Optimal. Leaf size=35 \[ -\frac {\sin (a) \text {Ci}(b x)}{b}-\frac {\cos (a) \text {Si}(b x)}{b}+\frac {\log (x) \sin (a+b x)}{b} \]
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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 = 6, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {2637, 2554, 12, 3303, 3299, 3302} \[ -\frac {\sin (a) \text {CosIntegral}(b x)}{b}-\frac {\cos (a) \text {Si}(b x)}{b}+\frac {\log (x) \sin (a+b x)}{b} \]
Antiderivative was successfully verified.
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Rule 12
Rule 2554
Rule 2637
Rule 3299
Rule 3302
Rule 3303
Rubi steps
\begin {align*} \int \cos (a+b x) \log (x) \, dx &=\frac {\log (x) \sin (a+b x)}{b}-\int \frac {\sin (a+b x)}{b x} \, dx\\ &=\frac {\log (x) \sin (a+b x)}{b}-\frac {\int \frac {\sin (a+b x)}{x} \, dx}{b}\\ &=\frac {\log (x) \sin (a+b x)}{b}-\frac {\cos (a) \int \frac {\sin (b x)}{x} \, dx}{b}-\frac {\sin (a) \int \frac {\cos (b x)}{x} \, dx}{b}\\ &=-\frac {\text {Ci}(b x) \sin (a)}{b}+\frac {\log (x) \sin (a+b x)}{b}-\frac {\cos (a) \text {Si}(b x)}{b}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 30, normalized size = 0.86 \[ -\frac {\sin (a) \text {Ci}(b x)+\cos (a) \text {Si}(b x)-\log (x) \sin (a+b x)}{b} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.70, size = 38, normalized size = 1.09 \[ \frac {2 \, \log \relax (x) \sin \left (b x + a\right ) - {\left (\operatorname {Ci}\left (b x\right ) + \operatorname {Ci}\left (-b x\right )\right )} \sin \relax (a) - 2 \, \cos \relax (a) \operatorname {Si}\left (b x\right )}{2 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [C] time = 0.19, size = 108, normalized size = 3.09 \[ \frac {\log \relax (x) \sin \left (b x + a\right )}{b} + \frac {\Im \left (\operatorname {Ci}\left (b x\right ) \right ) \tan \left (\frac {1}{2} \, a\right )^{2} - \Im \left (\operatorname {Ci}\left (-b x\right ) \right ) \tan \left (\frac {1}{2} \, a\right )^{2} + 2 \, \operatorname {Si}\left (b x\right ) \tan \left (\frac {1}{2} \, a\right )^{2} - 2 \, \Re \left (\operatorname {Ci}\left (b x\right ) \right ) \tan \left (\frac {1}{2} \, a\right ) - 2 \, \Re \left (\operatorname {Ci}\left (-b x\right ) \right ) \tan \left (\frac {1}{2} \, a\right ) - \Im \left (\operatorname {Ci}\left (b x\right ) \right ) + \Im \left (\operatorname {Ci}\left (-b x\right ) \right ) - 2 \, \operatorname {Si}\left (b x\right )}{2 \, {\left (b \tan \left (\frac {1}{2} \, a\right )^{2} + b\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 1.11, size = 79, normalized size = 2.26 \[ \frac {i \Ei \left (1, -i b x \right ) {\mathrm e}^{-i a}}{2 b}-\frac {i \Ei \left (1, -i b x \right ) {\mathrm e}^{i a}}{2 b}-\frac {\Si \left (b x \right ) {\mathrm e}^{-i a}}{b}+\frac {\pi \,\mathrm {csgn}\left (b x \right ) {\mathrm e}^{-i a}}{2 b}+\frac {\ln \relax (x ) \sin \left (b x +a \right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [C] time = 1.02, size = 55, normalized size = 1.57 \[ \frac {\log \relax (x) \sin \left (b x + a\right )}{b} + \frac {{\left (i \, E_{1}\left (i \, b x\right ) - i \, E_{1}\left (-i \, b x\right )\right )} \cos \relax (a) + {\left (E_{1}\left (i \, b x\right ) + E_{1}\left (-i \, b x\right )\right )} \sin \relax (a)}{2 \, b} \]
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 \[ \int \cos \left (a+b\,x\right )\,\ln \relax (x) \,d x \]
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 \[ \int \log {\relax (x )} \cos {\left (a + b x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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