Optimal. Leaf size=98 \[ -\frac {\text {Ci}(2 b x)}{b^3}-\frac {2 \text {Si}(b x) \sin (b x)}{b^3}+\frac {\log (x)}{b^3}-\frac {5 \sin ^2(b x)}{4 b^3}+\frac {2 x \text {Si}(b x) \cos (b x)}{b^2}+\frac {x \sin (b x) \cos (b x)}{2 b^2}+\frac {x^2 \text {Si}(b x) \sin (b x)}{b}-\frac {x^2}{4 b} \]
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Rubi [A] time = 0.12, antiderivative size = 98, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 9, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.750, Rules used = {6519, 12, 3310, 30, 6513, 2564, 6517, 3312, 3302} \[ -\frac {\text {CosIntegral}(2 b x)}{b^3}-\frac {2 \text {Si}(b x) \sin (b x)}{b^3}+\frac {2 x \text {Si}(b x) \cos (b x)}{b^2}+\frac {\log (x)}{b^3}-\frac {5 \sin ^2(b x)}{4 b^3}+\frac {x \sin (b x) \cos (b x)}{2 b^2}+\frac {x^2 \text {Si}(b x) \sin (b x)}{b}-\frac {x^2}{4 b} \]
Antiderivative was successfully verified.
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Rule 12
Rule 30
Rule 2564
Rule 3302
Rule 3310
Rule 3312
Rule 6513
Rule 6517
Rule 6519
Rubi steps
\begin {align*} \int x^2 \cos (b x) \text {Si}(b x) \, dx &=\frac {x^2 \sin (b x) \text {Si}(b x)}{b}-\frac {2 \int x \sin (b x) \text {Si}(b x) \, dx}{b}-\int \frac {x \sin ^2(b x)}{b} \, dx\\ &=\frac {2 x \cos (b x) \text {Si}(b x)}{b^2}+\frac {x^2 \sin (b x) \text {Si}(b x)}{b}-\frac {2 \int \cos (b x) \text {Si}(b x) \, dx}{b^2}-\frac {\int x \sin ^2(b x) \, dx}{b}-\frac {2 \int \frac {\cos (b x) \sin (b x)}{b} \, dx}{b}\\ &=\frac {x \cos (b x) \sin (b x)}{2 b^2}-\frac {\sin ^2(b x)}{4 b^3}+\frac {2 x \cos (b x) \text {Si}(b x)}{b^2}-\frac {2 \sin (b x) \text {Si}(b x)}{b^3}+\frac {x^2 \sin (b x) \text {Si}(b x)}{b}-\frac {2 \int \cos (b x) \sin (b x) \, dx}{b^2}+\frac {2 \int \frac {\sin ^2(b x)}{b x} \, dx}{b^2}-\frac {\int x \, dx}{2 b}\\ &=-\frac {x^2}{4 b}+\frac {x \cos (b x) \sin (b x)}{2 b^2}-\frac {\sin ^2(b x)}{4 b^3}+\frac {2 x \cos (b x) \text {Si}(b x)}{b^2}-\frac {2 \sin (b x) \text {Si}(b x)}{b^3}+\frac {x^2 \sin (b x) \text {Si}(b x)}{b}+\frac {2 \int \frac {\sin ^2(b x)}{x} \, dx}{b^3}-\frac {2 \operatorname {Subst}(\int x \, dx,x,\sin (b x))}{b^3}\\ &=-\frac {x^2}{4 b}+\frac {x \cos (b x) \sin (b x)}{2 b^2}-\frac {5 \sin ^2(b x)}{4 b^3}+\frac {2 x \cos (b x) \text {Si}(b x)}{b^2}-\frac {2 \sin (b x) \text {Si}(b x)}{b^3}+\frac {x^2 \sin (b x) \text {Si}(b x)}{b}+\frac {2 \int \left (\frac {1}{2 x}-\frac {\cos (2 b x)}{2 x}\right ) \, dx}{b^3}\\ &=-\frac {x^2}{4 b}+\frac {\log (x)}{b^3}+\frac {x \cos (b x) \sin (b x)}{2 b^2}-\frac {5 \sin ^2(b x)}{4 b^3}+\frac {2 x \cos (b x) \text {Si}(b x)}{b^2}-\frac {2 \sin (b x) \text {Si}(b x)}{b^3}+\frac {x^2 \sin (b x) \text {Si}(b x)}{b}-\frac {\int \frac {\cos (2 b x)}{x} \, dx}{b^3}\\ &=-\frac {x^2}{4 b}-\frac {\text {Ci}(2 b x)}{b^3}+\frac {\log (x)}{b^3}+\frac {x \cos (b x) \sin (b x)}{2 b^2}-\frac {5 \sin ^2(b x)}{4 b^3}+\frac {2 x \cos (b x) \text {Si}(b x)}{b^2}-\frac {2 \sin (b x) \text {Si}(b x)}{b^3}+\frac {x^2 \sin (b x) \text {Si}(b x)}{b}\\ \end {align*}
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Mathematica [A] time = 0.11, size = 72, normalized size = 0.73 \[ \frac {8 \text {Si}(b x) \left (\left (b^2 x^2-2\right ) \sin (b x)+2 b x \cos (b x)\right )-2 b^2 x^2-8 \text {Ci}(2 b x)+2 b x \sin (2 b x)+5 \cos (2 b x)+8 \log (x)}{8 b^3} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.61, size = 0, normalized size = 0.00 \[ {\rm integral}\left (x^{2} \cos \left (b x\right ) \operatorname {Si}\left (b x\right ), x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.23, size = 82, normalized size = 0.84 \[ {\left (\frac {2 \, x \cos \left (b x\right )}{b^{2}} + \frac {{\left (b^{2} x^{2} - 2\right )} \sin \left (b x\right )}{b^{3}}\right )} \operatorname {Si}\left (b x\right ) - \frac {2 \, b^{2} x^{2} - 2 \, b x \sin \left (2 \, b x\right ) - 5 \, \cos \left (2 \, b x\right ) + 4 \, \operatorname {Ci}\left (2 \, b x\right ) + 4 \, \operatorname {Ci}\left (-2 \, b x\right ) - 8 \, \log \relax (x)}{8 \, b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 105, normalized size = 1.07 \[ \frac {x^{2} \Si \left (b x \right ) \sin \left (b x \right )}{b}+\frac {2 x \cos \left (b x \right ) \Si \left (b x \right )}{b^{2}}-\frac {2 \Si \left (b x \right ) \sin \left (b x \right )}{b^{3}}+\frac {x \cos \left (b x \right ) \sin \left (b x \right )}{2 b^{2}}-\frac {x^{2}}{4 b}-\frac {\sin ^{2}\left (b x \right )}{4 b^{3}}+\frac {\ln \left (b x \right )}{b^{3}}-\frac {\Ci \left (2 b x \right )}{b^{3}}+\frac {\cos ^{2}\left (b x \right )}{b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{2} {\rm Si}\left (b x\right ) \cos \left (b x\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int x^2\,\mathrm {sinint}\left (b\,x\right )\,\cos \left (b\,x\right ) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{2} \cos {\left (b x \right )} \operatorname {Si}{\left (b x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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