Integrand size = 12, antiderivative size = 126 \[ \int x^3 \sin (b x) \text {Si}(b x) \, dx=-\frac {x^2}{b^2}-\frac {3 \operatorname {CosIntegral}(2 b x)}{b^4}+\frac {3 \log (x)}{b^4}+\frac {2 x \cos (b x) \sin (b x)}{b^3}-\frac {4 \sin ^2(b x)}{b^4}+\frac {x^2 \sin ^2(b x)}{2 b^2}+\frac {6 x \cos (b x) \text {Si}(b x)}{b^3}-\frac {x^3 \cos (b x) \text {Si}(b x)}{b}-\frac {6 \sin (b x) \text {Si}(b x)}{b^4}+\frac {3 x^2 \sin (b x) \text {Si}(b x)}{b^2} \]
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Time = 0.13 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.00, number of steps used = 18, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.833, Rules used = {6648, 12, 3524, 3391, 30, 6654, 2644, 6652, 3393, 3383} \[ \int x^3 \sin (b x) \text {Si}(b x) \, dx=-\frac {3 \operatorname {CosIntegral}(2 b x)}{b^4}-\frac {6 \text {Si}(b x) \sin (b x)}{b^4}+\frac {3 \log (x)}{b^4}-\frac {4 \sin ^2(b x)}{b^4}+\frac {6 x \text {Si}(b x) \cos (b x)}{b^3}+\frac {2 x \sin (b x) \cos (b x)}{b^3}+\frac {3 x^2 \text {Si}(b x) \sin (b x)}{b^2}-\frac {x^2}{b^2}+\frac {x^2 \sin ^2(b x)}{2 b^2}-\frac {x^3 \text {Si}(b x) \cos (b x)}{b} \]
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Rule 12
Rule 30
Rule 2644
Rule 3383
Rule 3391
Rule 3393
Rule 3524
Rule 6648
Rule 6652
Rule 6654
Rubi steps \begin{align*} \text {integral}& = -\frac {x^3 \cos (b x) \text {Si}(b x)}{b}+\frac {3 \int x^2 \cos (b x) \text {Si}(b x) \, dx}{b}+\int \frac {x^2 \cos (b x) \sin (b x)}{b} \, dx \\ & = -\frac {x^3 \cos (b x) \text {Si}(b x)}{b}+\frac {3 x^2 \sin (b x) \text {Si}(b x)}{b^2}-\frac {6 \int x \sin (b x) \text {Si}(b x) \, dx}{b^2}+\frac {\int x^2 \cos (b x) \sin (b x) \, dx}{b}-\frac {3 \int \frac {x \sin ^2(b x)}{b} \, dx}{b} \\ & = \frac {x^2 \sin ^2(b x)}{2 b^2}+\frac {6 x \cos (b x) \text {Si}(b x)}{b^3}-\frac {x^3 \cos (b x) \text {Si}(b x)}{b}+\frac {3 x^2 \sin (b x) \text {Si}(b x)}{b^2}-\frac {6 \int \cos (b x) \text {Si}(b x) \, dx}{b^3}-\frac {\int x \sin ^2(b x) \, dx}{b^2}-\frac {3 \int x \sin ^2(b x) \, dx}{b^2}-\frac {6 \int \frac {\cos (b x) \sin (b x)}{b} \, dx}{b^2} \\ & = \frac {2 x \cos (b x) \sin (b x)}{b^3}-\frac {\sin ^2(b x)}{b^4}+\frac {x^2 \sin ^2(b x)}{2 b^2}+\frac {6 x \cos (b x) \text {Si}(b x)}{b^3}-\frac {x^3 \cos (b x) \text {Si}(b x)}{b}-\frac {6 \sin (b x) \text {Si}(b x)}{b^4}+\frac {3 x^2 \sin (b x) \text {Si}(b x)}{b^2}-\frac {6 \int \cos (b x) \sin (b x) \, dx}{b^3}+\frac {6 \int \frac {\sin ^2(b x)}{b x} \, dx}{b^3}-\frac {\int x \, dx}{2 b^2}-\frac {3 \int x \, dx}{2 b^2} \\ & = -\frac {x^2}{b^2}+\frac {2 x \cos (b x) \sin (b x)}{b^3}-\frac {\sin ^2(b x)}{b^4}+\frac {x^2 \sin ^2(b x)}{2 b^2}+\frac {6 x \cos (b x) \text {Si}(b x)}{b^3}-\frac {x^3 \cos (b x) \text {Si}(b x)}{b}-\frac {6 \sin (b x) \text {Si}(b x)}{b^4}+\frac {3 x^2 \sin (b x) \text {Si}(b x)}{b^2}+\frac {6 \int \frac {\sin ^2(b x)}{x} \, dx}{b^4}-\frac {6 \text {Subst}(\int x \, dx,x,\sin (b x))}{b^4} \\ & = -\frac {x^2}{b^2}+\frac {2 x \cos (b x) \sin (b x)}{b^3}-\frac {4 \sin ^2(b x)}{b^4}+\frac {x^2 \sin ^2(b x)}{2 b^2}+\frac {6 x \cos (b x) \text {Si}(b x)}{b^3}-\frac {x^3 \cos (b x) \text {Si}(b x)}{b}-\frac {6 \sin (b x) \text {Si}(b x)}{b^4}+\frac {3 x^2 \sin (b x) \text {Si}(b x)}{b^2}+\frac {6 \int \left (\frac {1}{2 x}-\frac {\cos (2 b x)}{2 x}\right ) \, dx}{b^4} \\ & = -\frac {x^2}{b^2}+\frac {3 \log (x)}{b^4}+\frac {2 x \cos (b x) \sin (b x)}{b^3}-\frac {4 \sin ^2(b x)}{b^4}+\frac {x^2 \sin ^2(b x)}{2 b^2}+\frac {6 x \cos (b x) \text {Si}(b x)}{b^3}-\frac {x^3 \cos (b x) \text {Si}(b x)}{b}-\frac {6 \sin (b x) \text {Si}(b x)}{b^4}+\frac {3 x^2 \sin (b x) \text {Si}(b x)}{b^2}-\frac {3 \int \frac {\cos (2 b x)}{x} \, dx}{b^4} \\ & = -\frac {x^2}{b^2}-\frac {3 \operatorname {CosIntegral}(2 b x)}{b^4}+\frac {3 \log (x)}{b^4}+\frac {2 x \cos (b x) \sin (b x)}{b^3}-\frac {4 \sin ^2(b x)}{b^4}+\frac {x^2 \sin ^2(b x)}{2 b^2}+\frac {6 x \cos (b x) \text {Si}(b x)}{b^3}-\frac {x^3 \cos (b x) \text {Si}(b x)}{b}-\frac {6 \sin (b x) \text {Si}(b x)}{b^4}+\frac {3 x^2 \sin (b x) \text {Si}(b x)}{b^2} \\ \end{align*}
Time = 0.13 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.74 \[ \int x^3 \sin (b x) \text {Si}(b x) \, dx=-\frac {3 b^2 x^2-8 \cos (2 b x)+b^2 x^2 \cos (2 b x)+12 \operatorname {CosIntegral}(2 b x)-12 \log (x)-4 b x \sin (2 b x)+4 \left (b x \left (-6+b^2 x^2\right ) \cos (b x)-3 \left (-2+b^2 x^2\right ) \sin (b x)\right ) \text {Si}(b x)}{4 b^4} \]
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Time = 1.04 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.10
method | result | size |
derivativedivides | \(\frac {\operatorname {Si}\left (b x \right ) \left (-b^{3} x^{3} \cos \left (b x \right )+3 b^{2} x^{2} \sin \left (b x \right )-6 \sin \left (b x \right )+6 b x \cos \left (b x \right )\right )-\frac {b^{2} x^{2} \cos \left (b x \right )^{2}}{2}+b x \left (\frac {\sin \left (b x \right ) \cos \left (b x \right )}{2}+\frac {b x}{2}\right )+\frac {b^{2} x^{2}}{2}-\sin \left (b x \right )^{2}-3 b x \left (-\frac {\sin \left (b x \right ) \cos \left (b x \right )}{2}+\frac {b x}{2}\right )+3 \cos \left (b x \right )^{2}+3 \ln \left (b x \right )-3 \,\operatorname {Ci}\left (2 b x \right )}{b^{4}}\) | \(138\) |
default | \(\frac {\operatorname {Si}\left (b x \right ) \left (-b^{3} x^{3} \cos \left (b x \right )+3 b^{2} x^{2} \sin \left (b x \right )-6 \sin \left (b x \right )+6 b x \cos \left (b x \right )\right )-\frac {b^{2} x^{2} \cos \left (b x \right )^{2}}{2}+b x \left (\frac {\sin \left (b x \right ) \cos \left (b x \right )}{2}+\frac {b x}{2}\right )+\frac {b^{2} x^{2}}{2}-\sin \left (b x \right )^{2}-3 b x \left (-\frac {\sin \left (b x \right ) \cos \left (b x \right )}{2}+\frac {b x}{2}\right )+3 \cos \left (b x \right )^{2}+3 \ln \left (b x \right )-3 \,\operatorname {Ci}\left (2 b x \right )}{b^{4}}\) | \(138\) |
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Time = 0.27 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.73 \[ \int x^3 \sin (b x) \text {Si}(b x) \, dx=-\frac {b^{2} x^{2} + {\left (b^{2} x^{2} - 8\right )} \cos \left (b x\right )^{2} + 2 \, {\left (b^{3} x^{3} - 6 \, b x\right )} \cos \left (b x\right ) \operatorname {Si}\left (b x\right ) - 2 \, {\left (2 \, b x \cos \left (b x\right ) + 3 \, {\left (b^{2} x^{2} - 2\right )} \operatorname {Si}\left (b x\right )\right )} \sin \left (b x\right ) + 6 \, \operatorname {Ci}\left (2 \, b x\right ) - 6 \, \log \left (x\right )}{2 \, b^{4}} \]
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\[ \int x^3 \sin (b x) \text {Si}(b x) \, dx=\int x^{3} \sin {\left (b x \right )} \operatorname {Si}{\left (b x \right )}\, dx \]
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\[ \int x^3 \sin (b x) \text {Si}(b x) \, dx=\int { x^{3} \sin \left (b x\right ) \operatorname {Si}\left (b x\right ) \,d x } \]
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Time = 0.28 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.84 \[ \int x^3 \sin (b x) \text {Si}(b x) \, dx=-{\left (\frac {{\left (b^{3} x^{3} - 6 \, b x\right )} \cos \left (b x\right )}{b^{4}} - \frac {3 \, {\left (b^{2} x^{2} - 2\right )} \sin \left (b x\right )}{b^{4}}\right )} \operatorname {Si}\left (b x\right ) - \frac {b^{2} x^{2} \cos \left (2 \, b x\right ) + 3 \, b^{2} x^{2} - 4 \, b x \sin \left (2 \, b x\right ) - 8 \, \cos \left (2 \, b x\right ) + 6 \, \operatorname {Ci}\left (2 \, b x\right ) + 6 \, \operatorname {Ci}\left (-2 \, b x\right ) - 12 \, \log \left (x\right )}{4 \, b^{4}} \]
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Timed out. \[ \int x^3 \sin (b x) \text {Si}(b x) \, dx=\int x^3\,\mathrm {sinint}\left (b\,x\right )\,\sin \left (b\,x\right ) \,d x \]
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