Optimal. Leaf size=60 \[ -\frac {\text {Ci}(2 b x)}{2 b^2}+\frac {\text {Ci}(b x) \cos (b x)}{b^2}-\frac {\log (x)}{2 b^2}-\frac {\sin ^2(b x)}{2 b^2}+\frac {x \text {Ci}(b x) \sin (b x)}{b} \]
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Rubi [A] time = 0.08, antiderivative size = 60, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.700, Rules used = {6514, 12, 2564, 30, 6518, 3312, 3302} \[ -\frac {\text {CosIntegral}(2 b x)}{2 b^2}+\frac {\text {CosIntegral}(b x) \cos (b x)}{b^2}-\frac {\log (x)}{2 b^2}-\frac {\sin ^2(b x)}{2 b^2}+\frac {x \text {CosIntegral}(b x) \sin (b x)}{b} \]
Antiderivative was successfully verified.
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Rule 12
Rule 30
Rule 2564
Rule 3302
Rule 3312
Rule 6514
Rule 6518
Rubi steps
\begin {align*} \int x \cos (b x) \text {Ci}(b x) \, dx &=\frac {x \text {Ci}(b x) \sin (b x)}{b}-\frac {\int \text {Ci}(b x) \sin (b x) \, dx}{b}-\int \frac {\cos (b x) \sin (b x)}{b} \, dx\\ &=\frac {\cos (b x) \text {Ci}(b x)}{b^2}+\frac {x \text {Ci}(b x) \sin (b x)}{b}-\frac {\int \frac {\cos ^2(b x)}{b x} \, dx}{b}-\frac {\int \cos (b x) \sin (b x) \, dx}{b}\\ &=\frac {\cos (b x) \text {Ci}(b x)}{b^2}+\frac {x \text {Ci}(b x) \sin (b x)}{b}-\frac {\int \frac {\cos ^2(b x)}{x} \, dx}{b^2}-\frac {\operatorname {Subst}(\int x \, dx,x,\sin (b x))}{b^2}\\ &=\frac {\cos (b x) \text {Ci}(b x)}{b^2}+\frac {x \text {Ci}(b x) \sin (b x)}{b}-\frac {\sin ^2(b x)}{2 b^2}-\frac {\int \left (\frac {1}{2 x}+\frac {\cos (2 b x)}{2 x}\right ) \, dx}{b^2}\\ &=\frac {\cos (b x) \text {Ci}(b x)}{b^2}-\frac {\log (x)}{2 b^2}+\frac {x \text {Ci}(b x) \sin (b x)}{b}-\frac {\sin ^2(b x)}{2 b^2}-\frac {\int \frac {\cos (2 b x)}{x} \, dx}{2 b^2}\\ &=\frac {\cos (b x) \text {Ci}(b x)}{b^2}-\frac {\text {Ci}(2 b x)}{2 b^2}-\frac {\log (x)}{2 b^2}+\frac {x \text {Ci}(b x) \sin (b x)}{b}-\frac {\sin ^2(b x)}{2 b^2}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 42, normalized size = 0.70 \[ \frac {-2 \text {Ci}(2 b x)+4 \text {Ci}(b x) (b x \sin (b x)+\cos (b x))+\cos (2 b x)-2 \log (x)}{4 b^2} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.22, size = 0, normalized size = 0.00 \[ {\rm integral}\left (x \cos \left (b x\right ) \operatorname {Ci}\left (b x\right ), x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.28, size = 53, normalized size = 0.88 \[ {\left (\frac {x \sin \left (b x\right )}{b} + \frac {\cos \left (b x\right )}{b^{2}}\right )} \operatorname {Ci}\left (b x\right ) + \frac {\cos \left (2 \, b x\right ) - \operatorname {Ci}\left (2 \, b x\right ) - \operatorname {Ci}\left (-2 \, b x\right ) - 2 \, \log \relax (x)}{4 \, b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 57, normalized size = 0.95 \[ \frac {x \Ci \left (b x \right ) \sin \left (b x \right )}{b}+\frac {\Ci \left (b x \right ) \cos \left (b x \right )}{b^{2}}-\frac {\ln \left (b x \right )}{2 b^{2}}-\frac {\Ci \left (2 b x \right )}{2 b^{2}}+\frac {\cos ^{2}\left (b x \right )}{2 b^{2}} \]
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 \[ \int x {\rm Ci}\left (b x\right ) \cos \left (b x\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int x\,\mathrm {cosint}\left (b\,x\right )\,\cos \left (b\,x\right ) \,d x \]
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 \[ \int x \cos {\left (b x \right )} \operatorname {Ci}{\left (b x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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