Optimal. Leaf size=35 \[ \frac {\text {Ci}(2 b x)}{2 b}-\frac {\text {Ci}(b x) \cos (b x)}{b}+\frac {\log (x)}{2 b} \]
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Rubi [A] time = 0.06, antiderivative size = 35, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.444, Rules used = {6518, 12, 3312, 3302} \[ \frac {\text {CosIntegral}(2 b x)}{2 b}-\frac {\text {CosIntegral}(b x) \cos (b x)}{b}+\frac {\log (x)}{2 b} \]
Antiderivative was successfully verified.
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Rule 12
Rule 3302
Rule 3312
Rule 6518
Rubi steps
\begin {align*} \int \text {Ci}(b x) \sin (b x) \, dx &=-\frac {\cos (b x) \text {Ci}(b x)}{b}+\int \frac {\cos ^2(b x)}{b x} \, dx\\ &=-\frac {\cos (b x) \text {Ci}(b x)}{b}+\frac {\int \frac {\cos ^2(b x)}{x} \, dx}{b}\\ &=-\frac {\cos (b x) \text {Ci}(b x)}{b}+\frac {\int \left (\frac {1}{2 x}+\frac {\cos (2 b x)}{2 x}\right ) \, dx}{b}\\ &=-\frac {\cos (b x) \text {Ci}(b x)}{b}+\frac {\log (x)}{2 b}+\frac {\int \frac {\cos (2 b x)}{x} \, dx}{2 b}\\ &=-\frac {\cos (b x) \text {Ci}(b x)}{b}+\frac {\text {Ci}(2 b x)}{2 b}+\frac {\log (x)}{2 b}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 37, normalized size = 1.06 \[ \frac {\text {Ci}(2 b x)}{2 b}-\frac {\text {Ci}(b x) \cos (b x)}{b}+\frac {\log (b x)}{2 b} \]
Antiderivative was successfully verified.
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fricas [F] time = 2.02, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\operatorname {Ci}\left (b x\right ) \sin \left (b x\right ), x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.22, size = 34, normalized size = 0.97 \[ -\frac {\cos \left (b x\right ) \operatorname {Ci}\left (b x\right )}{b} + \frac {\operatorname {Ci}\left (2 \, b x\right ) + \operatorname {Ci}\left (-2 \, b x\right ) + 2 \, \log \relax (x)}{4 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 34, normalized size = 0.97 \[ -\frac {\Ci \left (b x \right ) \cos \left (b x \right )}{b}+\frac {\ln \left (b x \right )}{2 b}+\frac {\Ci \left (2 b x \right )}{2 b} \]
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 \[ \int {\rm Ci}\left (b x\right ) \sin \left (b x\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.03 \[ \frac {\ln \relax (x)}{2\,b}+\frac {\mathrm {cosint}\left (2\,b\,x\right )}{2\,b}-\frac {\mathrm {cosint}\left (b\,x\right )\,\cos \left (b\,x\right )}{b} \]
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 \[ \int \sin {\left (b x \right )} \operatorname {Ci}{\left (b x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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