Optimal. Leaf size=89 \[ \frac {3 \cos (a) \text {Ci}(b x)}{4 b}-\frac {\cos (3 a) \text {Ci}(3 b x)}{12 b}-\frac {3 \sin (a) \text {Si}(b x)}{4 b}+\frac {\sin (3 a) \text {Si}(3 b x)}{12 b}+\frac {\log (x) \cos ^3(a+b x)}{3 b}-\frac {\log (x) \cos (a+b x)}{b} \]
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Rubi [A] time = 0.52, antiderivative size = 89, normalized size of antiderivative = 1.00, 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 12
Rule 2554
Rule 2633
Rule 3299
Rule 3302
Rule 3303
Rule 3312
Rule 6742
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*}
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Mathematica [A] time = 0.12, size = 66, normalized size = 0.74 \[ \frac {9 \cos (a) \text {Ci}(b x)-\cos (3 a) \text {Ci}(3 b x)-9 \sin (a) \text {Si}(b x)+\sin (3 a) \text {Si}(3 b x)-9 \log (x) \cos (a+b x)+\log (x) \cos (3 (a+b x))}{12 b} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.72, size = 76, normalized size = 0.85 \[ -\frac {{\left (\operatorname {Ci}\left (3 \, b x\right ) + \operatorname {Ci}\left (-3 \, b x\right )\right )} \cos \left (3 \, a\right ) - 9 \, {\left (\operatorname {Ci}\left (b x\right ) + \operatorname {Ci}\left (-b x\right )\right )} \cos \relax (a) - 8 \, {\left (\cos \left (b x + a\right )^{3} - 3 \, \cos \left (b x + a\right )\right )} \log \relax (x) - 2 \, \sin \left (3 \, a\right ) \operatorname {Si}\left (3 \, b x\right ) + 18 \, \sin \relax (a) \operatorname {Si}\left (b x\right )}{24 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [C] time = 0.24, size = 454, normalized size = 5.10 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 1.40, size = 162, normalized size = 1.82 \[ \frac {\Ei \left (1, -3 i b x \right ) {\mathrm e}^{-3 i a}}{24 b}+\frac {\Ei \left (1, -3 i b x \right ) {\mathrm e}^{3 i a}}{24 b}-\frac {3 \Ei \left (1, -i b x \right ) {\mathrm e}^{-i a}}{8 b}-\frac {3 \Ei \left (1, -i b x \right ) {\mathrm e}^{i a}}{8 b}-\frac {3 i \Si \left (b x \right ) {\mathrm e}^{-i a}}{4 b}+\frac {i \Si \left (3 b x \right ) {\mathrm e}^{-3 i a}}{12 b}-\frac {3 \cos \left (b x +a \right ) \ln \relax (x )}{4 b}+\frac {\cos \left (3 b x +3 a \right ) \ln \relax (x )}{12 b}-\frac {i \pi \,\mathrm {csgn}\left (b x \right ) {\mathrm e}^{-3 i a}}{24 b}+\frac {3 i \pi \,\mathrm {csgn}\left (b x \right ) {\mathrm e}^{-i a}}{8 b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [C] time = 1.11, size = 110, normalized size = 1.24 \[ \frac {{\left (\cos \left (b x + a\right )^{3} - 3 \, \cos \left (b x + a\right )\right )} \log \relax (x)}{3 \, b} + \frac {{\left (E_{1}\left (3 i \, b x\right ) + E_{1}\left (-3 i \, b x\right )\right )} \cos \left (3 \, a\right ) - 9 \, {\left (E_{1}\left (i \, b x\right ) + E_{1}\left (-i \, b x\right )\right )} \cos \relax (a) - {\left (i \, E_{1}\left (3 i \, b x\right ) - i \, E_{1}\left (-3 i \, b x\right )\right )} \sin \left (3 \, a\right ) - {\left (-9 i \, E_{1}\left (i \, b x\right ) + 9 i \, E_{1}\left (-i \, b x\right )\right )} \sin \relax (a)}{24 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int {\sin \left (a+b\,x\right )}^3\,\ln \relax (x) \,d x \]
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 \[ \int \log {\relax (x )} \sin ^{3}{\left (a + b x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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