Optimal. Leaf size=147 \[ \frac{3 x^2 \text{CosIntegral}(b x) \sin (b x)}{b^2}-\frac{6 \text{CosIntegral}(b x) \sin (b x)}{b^4}+\frac{6 x \text{CosIntegral}(b x) \cos (b x)}{b^3}+\frac{3 \text{Si}(2 b x)}{b^4}+\frac{x^2 \sin (b x) \cos (b x)}{2 b^2}-\frac{5 x}{2 b^3}-\frac{3 x \sin ^2(b x)}{2 b^3}+\frac{x \cos ^2(b x)}{2 b^3}-\frac{4 \sin (b x) \cos (b x)}{b^4}-\frac{x^3 \text{CosIntegral}(b x) \cos (b x)}{b}+\frac{x^3}{6 b} \]
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Rubi [A] time = 0.182771, antiderivative size = 147, normalized size of antiderivative = 1., number of steps used = 20, number of rules used = 11, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.917, Rules used = {6520, 12, 3311, 30, 2635, 8, 6514, 3443, 6512, 4406, 3299} \[ \frac{3 x^2 \text{CosIntegral}(b x) \sin (b x)}{b^2}-\frac{6 \text{CosIntegral}(b x) \sin (b x)}{b^4}+\frac{6 x \text{CosIntegral}(b x) \cos (b x)}{b^3}+\frac{3 \text{Si}(2 b x)}{b^4}+\frac{x^2 \sin (b x) \cos (b x)}{2 b^2}-\frac{5 x}{2 b^3}-\frac{3 x \sin ^2(b x)}{2 b^3}+\frac{x \cos ^2(b x)}{2 b^3}-\frac{4 \sin (b x) \cos (b x)}{b^4}-\frac{x^3 \text{CosIntegral}(b x) \cos (b x)}{b}+\frac{x^3}{6 b} \]
Antiderivative was successfully verified.
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Rule 6520
Rule 12
Rule 3311
Rule 30
Rule 2635
Rule 8
Rule 6514
Rule 3443
Rule 6512
Rule 4406
Rule 3299
Rubi steps
\begin{align*} \int x^3 \text{Ci}(b x) \sin (b x) \, dx &=-\frac{x^3 \cos (b x) \text{Ci}(b x)}{b}+\frac{3 \int x^2 \cos (b x) \text{Ci}(b x) \, dx}{b}+\int \frac{x^2 \cos ^2(b x)}{b} \, dx\\ &=-\frac{x^3 \cos (b x) \text{Ci}(b x)}{b}+\frac{3 x^2 \text{Ci}(b x) \sin (b x)}{b^2}-\frac{6 \int x \text{Ci}(b x) \sin (b x) \, dx}{b^2}+\frac{\int x^2 \cos ^2(b x) \, dx}{b}-\frac{3 \int \frac{x \cos (b x) \sin (b x)}{b} \, dx}{b}\\ &=\frac{x \cos ^2(b x)}{2 b^3}+\frac{6 x \cos (b x) \text{Ci}(b x)}{b^3}-\frac{x^3 \cos (b x) \text{Ci}(b x)}{b}+\frac{x^2 \cos (b x) \sin (b x)}{2 b^2}+\frac{3 x^2 \text{Ci}(b x) \sin (b x)}{b^2}-\frac{\int \cos ^2(b x) \, dx}{2 b^3}-\frac{6 \int \cos (b x) \text{Ci}(b x) \, dx}{b^3}-\frac{3 \int x \cos (b x) \sin (b x) \, dx}{b^2}-\frac{6 \int \frac{\cos ^2(b x)}{b} \, dx}{b^2}+\frac{\int x^2 \, dx}{2 b}\\ &=\frac{x^3}{6 b}+\frac{x \cos ^2(b x)}{2 b^3}+\frac{6 x \cos (b x) \text{Ci}(b x)}{b^3}-\frac{x^3 \cos (b x) \text{Ci}(b x)}{b}-\frac{\cos (b x) \sin (b x)}{4 b^4}+\frac{x^2 \cos (b x) \sin (b x)}{2 b^2}-\frac{6 \text{Ci}(b x) \sin (b x)}{b^4}+\frac{3 x^2 \text{Ci}(b x) \sin (b x)}{b^2}-\frac{3 x \sin ^2(b x)}{2 b^3}-\frac{\int 1 \, dx}{4 b^3}+\frac{3 \int \sin ^2(b x) \, dx}{2 b^3}-\frac{6 \int \cos ^2(b x) \, dx}{b^3}+\frac{6 \int \frac{\cos (b x) \sin (b x)}{b x} \, dx}{b^3}\\ &=-\frac{x}{4 b^3}+\frac{x^3}{6 b}+\frac{x \cos ^2(b x)}{2 b^3}+\frac{6 x \cos (b x) \text{Ci}(b x)}{b^3}-\frac{x^3 \cos (b x) \text{Ci}(b x)}{b}-\frac{4 \cos (b x) \sin (b x)}{b^4}+\frac{x^2 \cos (b x) \sin (b x)}{2 b^2}-\frac{6 \text{Ci}(b x) \sin (b x)}{b^4}+\frac{3 x^2 \text{Ci}(b x) \sin (b x)}{b^2}-\frac{3 x \sin ^2(b x)}{2 b^3}+\frac{6 \int \frac{\cos (b x) \sin (b x)}{x} \, dx}{b^4}+\frac{3 \int 1 \, dx}{4 b^3}-\frac{3 \int 1 \, dx}{b^3}\\ &=-\frac{5 x}{2 b^3}+\frac{x^3}{6 b}+\frac{x \cos ^2(b x)}{2 b^3}+\frac{6 x \cos (b x) \text{Ci}(b x)}{b^3}-\frac{x^3 \cos (b x) \text{Ci}(b x)}{b}-\frac{4 \cos (b x) \sin (b x)}{b^4}+\frac{x^2 \cos (b x) \sin (b x)}{2 b^2}-\frac{6 \text{Ci}(b x) \sin (b x)}{b^4}+\frac{3 x^2 \text{Ci}(b x) \sin (b x)}{b^2}-\frac{3 x \sin ^2(b x)}{2 b^3}+\frac{6 \int \frac{\sin (2 b x)}{2 x} \, dx}{b^4}\\ &=-\frac{5 x}{2 b^3}+\frac{x^3}{6 b}+\frac{x \cos ^2(b x)}{2 b^3}+\frac{6 x \cos (b x) \text{Ci}(b x)}{b^3}-\frac{x^3 \cos (b x) \text{Ci}(b x)}{b}-\frac{4 \cos (b x) \sin (b x)}{b^4}+\frac{x^2 \cos (b x) \sin (b x)}{2 b^2}-\frac{6 \text{Ci}(b x) \sin (b x)}{b^4}+\frac{3 x^2 \text{Ci}(b x) \sin (b x)}{b^2}-\frac{3 x \sin ^2(b x)}{2 b^3}+\frac{3 \int \frac{\sin (2 b x)}{x} \, dx}{b^4}\\ &=-\frac{5 x}{2 b^3}+\frac{x^3}{6 b}+\frac{x \cos ^2(b x)}{2 b^3}+\frac{6 x \cos (b x) \text{Ci}(b x)}{b^3}-\frac{x^3 \cos (b x) \text{Ci}(b x)}{b}-\frac{4 \cos (b x) \sin (b x)}{b^4}+\frac{x^2 \cos (b x) \sin (b x)}{2 b^2}-\frac{6 \text{Ci}(b x) \sin (b x)}{b^4}+\frac{3 x^2 \text{Ci}(b x) \sin (b x)}{b^2}-\frac{3 x \sin ^2(b x)}{2 b^3}+\frac{3 \text{Si}(2 b x)}{b^4}\\ \end{align*}
Mathematica [A] time = 0.107106, size = 94, normalized size = 0.64 \[ \frac{-12 \text{CosIntegral}(b x) \left (b x \left (b^2 x^2-6\right ) \cos (b x)-3 \left (b^2 x^2-2\right ) \sin (b x)\right )+2 b^3 x^3+3 b^2 x^2 \sin (2 b x)+36 \text{Si}(2 b x)-36 b x-24 \sin (2 b x)+12 b x \cos (2 b x)}{12 b^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.076, size = 111, normalized size = 0.8 \begin{align*}{\frac{1}{{b}^{4}} \left ({\it Ci} \left ( bx \right ) \left ( -{b}^{3}{x}^{3}\cos \left ( bx \right ) +3\,{b}^{2}{x}^{2}\sin \left ( bx \right ) -6\,\sin \left ( bx \right ) +6\,bx\cos \left ( bx \right ) \right ) +{b}^{2}{x}^{2} \left ({\frac{\sin \left ( bx \right ) \cos \left ( bx \right ) }{2}}+{\frac{bx}{2}} \right ) +2\,bx \left ( \cos \left ( bx \right ) \right ) ^{2}-4\,\sin \left ( bx \right ) \cos \left ( bx \right ) -4\,bx-{\frac{{x}^{3}{b}^{3}}{3}}+3\,{\it Si} \left ( 2\,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 x^{3}{\rm Ci}\left (b x\right ) \sin \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 ) \sin \left (b x\right ), x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{3} \sin{\left (b x \right )} \operatorname{Ci}{\left (b x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [C] time = 1.23702, size = 116, normalized size = 0.79 \begin{align*} -{\left (\frac{{\left (b^{3} x^{3} - 6 \, b x\right )} \cos \left (b x\right )}{b^{4}} - \frac{3 \,{\left (b^{2} x^{2} - 2\right )} \sin \left (b x\right )}{b^{4}}\right )} \operatorname{Ci}\left (b x\right ) + \frac{b^{3} x^{3} - 18 \, b x + 9 \, \Im \left ( \operatorname{Ci}\left (2 \, b x\right ) \right ) - 9 \, \Im \left ( \operatorname{Ci}\left (-2 \, b x\right ) \right ) + 18 \, \operatorname{Si}\left (2 \, b x\right )}{6 \, b^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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