Integrand size = 10, antiderivative size = 112 \[ \int x^2 \text {Si}(b x)^2 \, dx=\frac {5 x}{6 b^2}-\frac {5 \cos (b x) \sin (b x)}{6 b^3}-\frac {x \sin ^2(b x)}{3 b^2}-\frac {4 \cos (b x) \text {Si}(b x)}{3 b^3}+\frac {2 x^2 \cos (b x) \text {Si}(b x)}{3 b}-\frac {4 x \sin (b x) \text {Si}(b x)}{3 b^2}+\frac {1}{3} x^3 \text {Si}(b x)^2+\frac {2 \text {Si}(2 b x)}{3 b^3} \]
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Time = 0.11 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {6642, 6648, 12, 3524, 2715, 8, 6654, 6646, 4491, 3380} \[ \int x^2 \text {Si}(b x)^2 \, dx=\frac {2 \text {Si}(2 b x)}{3 b^3}-\frac {4 \text {Si}(b x) \cos (b x)}{3 b^3}-\frac {5 \sin (b x) \cos (b x)}{6 b^3}-\frac {4 x \text {Si}(b x) \sin (b x)}{3 b^2}+\frac {5 x}{6 b^2}-\frac {x \sin ^2(b x)}{3 b^2}+\frac {1}{3} x^3 \text {Si}(b x)^2+\frac {2 x^2 \text {Si}(b x) \cos (b x)}{3 b} \]
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Rule 8
Rule 12
Rule 2715
Rule 3380
Rule 3524
Rule 4491
Rule 6642
Rule 6646
Rule 6648
Rule 6654
Rubi steps \begin{align*} \text {integral}& = \frac {1}{3} x^3 \text {Si}(b x)^2-\frac {2}{3} \int x^2 \sin (b x) \text {Si}(b x) \, dx \\ & = \frac {2 x^2 \cos (b x) \text {Si}(b x)}{3 b}+\frac {1}{3} x^3 \text {Si}(b x)^2-\frac {2}{3} \int \frac {x \cos (b x) \sin (b x)}{b} \, dx-\frac {4 \int x \cos (b x) \text {Si}(b x) \, dx}{3 b} \\ & = \frac {2 x^2 \cos (b x) \text {Si}(b x)}{3 b}-\frac {4 x \sin (b x) \text {Si}(b x)}{3 b^2}+\frac {1}{3} x^3 \text {Si}(b x)^2+\frac {4 \int \sin (b x) \text {Si}(b x) \, dx}{3 b^2}-\frac {2 \int x \cos (b x) \sin (b x) \, dx}{3 b}+\frac {4 \int \frac {\sin ^2(b x)}{b} \, dx}{3 b} \\ & = -\frac {x \sin ^2(b x)}{3 b^2}-\frac {4 \cos (b x) \text {Si}(b x)}{3 b^3}+\frac {2 x^2 \cos (b x) \text {Si}(b x)}{3 b}-\frac {4 x \sin (b x) \text {Si}(b x)}{3 b^2}+\frac {1}{3} x^3 \text {Si}(b x)^2+\frac {\int \sin ^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 \int \sin ^2(b x) \, dx}{3 b^2} \\ & = -\frac {5 \cos (b x) \sin (b x)}{6 b^3}-\frac {x \sin ^2(b x)}{3 b^2}-\frac {4 \cos (b x) \text {Si}(b x)}{3 b^3}+\frac {2 x^2 \cos (b x) \text {Si}(b x)}{3 b}-\frac {4 x \sin (b x) \text {Si}(b x)}{3 b^2}+\frac {1}{3} x^3 \text {Si}(b x)^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 {5 x}{6 b^2}-\frac {5 \cos (b x) \sin (b x)}{6 b^3}-\frac {x \sin ^2(b x)}{3 b^2}-\frac {4 \cos (b x) \text {Si}(b x)}{3 b^3}+\frac {2 x^2 \cos (b x) \text {Si}(b x)}{3 b}-\frac {4 x \sin (b x) \text {Si}(b x)}{3 b^2}+\frac {1}{3} x^3 \text {Si}(b x)^2+\frac {4 \int \frac {\sin (2 b x)}{2 x} \, dx}{3 b^3} \\ & = \frac {5 x}{6 b^2}-\frac {5 \cos (b x) \sin (b x)}{6 b^3}-\frac {x \sin ^2(b x)}{3 b^2}-\frac {4 \cos (b x) \text {Si}(b x)}{3 b^3}+\frac {2 x^2 \cos (b x) \text {Si}(b x)}{3 b}-\frac {4 x \sin (b x) \text {Si}(b x)}{3 b^2}+\frac {1}{3} x^3 \text {Si}(b x)^2+\frac {2 \int \frac {\sin (2 b x)}{x} \, dx}{3 b^3} \\ & = \frac {5 x}{6 b^2}-\frac {5 \cos (b x) \sin (b x)}{6 b^3}-\frac {x \sin ^2(b x)}{3 b^2}-\frac {4 \cos (b x) \text {Si}(b x)}{3 b^3}+\frac {2 x^2 \cos (b x) \text {Si}(b x)}{3 b}-\frac {4 x \sin (b x) \text {Si}(b x)}{3 b^2}+\frac {1}{3} x^3 \text {Si}(b x)^2+\frac {2 \text {Si}(2 b x)}{3 b^3} \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.70 \[ \int x^2 \text {Si}(b x)^2 \, dx=\frac {8 b x+2 b x \cos (2 b x)-5 \sin (2 b x)+8 \left (\left (-2+b^2 x^2\right ) \cos (b x)-2 b x \sin (b x)\right ) \text {Si}(b x)+4 b^3 x^3 \text {Si}(b x)^2+8 \text {Si}(2 b x)}{12 b^3} \]
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Time = 0.58 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.75
method | result | size |
derivativedivides | \(\frac {\frac {b^{3} x^{3} \operatorname {Si}\left (b x \right )^{2}}{3}-2 \,\operatorname {Si}\left (b x \right ) \left (-\frac {b^{2} x^{2} \cos \left (b x \right )}{3}+\frac {2 \cos \left (b x \right )}{3}+\frac {2 b x \sin \left (b x \right )}{3}\right )+\frac {b x \cos \left (b x \right )^{2}}{3}-\frac {5 \sin \left (b x \right ) \cos \left (b x \right )}{6}+\frac {b x}{2}+\frac {2 \,\operatorname {Si}\left (2 b x \right )}{3}}{b^{3}}\) | \(84\) |
default | \(\frac {\frac {b^{3} x^{3} \operatorname {Si}\left (b x \right )^{2}}{3}-2 \,\operatorname {Si}\left (b x \right ) \left (-\frac {b^{2} x^{2} \cos \left (b x \right )}{3}+\frac {2 \cos \left (b x \right )}{3}+\frac {2 b x \sin \left (b x \right )}{3}\right )+\frac {b x \cos \left (b x \right )^{2}}{3}-\frac {5 \sin \left (b x \right ) \cos \left (b x \right )}{6}+\frac {b x}{2}+\frac {2 \,\operatorname {Si}\left (2 b x \right )}{3}}{b^{3}}\) | \(84\) |
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Time = 0.27 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.72 \[ \int x^2 \text {Si}(b x)^2 \, dx=\frac {2 \, b^{3} x^{3} \operatorname {Si}\left (b x\right )^{2} + 2 \, b x \cos \left (b x\right )^{2} + 4 \, {\left (b^{2} x^{2} - 2\right )} \cos \left (b x\right ) \operatorname {Si}\left (b x\right ) + 3 \, b x - {\left (8 \, b x \operatorname {Si}\left (b x\right ) + 5 \, \cos \left (b x\right )\right )} \sin \left (b x\right ) + 4 \, \operatorname {Si}\left (2 \, b x\right )}{6 \, b^{3}} \]
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\[ \int x^2 \text {Si}(b x)^2 \, dx=\int x^{2} \operatorname {Si}^{2}{\left (b x \right )}\, dx \]
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\[ \int x^2 \text {Si}(b x)^2 \, dx=\int { x^{2} \operatorname {Si}\left (b x\right )^{2} \,d x } \]
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.29 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.34 \[ \int x^2 \text {Si}(b x)^2 \, dx=\frac {1}{3} \, x^{3} \operatorname {Si}\left (b x\right )^{2} - \frac {2}{3} \, {\left (\frac {2 \, x \sin \left (b x\right )}{b^{2}} - \frac {{\left (b^{2} x^{2} - 2\right )} \cos \left (b x\right )}{b^{3}}\right )} \operatorname {Si}\left (b x\right ) + \frac {3 \, b x \tan \left (b x\right )^{2} + 2 \, \Im \left ( \operatorname {Ci}\left (2 \, b x\right ) \right ) \tan \left (b x\right )^{2} - 2 \, \Im \left ( \operatorname {Ci}\left (-2 \, b x\right ) \right ) \tan \left (b x\right )^{2} + 4 \, \operatorname {Si}\left (2 \, b x\right ) \tan \left (b x\right )^{2} + 5 \, b x + 2 \, \Im \left ( \operatorname {Ci}\left (2 \, b x\right ) \right ) - 2 \, \Im \left ( \operatorname {Ci}\left (-2 \, b x\right ) \right ) + 4 \, \operatorname {Si}\left (2 \, b x\right ) - 5 \, \tan \left (b x\right )}{6 \, {\left (b^{3} \tan \left (b x\right )^{2} + b^{3}\right )}} \]
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Timed out. \[ \int x^2 \text {Si}(b x)^2 \, dx=\int x^2\,{\mathrm {sinint}\left (b\,x\right )}^2 \,d x \]
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