Optimal. Leaf size=147 \[ -\frac {6 \text {Ci}(b x) \sin (b x)}{b^4}+\frac {3 \text {Si}(2 b x)}{b^4}-\frac {4 \sin (b x) \cos (b x)}{b^4}+\frac {6 x \text {Ci}(b x) \cos (b x)}{b^3}-\frac {5 x}{2 b^3}-\frac {3 x \sin ^2(b x)}{2 b^3}+\frac {x \cos ^2(b x)}{2 b^3}+\frac {3 x^2 \text {Ci}(b x) \sin (b x)}{b^2}+\frac {x^2 \sin (b x) \cos (b x)}{2 b^2}-\frac {x^3 \text {Ci}(b x) \cos (b x)}{b}+\frac {x^3}{6 b} \]
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Rubi [A] time = 0.18, antiderivative size = 147, normalized size of antiderivative = 1.00, number of steps used = 20, number of rules used = 11, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.917, Rules used = {6520, 12, 3311, 30, 2635, 8, 6514, 3443, 6512, 4406, 3299} \[ \frac {3 x^2 \text {CosIntegral}(b x) \sin (b x)}{b^2}-\frac {6 \text {CosIntegral}(b x) \sin (b x)}{b^4}+\frac {6 x \text {CosIntegral}(b x) \cos (b x)}{b^3}+\frac {3 \text {Si}(2 b x)}{b^4}+\frac {x^2 \sin (b x) \cos (b x)}{2 b^2}-\frac {5 x}{2 b^3}-\frac {3 x \sin ^2(b x)}{2 b^3}+\frac {x \cos ^2(b x)}{2 b^3}-\frac {4 \sin (b x) \cos (b x)}{b^4}-\frac {x^3 \text {CosIntegral}(b x) \cos (b x)}{b}+\frac {x^3}{6 b} \]
Antiderivative was successfully verified.
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Rule 8
Rule 12
Rule 30
Rule 2635
Rule 3299
Rule 3311
Rule 3443
Rule 4406
Rule 6512
Rule 6514
Rule 6520
Rubi steps
\begin {align*} \int x^3 \text {Ci}(b x) \sin (b x) \, dx &=-\frac {x^3 \cos (b x) \text {Ci}(b x)}{b}+\frac {3 \int x^2 \cos (b x) \text {Ci}(b x) \, dx}{b}+\int \frac {x^2 \cos ^2(b x)}{b} \, dx\\ &=-\frac {x^3 \cos (b x) \text {Ci}(b x)}{b}+\frac {3 x^2 \text {Ci}(b x) \sin (b x)}{b^2}-\frac {6 \int x \text {Ci}(b x) \sin (b x) \, dx}{b^2}+\frac {\int x^2 \cos ^2(b x) \, dx}{b}-\frac {3 \int \frac {x \cos (b x) \sin (b x)}{b} \, dx}{b}\\ &=\frac {x \cos ^2(b x)}{2 b^3}+\frac {6 x \cos (b x) \text {Ci}(b x)}{b^3}-\frac {x^3 \cos (b x) \text {Ci}(b x)}{b}+\frac {x^2 \cos (b x) \sin (b x)}{2 b^2}+\frac {3 x^2 \text {Ci}(b x) \sin (b x)}{b^2}-\frac {\int \cos ^2(b x) \, dx}{2 b^3}-\frac {6 \int \cos (b x) \text {Ci}(b x) \, dx}{b^3}-\frac {3 \int x \cos (b x) \sin (b x) \, dx}{b^2}-\frac {6 \int \frac {\cos ^2(b x)}{b} \, dx}{b^2}+\frac {\int x^2 \, dx}{2 b}\\ &=\frac {x^3}{6 b}+\frac {x \cos ^2(b x)}{2 b^3}+\frac {6 x \cos (b x) \text {Ci}(b x)}{b^3}-\frac {x^3 \cos (b x) \text {Ci}(b x)}{b}-\frac {\cos (b x) \sin (b x)}{4 b^4}+\frac {x^2 \cos (b x) \sin (b x)}{2 b^2}-\frac {6 \text {Ci}(b x) \sin (b x)}{b^4}+\frac {3 x^2 \text {Ci}(b x) \sin (b x)}{b^2}-\frac {3 x \sin ^2(b x)}{2 b^3}-\frac {\int 1 \, dx}{4 b^3}+\frac {3 \int \sin ^2(b x) \, dx}{2 b^3}-\frac {6 \int \cos ^2(b x) \, dx}{b^3}+\frac {6 \int \frac {\cos (b x) \sin (b x)}{b x} \, dx}{b^3}\\ &=-\frac {x}{4 b^3}+\frac {x^3}{6 b}+\frac {x \cos ^2(b x)}{2 b^3}+\frac {6 x \cos (b x) \text {Ci}(b x)}{b^3}-\frac {x^3 \cos (b x) \text {Ci}(b x)}{b}-\frac {4 \cos (b x) \sin (b x)}{b^4}+\frac {x^2 \cos (b x) \sin (b x)}{2 b^2}-\frac {6 \text {Ci}(b x) \sin (b x)}{b^4}+\frac {3 x^2 \text {Ci}(b x) \sin (b x)}{b^2}-\frac {3 x \sin ^2(b x)}{2 b^3}+\frac {6 \int \frac {\cos (b x) \sin (b x)}{x} \, dx}{b^4}+\frac {3 \int 1 \, dx}{4 b^3}-\frac {3 \int 1 \, dx}{b^3}\\ &=-\frac {5 x}{2 b^3}+\frac {x^3}{6 b}+\frac {x \cos ^2(b x)}{2 b^3}+\frac {6 x \cos (b x) \text {Ci}(b x)}{b^3}-\frac {x^3 \cos (b x) \text {Ci}(b x)}{b}-\frac {4 \cos (b x) \sin (b x)}{b^4}+\frac {x^2 \cos (b x) \sin (b x)}{2 b^2}-\frac {6 \text {Ci}(b x) \sin (b x)}{b^4}+\frac {3 x^2 \text {Ci}(b x) \sin (b x)}{b^2}-\frac {3 x \sin ^2(b x)}{2 b^3}+\frac {6 \int \frac {\sin (2 b x)}{2 x} \, dx}{b^4}\\ &=-\frac {5 x}{2 b^3}+\frac {x^3}{6 b}+\frac {x \cos ^2(b x)}{2 b^3}+\frac {6 x \cos (b x) \text {Ci}(b x)}{b^3}-\frac {x^3 \cos (b x) \text {Ci}(b x)}{b}-\frac {4 \cos (b x) \sin (b x)}{b^4}+\frac {x^2 \cos (b x) \sin (b x)}{2 b^2}-\frac {6 \text {Ci}(b x) \sin (b x)}{b^4}+\frac {3 x^2 \text {Ci}(b x) \sin (b x)}{b^2}-\frac {3 x \sin ^2(b x)}{2 b^3}+\frac {3 \int \frac {\sin (2 b x)}{x} \, dx}{b^4}\\ &=-\frac {5 x}{2 b^3}+\frac {x^3}{6 b}+\frac {x \cos ^2(b x)}{2 b^3}+\frac {6 x \cos (b x) \text {Ci}(b x)}{b^3}-\frac {x^3 \cos (b x) \text {Ci}(b x)}{b}-\frac {4 \cos (b x) \sin (b x)}{b^4}+\frac {x^2 \cos (b x) \sin (b x)}{2 b^2}-\frac {6 \text {Ci}(b x) \sin (b x)}{b^4}+\frac {3 x^2 \text {Ci}(b x) \sin (b x)}{b^2}-\frac {3 x \sin ^2(b x)}{2 b^3}+\frac {3 \text {Si}(2 b x)}{b^4}\\ \end {align*}
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Mathematica [A] time = 0.10, size = 94, normalized size = 0.64 \[ \frac {2 b^3 x^3-12 \text {Ci}(b x) \left (b x \left (b^2 x^2-6\right ) \cos (b x)-3 \left (b^2 x^2-2\right ) \sin (b x)\right )+3 b^2 x^2 \sin (2 b x)+36 \text {Si}(2 b x)-36 b x-24 \sin (2 b x)+12 b x \cos (2 b x)}{12 b^4} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.70, size = 0, normalized size = 0.00 \[ {\rm integral}\left (x^{3} \operatorname {Ci}\left (b x\right ) \sin \left (b x\right ), x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [C] time = 0.29, size = 181, normalized size = 1.23 \[ -{\left (\frac {{\left (b^{3} x^{3} - 6 \, b x\right )} \cos \left (b x\right )}{b^{4}} - \frac {3 \, {\left (b^{2} x^{2} - 2\right )} \sin \left (b x\right )}{b^{4}}\right )} \operatorname {Ci}\left (b x\right ) + \frac {b^{3} x^{3} \tan \left (b x\right )^{2} + b^{3} x^{3} + 3 \, b^{2} x^{2} \tan \left (b x\right ) - 24 \, b x \tan \left (b x\right )^{2} + 9 \, \Im \left (\operatorname {Ci}\left (2 \, b x\right ) \right ) \tan \left (b x\right )^{2} - 9 \, \Im \left (\operatorname {Ci}\left (-2 \, b x\right ) \right ) \tan \left (b x\right )^{2} + 18 \, \operatorname {Si}\left (2 \, b x\right ) \tan \left (b x\right )^{2} - 12 \, b x + 9 \, \Im \left (\operatorname {Ci}\left (2 \, b x\right ) \right ) - 9 \, \Im \left (\operatorname {Ci}\left (-2 \, b x\right ) \right ) + 18 \, \operatorname {Si}\left (2 \, b x\right ) - 24 \, \tan \left (b x\right )}{6 \, {\left (b^{4} \tan \left (b x\right )^{2} + b^{4}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 111, normalized size = 0.76 \[ \frac {\Ci \left (b x \right ) \left (-b^{3} x^{3} \cos \left (b x \right )+3 b^{2} x^{2} \sin \left (b x \right )-6 \sin \left (b x \right )+6 b x \cos \left (b x \right )\right )+b^{2} x^{2} \left (\frac {\sin \left (b x \right ) \cos \left (b x \right )}{2}+\frac {b x}{2}\right )+2 b x \left (\cos ^{2}\left (b x \right )\right )-4 \sin \left (b x \right ) \cos \left (b x \right )-4 b x -\frac {b^{3} x^{3}}{3}+3 \Si \left (2 b x \right )}{b^{4}} \]
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 \[ \int x^{3} {\rm Ci}\left (b x\right ) \sin \left (b x\right )\,{d x} \]
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 \[ \int x^3\,\mathrm {cosint}\left (b\,x\right )\,\sin \left (b\,x\right ) \,d x \]
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 \[ \int x^{3} \sin {\left (b x \right )} \operatorname {Ci}{\left (b x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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