Optimal. Leaf size=35 \[ \frac{\cos (a) \text{CosIntegral}(b x)}{b}-\frac{\sin (a) \text{Si}(b x)}{b}-\frac{\log (x) \cos (a+b x)}{b} \]
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Rubi [A] time = 0.0780172, antiderivative size = 35, normalized size of antiderivative = 1., 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 = {2638, 2554, 12, 3303, 3299, 3302} \[ \frac{\cos (a) \text{CosIntegral}(b x)}{b}-\frac{\sin (a) \text{Si}(b x)}{b}-\frac{\log (x) \cos (a+b x)}{b} \]
Antiderivative was successfully verified.
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Rule 2638
Rule 2554
Rule 12
Rule 3303
Rule 3299
Rule 3302
Rubi steps
\begin{align*} \int \log (x) \sin (a+b x) \, dx &=-\frac{\cos (a+b x) \log (x)}{b}+\int \frac{\cos (a+b x)}{b x} \, dx\\ &=-\frac{\cos (a+b x) \log (x)}{b}+\frac{\int \frac{\cos (a+b x)}{x} \, dx}{b}\\ &=-\frac{\cos (a+b x) \log (x)}{b}+\frac{\cos (a) \int \frac{\cos (b x)}{x} \, dx}{b}-\frac{\sin (a) \int \frac{\sin (b x)}{x} \, dx}{b}\\ &=\frac{\cos (a) \text{Ci}(b x)}{b}-\frac{\cos (a+b x) \log (x)}{b}-\frac{\sin (a) \text{Si}(b x)}{b}\\ \end{align*}
Mathematica [A] time = 0.0423173, size = 30, normalized size = 0.86 \[ \frac{\cos (a) \text{CosIntegral}(b x)-\sin (a) \text{Si}(b x)-\log (x) \cos (a+b x)}{b} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.107, size = 80, normalized size = 2.3 \begin{align*} -{\frac{\cos \left ( bx+a \right ) \ln \left ( x \right ) }{b}}+{\frac{{\frac{i}{2}}{{\rm e}^{-ia}}\pi \,{\it csgn} \left ( bx \right ) }{b}}-{\frac{i{{\rm e}^{-ia}}{\it Si} \left ( bx \right ) }{b}}-{\frac{{{\rm e}^{-ia}}{\it Ei} \left ( 1,-ibx \right ) }{2\,b}}-{\frac{{{\rm e}^{ia}}{\it Ei} \left ( 1,-ibx \right ) }{2\,b}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [C] time = 1.15679, size = 77, normalized size = 2.2 \begin{align*} -\frac{\cos \left (b x + a\right ) \log \left (x\right )}{b} - \frac{{\left (E_{1}\left (i \, b x\right ) + E_{1}\left (-i \, b x\right )\right )} \cos \left (a\right ) -{\left (i \, E_{1}\left (i \, b x\right ) - i \, E_{1}\left (-i \, b x\right )\right )} \sin \left (a\right )}{2 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.28442, size = 149, normalized size = 4.26 \begin{align*} \frac{{\left (\operatorname{Ci}\left (b x\right ) + \operatorname{Ci}\left (-b x\right )\right )} \cos \left (a\right ) - 2 \, \cos \left (b x + a\right ) \log \left (x\right ) - 2 \, \sin \left (a\right ) \operatorname{Si}\left (b x\right )}{2 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \log{\left (x \right )} \sin{\left (a + b x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [C] time = 1.30149, size = 138, normalized size = 3.94 \begin{align*} -\frac{\cos \left (b x + a\right ) \log \left (x\right )}{b} - \frac{\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 )^{2} + 2 \, \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 ) + 4 \, \operatorname{Si}\left (b x\right ) \tan \left (\frac{1}{2} \, a\right ) - \Re \left ( \operatorname{Ci}\left (b x\right ) \right ) - \Re \left ( \operatorname{Ci}\left (-b x\right ) \right )}{2 \,{\left (b \tan \left (\frac{1}{2} \, a\right )^{2} + b\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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