Optimal. Leaf size=91 \[ -\frac{\text{Si}(2 b x)}{b^3}+\frac{2 x \text{Si}(b x) \sin (b x)}{b^2}+\frac{2 \text{Si}(b x) \cos (b x)}{b^3}-\frac{5 x}{4 b^2}+\frac{x \sin ^2(b x)}{2 b^2}+\frac{5 \sin (b x) \cos (b x)}{4 b^3}-\frac{x^2 \text{Si}(b x) \cos (b x)}{b} \]
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Rubi [A] time = 0.111096, antiderivative size = 91, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 9, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.75, Rules used = {6513, 12, 3443, 2635, 8, 6519, 6511, 4406, 3299} \[ -\frac{\text{Si}(2 b x)}{b^3}+\frac{2 x \text{Si}(b x) \sin (b x)}{b^2}+\frac{2 \text{Si}(b x) \cos (b x)}{b^3}-\frac{5 x}{4 b^2}+\frac{x \sin ^2(b x)}{2 b^2}+\frac{5 \sin (b x) \cos (b x)}{4 b^3}-\frac{x^2 \text{Si}(b x) \cos (b x)}{b} \]
Antiderivative was successfully verified.
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Rule 6513
Rule 12
Rule 3443
Rule 2635
Rule 8
Rule 6519
Rule 6511
Rule 4406
Rule 3299
Rubi steps
\begin{align*} \int x^2 \sin (b x) \text{Si}(b x) \, dx &=-\frac{x^2 \cos (b x) \text{Si}(b x)}{b}+\frac{2 \int x \cos (b x) \text{Si}(b x) \, dx}{b}+\int \frac{x \cos (b x) \sin (b x)}{b} \, dx\\ &=-\frac{x^2 \cos (b x) \text{Si}(b x)}{b}+\frac{2 x \sin (b x) \text{Si}(b x)}{b^2}-\frac{2 \int \sin (b x) \text{Si}(b x) \, dx}{b^2}+\frac{\int x \cos (b x) \sin (b x) \, dx}{b}-\frac{2 \int \frac{\sin ^2(b x)}{b} \, dx}{b}\\ &=\frac{x \sin ^2(b x)}{2 b^2}+\frac{2 \cos (b x) \text{Si}(b x)}{b^3}-\frac{x^2 \cos (b x) \text{Si}(b x)}{b}+\frac{2 x \sin (b x) \text{Si}(b x)}{b^2}-\frac{\int \sin ^2(b x) \, dx}{2 b^2}-\frac{2 \int \frac{\cos (b x) \sin (b x)}{b x} \, dx}{b^2}-\frac{2 \int \sin ^2(b x) \, dx}{b^2}\\ &=\frac{5 \cos (b x) \sin (b x)}{4 b^3}+\frac{x \sin ^2(b x)}{2 b^2}+\frac{2 \cos (b x) \text{Si}(b x)}{b^3}-\frac{x^2 \cos (b x) \text{Si}(b x)}{b}+\frac{2 x \sin (b x) \text{Si}(b x)}{b^2}-\frac{2 \int \frac{\cos (b x) \sin (b x)}{x} \, dx}{b^3}-\frac{\int 1 \, dx}{4 b^2}-\frac{\int 1 \, dx}{b^2}\\ &=-\frac{5 x}{4 b^2}+\frac{5 \cos (b x) \sin (b x)}{4 b^3}+\frac{x \sin ^2(b x)}{2 b^2}+\frac{2 \cos (b x) \text{Si}(b x)}{b^3}-\frac{x^2 \cos (b x) \text{Si}(b x)}{b}+\frac{2 x \sin (b x) \text{Si}(b x)}{b^2}-\frac{2 \int \frac{\sin (2 b x)}{2 x} \, dx}{b^3}\\ &=-\frac{5 x}{4 b^2}+\frac{5 \cos (b x) \sin (b x)}{4 b^3}+\frac{x \sin ^2(b x)}{2 b^2}+\frac{2 \cos (b x) \text{Si}(b x)}{b^3}-\frac{x^2 \cos (b x) \text{Si}(b x)}{b}+\frac{2 x \sin (b x) \text{Si}(b x)}{b^2}-\frac{\int \frac{\sin (2 b x)}{x} \, dx}{b^3}\\ &=-\frac{5 x}{4 b^2}+\frac{5 \cos (b x) \sin (b x)}{4 b^3}+\frac{x \sin ^2(b x)}{2 b^2}+\frac{2 \cos (b x) \text{Si}(b x)}{b^3}-\frac{x^2 \cos (b x) \text{Si}(b x)}{b}+\frac{2 x \sin (b x) \text{Si}(b x)}{b^2}-\frac{\text{Si}(2 b x)}{b^3}\\ \end{align*}
Mathematica [A] time = 0.103689, size = 64, normalized size = 0.7 \[ -\frac{8 \text{Si}(b x) \left (\left (b^2 x^2-2\right ) \cos (b x)-2 b x \sin (b x)\right )+8 \text{Si}(2 b x)+8 b x-5 \sin (2 b x)+2 b x \cos (2 b x)}{8 b^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.059, size = 69, normalized size = 0.8 \begin{align*}{\frac{1}{{b}^{3}} \left ({\it Si} \left ( bx \right ) \left ( -{b}^{2}{x}^{2}\cos \left ( bx \right ) +2\,\cos \left ( bx \right ) +2\,\sin \left ( bx \right ) bx \right ) -{\frac{bx \left ( \cos \left ( bx \right ) \right ) ^{2}}{2}}+{\frac{5\,\sin \left ( bx \right ) \cos \left ( bx \right ) }{4}}-{\frac{3\,bx}{4}}-{\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^{2}{\rm Si}\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^{2} \sin \left (b x\right ) \operatorname{Si}\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^{2} \sin{\left (b x \right )} \operatorname{Si}{\left (b x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [C] time = 1.16074, size = 89, normalized size = 0.98 \begin{align*}{\left (\frac{2 \, x \sin \left (b x\right )}{b^{2}} - \frac{{\left (b^{2} x^{2} - 2\right )} \cos \left (b x\right )}{b^{3}}\right )} \operatorname{Si}\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 )}{2 \, b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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