} \, dx\\ &=\frac{1}{4} x^4 \text{Ci}(b x)-\frac{1}{4} \int x^3 \cos (b x) \, dx\\ &=\frac{1}{4} x^4 \text{Ci}(b x)-\frac{x^3 \sin (b x)}{4 b}+\frac{3 \int x^2 \sin (b x) \, dx}{4 b}\\ &=-\frac{3 x^2 \cos (b x)}{4 b^2}+\frac{1}{4} x^4 \text{Ci}(b x)-\frac{x^3 \sin (b x)}{4 b}+\frac{3 \int x \cos (b x) \, dx}{2 b^2}\\ &=-\frac{3 x^2 \cos (b x)}{4 b^2}+\frac{1}{4} x^4 \text{Ci}(b x)+\frac{3 x \sin (b x)}{2 b^3}-\frac{x^3 \sin (b x)}{4 b}-\frac{3 \int \sin (b x) \, dx}{2 b^3}\\ &=\frac{3 \cos (b x)}{2 b^4}-\frac{3 x^2 \cos (b x)}{4 b^2}+\frac{1}{4} x^4 \text{Ci}(b x)+\frac{3 x \sin (b x)}{2 b^3}-\frac{x^3 \sin (b x)}{4 b}\\ \end{align*}
Mathematica [A] time = 0.0398461, size = 53, normalized size = 0.84 \[ -\frac{x \left (b^2 x^2-6\right ) \sin (b x)}{4 b^3}-\frac{3 \left (b^2 x^2-2\right ) \cos (b x)}{4 b^4}+\frac{1}{4} x^4 \text{CosIntegral}(b x) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.048, size = 56, normalized size = 0.9 \begin{align*}{\frac{1}{{b}^{4}} \left ({\frac{{b}^{4}{x}^{4}{\it Ci} \left ( bx \right ) }{4}}-{\frac{\sin \left ( bx \right ){b}^{3}{x}^{3}}{4}}-{\frac{3\,{b}^{2}{x}^{2}\cos \left ( bx \right ) }{4}}+{\frac{3\,\cos \left ( bx \right ) }{2}}+{\frac{3\,\sin \left ( bx \right ) bx}{2}} \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^{3}{\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^{3} \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 = 2.9055, size = 85, normalized size = 1.35 \begin{align*} - \frac{x^{4} \log{\left (b x \right )}}{4} + \frac{x^{4} \log{\left (b^{2} x^{2} \right )}}{8} + \frac{x^{4} \operatorname{Ci}{\left (b x \right )}}{4} - \frac{x^{3} \sin{\left (b x \right )}}{4 b} - \frac{3 x^{2} \cos{\left (b x \right )}}{4 b^{2}} + \frac{3 x \sin{\left (b x \right )}}{2 b^{3}} + \frac{3 \cos{\left (b x \right )}}{2 b^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.19487, size = 66, normalized size = 1.05 \begin{align*} \frac{1}{4} \, x^{4} \operatorname{Ci}\left (b x\right ) - \frac{3 \,{\left (b^{2} x^{2} - 2\right )} \cos \left (b x\right )}{4 \, b^{4}} - \frac{{\left (b^{3} x^{3} - 6 \, b x\right )} \sin \left (b x\right )}{4 \, b^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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