Optimal. Leaf size=89 \[ \frac{3 \cos (a) \text{CosIntegral}(b x)}{4 b}-\frac{\cos (3 a) \text{CosIntegral}(3 b x)}{12 b}-\frac{3 \sin (a) \text{Si}(b x)}{4 b}+\frac{\sin (3 a) \text{Si}(3 b x)}{12 b}+\frac{\log (x) \cos ^3(a+b x)}{3 b}-\frac{\log (x) \cos (a+b x)}{b} \]
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Rubi [A] time = 0.517726, antiderivative size = 89, normalized size of antiderivative = 1., number of steps used = 15, number of rules used = 8, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.727, Rules used = {2633, 2554, 12, 6742, 3303, 3299, 3302, 3312} \[ \frac{3 \cos (a) \text{CosIntegral}(b x)}{4 b}-\frac{\cos (3 a) \text{CosIntegral}(3 b x)}{12 b}-\frac{3 \sin (a) \text{Si}(b x)}{4 b}+\frac{\sin (3 a) \text{Si}(3 b x)}{12 b}+\frac{\log (x) \cos ^3(a+b x)}{3 b}-\frac{\log (x) \cos (a+b x)}{b} \]
Antiderivative was successfully verified.
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Rule 2633
Rule 2554
Rule 12
Rule 6742
Rule 3303
Rule 3299
Rule 3302
Rule 3312
Rubi steps
\begin{align*} \int \log (x) \sin ^3(a+b x) \, dx &=-\frac{\cos (a+b x) \log (x)}{b}+\frac{\cos ^3(a+b x) \log (x)}{3 b}-\int \frac{\cos (a+b x) \left (-3+\cos ^2(a+b x)\right )}{3 b x} \, dx\\ &=-\frac{\cos (a+b x) \log (x)}{b}+\frac{\cos ^3(a+b x) \log (x)}{3 b}-\frac{\int \frac{\cos (a+b x) \left (-3+\cos ^2(a+b x)\right )}{x} \, dx}{3 b}\\ &=-\frac{\cos (a+b x) \log (x)}{b}+\frac{\cos ^3(a+b x) \log (x)}{3 b}-\frac{\int \left (-\frac{3 \cos (a+b x)}{x}+\frac{\cos ^3(a+b x)}{x}\right ) \, dx}{3 b}\\ &=-\frac{\cos (a+b x) \log (x)}{b}+\frac{\cos ^3(a+b x) \log (x)}{3 b}-\frac{\int \frac{\cos ^3(a+b x)}{x} \, dx}{3 b}+\frac{\int \frac{\cos (a+b x)}{x} \, dx}{b}\\ &=-\frac{\cos (a+b x) \log (x)}{b}+\frac{\cos ^3(a+b x) \log (x)}{3 b}-\frac{\int \left (\frac{3 \cos (a+b x)}{4 x}+\frac{\cos (3 a+3 b x)}{4 x}\right ) \, dx}{3 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{\cos ^3(a+b x) \log (x)}{3 b}-\frac{\sin (a) \text{Si}(b x)}{b}-\frac{\int \frac{\cos (3 a+3 b x)}{x} \, dx}{12 b}-\frac{\int \frac{\cos (a+b x)}{x} \, dx}{4 b}\\ &=\frac{\cos (a) \text{Ci}(b x)}{b}-\frac{\cos (a+b x) \log (x)}{b}+\frac{\cos ^3(a+b x) \log (x)}{3 b}-\frac{\sin (a) \text{Si}(b x)}{b}-\frac{\cos (a) \int \frac{\cos (b x)}{x} \, dx}{4 b}-\frac{\cos (3 a) \int \frac{\cos (3 b x)}{x} \, dx}{12 b}+\frac{\sin (a) \int \frac{\sin (b x)}{x} \, dx}{4 b}+\frac{\sin (3 a) \int \frac{\sin (3 b x)}{x} \, dx}{12 b}\\ &=\frac{3 \cos (a) \text{Ci}(b x)}{4 b}-\frac{\cos (3 a) \text{Ci}(3 b x)}{12 b}-\frac{\cos (a+b x) \log (x)}{b}+\frac{\cos ^3(a+b x) \log (x)}{3 b}-\frac{3 \sin (a) \text{Si}(b x)}{4 b}+\frac{\sin (3 a) \text{Si}(3 b x)}{12 b}\\ \end{align*}
Mathematica [A] time = 0.105273, size = 66, normalized size = 0.74 \[ \frac{9 \cos (a) \text{CosIntegral}(b x)-\cos (3 a) \text{CosIntegral}(3 b x)-9 \sin (a) \text{Si}(b x)+\sin (3 a) \text{Si}(3 b x)-9 \log (x) \cos (a+b x)+\log (x) \cos (3 (a+b x))}{12 b} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.129, size = 162, normalized size = 1.8 \begin{align*} -{\frac{3\,\cos \left ( bx+a \right ) \ln \left ( x \right ) }{4\,b}}+{\frac{\ln \left ( x \right ) \cos \left ( 3\,bx+3\,a \right ) }{12\,b}}-{\frac{{\frac{i}{24}}{{\rm e}^{-3\,ia}}\pi \,{\it csgn} \left ( bx \right ) }{b}}+{\frac{{\frac{i}{12}}{{\rm e}^{-3\,ia}}{\it Si} \left ( 3\,bx \right ) }{b}}+{\frac{{{\rm e}^{-3\,ia}}{\it Ei} \left ( 1,-3\,ibx \right ) }{24\,b}}+{\frac{{\frac{3\,i}{8}}{{\rm e}^{-ia}}\pi \,{\it csgn} \left ( bx \right ) }{b}}-{\frac{{\frac{3\,i}{4}}{{\rm e}^{-ia}}{\it Si} \left ( bx \right ) }{b}}-{\frac{3\,{{\rm e}^{-ia}}{\it Ei} \left ( 1,-ibx \right ) }{8\,b}}-{\frac{3\,{{\rm e}^{ia}}{\it Ei} \left ( 1,-ibx \right ) }{8\,b}}+{\frac{{{\rm e}^{3\,ia}}{\it Ei} \left ( 1,-3\,ibx \right ) }{24\,b}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [C] time = 1.25428, size = 149, normalized size = 1.67 \begin{align*} \frac{{\left (\cos \left (b x + a\right )^{3} - 3 \, \cos \left (b x + a\right )\right )} \log \left (x\right )}{3 \, b} + \frac{{\left (E_{1}\left (3 i \, b x\right ) + E_{1}\left (-3 i \, b x\right )\right )} \cos \left (3 \, a\right ) - 9 \,{\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 (3 i \, b x\right ) - i \, E_{1}\left (-3 i \, b x\right )\right )} \sin \left (3 \, a\right ) -{\left (-9 i \, E_{1}\left (i \, b x\right ) + 9 i \, E_{1}\left (-i \, b x\right )\right )} \sin \left (a\right )}{24 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.27604, size = 304, normalized size = 3.42 \begin{align*} -\frac{{\left (\operatorname{Ci}\left (3 \, b x\right ) + \operatorname{Ci}\left (-3 \, b x\right )\right )} \cos \left (3 \, a\right ) - 9 \,{\left (\operatorname{Ci}\left (b x\right ) + \operatorname{Ci}\left (-b x\right )\right )} \cos \left (a\right ) - 8 \,{\left (\cos \left (b x + a\right )^{3} - 3 \, \cos \left (b x + a\right )\right )} \log \left (x\right ) - 2 \, \sin \left (3 \, a\right ) \operatorname{Si}\left (3 \, b x\right ) + 18 \, \sin \left (a\right ) \operatorname{Si}\left (b x\right )}{24 \, 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 ^{3}{\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 = 1.33762, size = 613, normalized size = 6.89 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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