Optimal. Leaf size=142 \[ \frac{3 x^2 \text{CosIntegral}(b x) \cos (b x)}{b^2}+\frac{3 \text{CosIntegral}(2 b x)}{b^4}-\frac{6 x \text{CosIntegral}(b x) \sin (b x)}{b^3}-\frac{6 \text{CosIntegral}(b x) \cos (b x)}{b^4}-\frac{x^2}{2 b^2}-\frac{x^2 \sin ^2(b x)}{2 b^2}+\frac{3 \log (x)}{b^4}+\frac{13 \sin ^2(b x)}{4 b^4}-\frac{3 \cos ^2(b x)}{4 b^4}-\frac{2 x \sin (b x) \cos (b x)}{b^3}+\frac{x^3 \text{CosIntegral}(b x) \sin (b x)}{b} \]
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Rubi [A] time = 0.18973, antiderivative size = 142, normalized size of antiderivative = 1., number of steps used = 18, number of rules used = 10, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.833, Rules used = {6514, 12, 3443, 3310, 30, 6520, 2564, 6518, 3312, 3302} \[ \frac{3 x^2 \text{CosIntegral}(b x) \cos (b x)}{b^2}+\frac{3 \text{CosIntegral}(2 b x)}{b^4}-\frac{6 x \text{CosIntegral}(b x) \sin (b x)}{b^3}-\frac{6 \text{CosIntegral}(b x) \cos (b x)}{b^4}-\frac{x^2}{2 b^2}-\frac{x^2 \sin ^2(b x)}{2 b^2}+\frac{3 \log (x)}{b^4}+\frac{13 \sin ^2(b x)}{4 b^4}-\frac{3 \cos ^2(b x)}{4 b^4}-\frac{2 x \sin (b x) \cos (b x)}{b^3}+\frac{x^3 \text{CosIntegral}(b x) \sin (b x)}{b} \]
Antiderivative was successfully verified.
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Rule 6514
Rule 12
Rule 3443
Rule 3310
Rule 30
Rule 6520
Rule 2564
Rule 6518
Rule 3312
Rule 3302
Rubi steps
\begin{align*} \int x^3 \cos (b x) \text{Ci}(b x) \, dx &=\frac{x^3 \text{Ci}(b x) \sin (b x)}{b}-\frac{3 \int x^2 \text{Ci}(b x) \sin (b x) \, dx}{b}-\int \frac{x^2 \cos (b x) \sin (b x)}{b} \, dx\\ &=\frac{3 x^2 \cos (b x) \text{Ci}(b x)}{b^2}+\frac{x^3 \text{Ci}(b x) \sin (b x)}{b}-\frac{6 \int x \cos (b x) \text{Ci}(b x) \, dx}{b^2}-\frac{\int x^2 \cos (b x) \sin (b x) \, dx}{b}-\frac{3 \int \frac{x \cos ^2(b x)}{b} \, dx}{b}\\ &=\frac{3 x^2 \cos (b x) \text{Ci}(b x)}{b^2}-\frac{6 x \text{Ci}(b x) \sin (b x)}{b^3}+\frac{x^3 \text{Ci}(b x) \sin (b x)}{b}-\frac{x^2 \sin ^2(b x)}{2 b^2}+\frac{6 \int \text{Ci}(b x) \sin (b x) \, dx}{b^3}+\frac{\int x \sin ^2(b x) \, dx}{b^2}-\frac{3 \int x \cos ^2(b x) \, dx}{b^2}+\frac{6 \int \frac{\cos (b x) \sin (b x)}{b} \, dx}{b^2}\\ &=-\frac{3 \cos ^2(b x)}{4 b^4}-\frac{6 \cos (b x) \text{Ci}(b x)}{b^4}+\frac{3 x^2 \cos (b x) \text{Ci}(b x)}{b^2}-\frac{2 x \cos (b x) \sin (b x)}{b^3}-\frac{6 x \text{Ci}(b x) \sin (b x)}{b^3}+\frac{x^3 \text{Ci}(b x) \sin (b x)}{b}+\frac{\sin ^2(b x)}{4 b^4}-\frac{x^2 \sin ^2(b x)}{2 b^2}+\frac{6 \int \frac{\cos ^2(b x)}{b x} \, dx}{b^3}+\frac{6 \int \cos (b x) \sin (b x) \, dx}{b^3}+\frac{\int x \, dx}{2 b^2}-\frac{3 \int x \, dx}{2 b^2}\\ &=-\frac{x^2}{2 b^2}-\frac{3 \cos ^2(b x)}{4 b^4}-\frac{6 \cos (b x) \text{Ci}(b x)}{b^4}+\frac{3 x^2 \cos (b x) \text{Ci}(b x)}{b^2}-\frac{2 x \cos (b x) \sin (b x)}{b^3}-\frac{6 x \text{Ci}(b x) \sin (b x)}{b^3}+\frac{x^3 \text{Ci}(b x) \sin (b x)}{b}+\frac{\sin ^2(b x)}{4 b^4}-\frac{x^2 \sin ^2(b x)}{2 b^2}+\frac{6 \int \frac{\cos ^2(b x)}{x} \, dx}{b^4}+\frac{6 \operatorname{Subst}(\int x \, dx,x,\sin (b x))}{b^4}\\ &=-\frac{x^2}{2 b^2}-\frac{3 \cos ^2(b x)}{4 b^4}-\frac{6 \cos (b x) \text{Ci}(b x)}{b^4}+\frac{3 x^2 \cos (b x) \text{Ci}(b x)}{b^2}-\frac{2 x \cos (b x) \sin (b x)}{b^3}-\frac{6 x \text{Ci}(b x) \sin (b x)}{b^3}+\frac{x^3 \text{Ci}(b x) \sin (b x)}{b}+\frac{13 \sin ^2(b x)}{4 b^4}-\frac{x^2 \sin ^2(b x)}{2 b^2}+\frac{6 \int \left (\frac{1}{2 x}+\frac{\cos (2 b x)}{2 x}\right ) \, dx}{b^4}\\ &=-\frac{x^2}{2 b^2}-\frac{3 \cos ^2(b x)}{4 b^4}-\frac{6 \cos (b x) \text{Ci}(b x)}{b^4}+\frac{3 x^2 \cos (b x) \text{Ci}(b x)}{b^2}+\frac{3 \log (x)}{b^4}-\frac{2 x \cos (b x) \sin (b x)}{b^3}-\frac{6 x \text{Ci}(b x) \sin (b x)}{b^3}+\frac{x^3 \text{Ci}(b x) \sin (b x)}{b}+\frac{13 \sin ^2(b x)}{4 b^4}-\frac{x^2 \sin ^2(b x)}{2 b^2}+\frac{3 \int \frac{\cos (2 b x)}{x} \, dx}{b^4}\\ &=-\frac{x^2}{2 b^2}-\frac{3 \cos ^2(b x)}{4 b^4}-\frac{6 \cos (b x) \text{Ci}(b x)}{b^4}+\frac{3 x^2 \cos (b x) \text{Ci}(b x)}{b^2}+\frac{3 \text{Ci}(2 b x)}{b^4}+\frac{3 \log (x)}{b^4}-\frac{2 x \cos (b x) \sin (b x)}{b^3}-\frac{6 x \text{Ci}(b x) \sin (b x)}{b^3}+\frac{x^3 \text{Ci}(b x) \sin (b x)}{b}+\frac{13 \sin ^2(b x)}{4 b^4}-\frac{x^2 \sin ^2(b x)}{2 b^2}\\ \end{align*}
Mathematica [A] time = 0.105215, size = 93, normalized size = 0.65 \[ \frac{4 \text{CosIntegral}(b x) \left (b x \left (b^2 x^2-6\right ) \sin (b x)+3 \left (b^2 x^2-2\right ) \cos (b x)\right )-3 b^2 x^2+b^2 x^2 \cos (2 b x)+12 \text{CosIntegral}(2 b x)-4 b x \sin (2 b x)-8 \cos (2 b x)+12 \log (x)}{4 b^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.065, size = 136, normalized size = 1. \begin{align*}{\frac{{x}^{3}{\it Ci} \left ( bx \right ) \sin \left ( bx \right ) }{b}}+3\,{\frac{{x}^{2}{\it Ci} \left ( bx \right ) \cos \left ( bx \right ) }{{b}^{2}}}-6\,{\frac{x{\it Ci} \left ( bx \right ) \sin \left ( bx \right ) }{{b}^{3}}}-6\,{\frac{{\it Ci} \left ( bx \right ) \cos \left ( bx \right ) }{{b}^{4}}}+{\frac{{x}^{2} \left ( \cos \left ( bx \right ) \right ) ^{2}}{2\,{b}^{2}}}-2\,{\frac{x\cos \left ( bx \right ) \sin \left ( bx \right ) }{{b}^{3}}}-{\frac{{x}^{2}}{{b}^{2}}}+{\frac{ \left ( \sin \left ( bx \right ) \right ) ^{2}}{{b}^{4}}}+3\,{\frac{\ln \left ( bx \right ) }{{b}^{4}}}+3\,{\frac{{\it Ci} \left ( 2\,bx \right ) }{{b}^{4}}}-3\,{\frac{ \left ( \cos \left ( bx \right ) \right ) ^{2}}{{b}^{4}}} \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 ) \cos \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} \cos \left (b x\right ) \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 [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{3} \cos{\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 [A] time = 1.26308, size = 103, normalized size = 0.73 \begin{align*}{\left (\frac{3 \,{\left (b^{2} x^{2} - 2\right )} \cos \left (b x\right )}{b^{4}} + \frac{{\left (b^{3} x^{3} - 6 \, b x\right )} \sin \left (b x\right )}{b^{4}}\right )} \operatorname{Ci}\left (b x\right ) - \frac{3 \,{\left (b^{2} x^{2} - 2 \, \operatorname{Ci}\left (2 \, b x\right ) - 2 \, \operatorname{Ci}\left (-2 \, b x\right ) - 4 \, \log \left (x\right )\right )}}{4 \, b^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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