Optimal. Leaf size=25 \[ \frac {\text {Ci}(b x) \sin (b x)}{b}-\frac {\text {Si}(2 b x)}{2 b} \]
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Rubi [A] time = 0.04, antiderivative size = 25, 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 = {6512, 12, 4406, 3299} \[ \frac {\text {CosIntegral}(b x) \sin (b x)}{b}-\frac {\text {Si}(2 b x)}{2 b} \]
Antiderivative was successfully verified.
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Rule 12
Rule 3299
Rule 4406
Rule 6512
Rubi steps
\begin {align*} \int \cos (b x) \text {Ci}(b x) \, dx &=\frac {\text {Ci}(b x) \sin (b x)}{b}-\int \frac {\cos (b x) \sin (b x)}{b x} \, dx\\ &=\frac {\text {Ci}(b x) \sin (b x)}{b}-\frac {\int \frac {\cos (b x) \sin (b x)}{x} \, dx}{b}\\ &=\frac {\text {Ci}(b x) \sin (b x)}{b}-\frac {\int \frac {\sin (2 b x)}{2 x} \, dx}{b}\\ &=\frac {\text {Ci}(b x) \sin (b x)}{b}-\frac {\int \frac {\sin (2 b x)}{x} \, dx}{2 b}\\ &=\frac {\text {Ci}(b x) \sin (b x)}{b}-\frac {\text {Si}(2 b x)}{2 b}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 25, normalized size = 1.00 \[ \frac {\text {Ci}(b x) \sin (b x)}{b}-\frac {\text {Si}(2 b x)}{2 b} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.91, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\cos \left (b x\right ) \operatorname {Ci}\left (b x\right ), x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [C] time = 0.26, size = 40, normalized size = 1.60 \[ \frac {\operatorname {Ci}\left (b x\right ) \sin \left (b x\right )}{b} - \frac {\Im \left (\operatorname {Ci}\left (2 \, b x\right ) \right ) - \Im \left (\operatorname {Ci}\left (-2 \, b x\right ) \right ) + 2 \, \operatorname {Si}\left (2 \, b x\right )}{4 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 22, normalized size = 0.88 \[ \frac {\Ci \left (b x \right ) \sin \left (b x \right )-\frac {\Si \left (2 b x \right )}{2}}{b} \]
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 \[ \int {\rm Ci}\left (b x\right ) \cos \left (b x\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.04 \[ \int \mathrm {cosint}\left (b\,x\right )\,\cos \left (b\,x\right ) \,d x \]
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 \[ \int \cos {\left (b x \right )} \operatorname {Ci}{\left (b x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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