Optimal. Leaf size=89 \[ -\frac {2 \text {Ci}(b x) \sin (b x)}{b^3}+\frac {\text {Si}(2 b x)}{b^3}-\frac {5 \sin (b x) \cos (b x)}{4 b^3}+\frac {2 x \text {Ci}(b x) \cos (b x)}{b^2}-\frac {3 x}{4 b^2}-\frac {x \sin ^2(b x)}{2 b^2}+\frac {x^2 \text {Ci}(b x) \sin (b x)}{b} \]
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Rubi [A] time = 0.11, antiderivative size = 89, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 9, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.750, Rules used = {6514, 12, 3443, 2635, 8, 6520, 6512, 4406, 3299} \[ -\frac {2 \text {CosIntegral}(b x) \sin (b x)}{b^3}+\frac {2 x \text {CosIntegral}(b x) \cos (b x)}{b^2}+\frac {\text {Si}(2 b x)}{b^3}-\frac {3 x}{4 b^2}-\frac {x \sin ^2(b x)}{2 b^2}-\frac {5 \sin (b x) \cos (b x)}{4 b^3}+\frac {x^2 \text {CosIntegral}(b x) \sin (b x)}{b} \]
Antiderivative was successfully verified.
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Rule 8
Rule 12
Rule 2635
Rule 3299
Rule 3443
Rule 4406
Rule 6512
Rule 6514
Rule 6520
Rubi steps
\begin {align*} \int x^2 \cos (b x) \text {Ci}(b x) \, dx &=\frac {x^2 \text {Ci}(b x) \sin (b x)}{b}-\frac {2 \int x \text {Ci}(b x) \sin (b x) \, dx}{b}-\int \frac {x \cos (b x) \sin (b x)}{b} \, dx\\ &=\frac {2 x \cos (b x) \text {Ci}(b x)}{b^2}+\frac {x^2 \text {Ci}(b x) \sin (b x)}{b}-\frac {2 \int \cos (b x) \text {Ci}(b x) \, dx}{b^2}-\frac {\int x \cos (b x) \sin (b x) \, dx}{b}-\frac {2 \int \frac {\cos ^2(b x)}{b} \, dx}{b}\\ &=\frac {2 x \cos (b x) \text {Ci}(b x)}{b^2}-\frac {2 \text {Ci}(b x) \sin (b x)}{b^3}+\frac {x^2 \text {Ci}(b x) \sin (b x)}{b}-\frac {x \sin ^2(b x)}{2 b^2}+\frac {\int \sin ^2(b x) \, dx}{2 b^2}-\frac {2 \int \cos ^2(b x) \, dx}{b^2}+\frac {2 \int \frac {\cos (b x) \sin (b x)}{b x} \, dx}{b^2}\\ &=\frac {2 x \cos (b x) \text {Ci}(b x)}{b^2}-\frac {5 \cos (b x) \sin (b x)}{4 b^3}-\frac {2 \text {Ci}(b x) \sin (b x)}{b^3}+\frac {x^2 \text {Ci}(b x) \sin (b x)}{b}-\frac {x \sin ^2(b x)}{2 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 {3 x}{4 b^2}+\frac {2 x \cos (b x) \text {Ci}(b x)}{b^2}-\frac {5 \cos (b x) \sin (b x)}{4 b^3}-\frac {2 \text {Ci}(b x) \sin (b x)}{b^3}+\frac {x^2 \text {Ci}(b x) \sin (b x)}{b}-\frac {x \sin ^2(b x)}{2 b^2}+\frac {2 \int \frac {\sin (2 b x)}{2 x} \, dx}{b^3}\\ &=-\frac {3 x}{4 b^2}+\frac {2 x \cos (b x) \text {Ci}(b x)}{b^2}-\frac {5 \cos (b x) \sin (b x)}{4 b^3}-\frac {2 \text {Ci}(b x) \sin (b x)}{b^3}+\frac {x^2 \text {Ci}(b x) \sin (b x)}{b}-\frac {x \sin ^2(b x)}{2 b^2}+\frac {\int \frac {\sin (2 b x)}{x} \, dx}{b^3}\\ &=-\frac {3 x}{4 b^2}+\frac {2 x \cos (b x) \text {Ci}(b x)}{b^2}-\frac {5 \cos (b x) \sin (b x)}{4 b^3}-\frac {2 \text {Ci}(b x) \sin (b x)}{b^3}+\frac {x^2 \text {Ci}(b x) \sin (b x)}{b}-\frac {x \sin ^2(b x)}{2 b^2}+\frac {\text {Si}(2 b x)}{b^3}\\ \end {align*}
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Mathematica [A] time = 0.09, size = 64, normalized size = 0.72 \[ \frac {8 \text {Ci}(b x) \left (\left (b^2 x^2-2\right ) \sin (b x)+2 b x \cos (b x)\right )+8 \text {Si}(2 b x)-8 b x-5 \sin (2 b x)+2 b x \cos (2 b x)}{8 b^3} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.87, size = 0, normalized size = 0.00 \[ {\rm integral}\left (x^{2} \cos \left (b x\right ) \operatorname {Ci}\left (b x\right ), x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [C] time = 2.83, size = 137, normalized size = 1.54 \[ {\left (\frac {2 \, x \cos \left (b x\right )}{b^{2}} + \frac {{\left (b^{2} x^{2} - 2\right )} \sin \left (b x\right )}{b^{3}}\right )} \operatorname {Ci}\left (b x\right ) - \frac {5 \, b x \tan \left (b x\right )^{2} - 2 \, \Im \left (\operatorname {Ci}\left (2 \, b x\right ) \right ) \tan \left (b x\right )^{2} + 2 \, \Im \left (\operatorname {Ci}\left (-2 \, b x\right ) \right ) \tan \left (b x\right )^{2} - 4 \, \operatorname {Si}\left (2 \, b x\right ) \tan \left (b x\right )^{2} + 3 \, b x - 2 \, \Im \left (\operatorname {Ci}\left (2 \, b x\right ) \right ) + 2 \, \Im \left (\operatorname {Ci}\left (-2 \, b x\right ) \right ) - 4 \, \operatorname {Si}\left (2 \, b x\right ) + 5 \, \tan \left (b x\right )}{4 \, {\left (b^{3} \tan \left (b x\right )^{2} + b^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 66, normalized size = 0.74 \[ \frac {\Ci \left (b x \right ) \left (b^{2} x^{2} \sin \left (b x \right )-2 \sin \left (b x \right )+2 b x \cos \left (b x \right )\right )+\frac {b x \left (\cos ^{2}\left (b x \right )\right )}{2}-\frac {5 \sin \left (b x \right ) \cos \left (b x \right )}{4}-\frac {5 b x}{4}+\Si \left (2 b x \right )}{b^{3}} \]
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 \[ \int x^{2} {\rm Ci}\left (b x\right ) \cos \left (b x\right )\,{d x} \]
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 \[ \int x^2\,\mathrm {cosint}\left (b\,x\right )\,\cos \left (b\,x\right ) \,d x \]
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 \[ \int x^{2} \cos {\left (b x \right )} \operatorname {Ci}{\left (b x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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