Optimal. Leaf size=112 \[ \frac{4 \text{CosIntegral}(b x) \sin (b x)}{3 b^3}-\frac{4 x \text{CosIntegral}(b x) \cos (b x)}{3 b^2}-\frac{2 \text{Si}(2 b x)}{3 b^3}+\frac{x}{2 b^2}+\frac{x \sin ^2(b x)}{3 b^2}+\frac{5 \sin (b x) \cos (b x)}{6 b^3}+\frac{1}{3} x^3 \text{CosIntegral}(b x)^2-\frac{2 x^2 \text{CosIntegral}(b x) \sin (b x)}{3 b} \]
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Rubi [A] time = 0.138129, antiderivative size = 112, normalized size of antiderivative = 1., number of steps used = 15, number of rules used = 10, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 1., Rules used = {6508, 6514, 12, 3443, 2635, 8, 6520, 6512, 4406, 3299} \[ \frac{4 \text{CosIntegral}(b x) \sin (b x)}{3 b^3}-\frac{4 x \text{CosIntegral}(b x) \cos (b x)}{3 b^2}-\frac{2 \text{Si}(2 b x)}{3 b^3}+\frac{x}{2 b^2}+\frac{x \sin ^2(b x)}{3 b^2}+\frac{5 \sin (b x) \cos (b x)}{6 b^3}+\frac{1}{3} x^3 \text{CosIntegral}(b x)^2-\frac{2 x^2 \text{CosIntegral}(b x) \sin (b x)}{3 b} \]
Antiderivative was successfully verified.
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Rule 6508
Rule 6514
Rule 12
Rule 3443
Rule 2635
Rule 8
Rule 6520
Rule 6512
Rule 4406
Rule 3299
Rubi steps
\begin{align*} \int x^2 \text{Ci}(b x)^2 \, dx &=\frac{1}{3} x^3 \text{Ci}(b x)^2-\frac{2}{3} \int x^2 \cos (b x) \text{Ci}(b x) \, dx\\ &=\frac{1}{3} x^3 \text{Ci}(b x)^2-\frac{2 x^2 \text{Ci}(b x) \sin (b x)}{3 b}+\frac{2}{3} \int \frac{x \cos (b x) \sin (b x)}{b} \, dx+\frac{4 \int x \text{Ci}(b x) \sin (b x) \, dx}{3 b}\\ &=-\frac{4 x \cos (b x) \text{Ci}(b x)}{3 b^2}+\frac{1}{3} x^3 \text{Ci}(b x)^2-\frac{2 x^2 \text{Ci}(b x) \sin (b x)}{3 b}+\frac{4 \int \cos (b x) \text{Ci}(b x) \, dx}{3 b^2}+\frac{2 \int x \cos (b x) \sin (b x) \, dx}{3 b}+\frac{4 \int \frac{\cos ^2(b x)}{b} \, dx}{3 b}\\ &=-\frac{4 x \cos (b x) \text{Ci}(b x)}{3 b^2}+\frac{1}{3} x^3 \text{Ci}(b x)^2+\frac{4 \text{Ci}(b x) \sin (b x)}{3 b^3}-\frac{2 x^2 \text{Ci}(b x) \sin (b x)}{3 b}+\frac{x \sin ^2(b x)}{3 b^2}-\frac{\int \sin ^2(b x) \, dx}{3 b^2}+\frac{4 \int \cos ^2(b x) \, dx}{3 b^2}-\frac{4 \int \frac{\cos (b x) \sin (b x)}{b x} \, dx}{3 b^2}\\ &=-\frac{4 x \cos (b x) \text{Ci}(b x)}{3 b^2}+\frac{1}{3} x^3 \text{Ci}(b x)^2+\frac{5 \cos (b x) \sin (b x)}{6 b^3}+\frac{4 \text{Ci}(b x) \sin (b x)}{3 b^3}-\frac{2 x^2 \text{Ci}(b x) \sin (b x)}{3 b}+\frac{x \sin ^2(b x)}{3 b^2}-\frac{4 \int \frac{\cos (b x) \sin (b x)}{x} \, dx}{3 b^3}-\frac{\int 1 \, dx}{6 b^2}+\frac{2 \int 1 \, dx}{3 b^2}\\ &=\frac{x}{2 b^2}-\frac{4 x \cos (b x) \text{Ci}(b x)}{3 b^2}+\frac{1}{3} x^3 \text{Ci}(b x)^2+\frac{5 \cos (b x) \sin (b x)}{6 b^3}+\frac{4 \text{Ci}(b x) \sin (b x)}{3 b^3}-\frac{2 x^2 \text{Ci}(b x) \sin (b x)}{3 b}+\frac{x \sin ^2(b x)}{3 b^2}-\frac{4 \int \frac{\sin (2 b x)}{2 x} \, dx}{3 b^3}\\ &=\frac{x}{2 b^2}-\frac{4 x \cos (b x) \text{Ci}(b x)}{3 b^2}+\frac{1}{3} x^3 \text{Ci}(b x)^2+\frac{5 \cos (b x) \sin (b x)}{6 b^3}+\frac{4 \text{Ci}(b x) \sin (b x)}{3 b^3}-\frac{2 x^2 \text{Ci}(b x) \sin (b x)}{3 b}+\frac{x \sin ^2(b x)}{3 b^2}-\frac{2 \int \frac{\sin (2 b x)}{x} \, dx}{3 b^3}\\ &=\frac{x}{2 b^2}-\frac{4 x \cos (b x) \text{Ci}(b x)}{3 b^2}+\frac{1}{3} x^3 \text{Ci}(b x)^2+\frac{5 \cos (b x) \sin (b x)}{6 b^3}+\frac{4 \text{Ci}(b x) \sin (b x)}{3 b^3}-\frac{2 x^2 \text{Ci}(b x) \sin (b x)}{3 b}+\frac{x \sin ^2(b x)}{3 b^2}-\frac{2 \text{Si}(2 b x)}{3 b^3}\\ \end{align*}
Mathematica [A] time = 0.0805696, size = 78, normalized size = 0.7 \[ \frac{4 b^3 x^3 \text{CosIntegral}(b x)^2-8 \text{CosIntegral}(b x) \left (\left (b^2 x^2-2\right ) \sin (b x)+2 b x \cos (b x)\right )-8 \text{Si}(2 b x)+8 b x+5 \sin (2 b x)-2 b x \cos (2 b x)}{12 b^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.073, size = 84, normalized size = 0.8 \begin{align*}{\frac{1}{{b}^{3}} \left ({\frac{{b}^{3}{x}^{3} \left ({\it Ci} \left ( bx \right ) \right ) ^{2}}{3}}-2\,{\it Ci} \left ( bx \right ) \left ( 1/3\,{b}^{2}{x}^{2}\sin \left ( bx \right ) -2/3\,\sin \left ( bx \right ) +2/3\,bx\cos \left ( bx \right ) \right ) -{\frac{bx \left ( \cos \left ( bx \right ) \right ) ^{2}}{3}}+{\frac{5\,\sin \left ( bx \right ) \cos \left ( bx \right ) }{6}}+{\frac{5\,bx}{6}}-{\frac{2\,{\it Si} \left ( 2\,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 )^{2}\,{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 )^{2}, 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^{2} \operatorname{Ci}^{2}{\left (b x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [C] time = 1.22111, size = 104, normalized size = 0.93 \begin{align*} \frac{1}{3} \, x^{3} \operatorname{Ci}\left (b x\right )^{2} - \frac{2}{3} \,{\left (\frac{2 \, x \cos \left (b x\right )}{b^{2}} + \frac{{\left (b^{2} x^{2} - 2\right )} \sin \left (b x\right )}{b^{3}}\right )} \operatorname{Ci}\left (b x\right ) + \frac{2 \, b x - \Im \left ( \operatorname{Ci}\left (2 \, b x\right ) \right ) + \Im \left ( \operatorname{Ci}\left (-2 \, b x\right ) \right ) - 2 \, \operatorname{Si}\left (2 \, b x\right )}{3 \, b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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