Integrand size = 11, antiderivative size = 66 \[ \int \log (x) \sin ^2(a+b x) \, dx=-\frac {x}{2}+\frac {1}{2} x \log (x)+\frac {\operatorname {CosIntegral}(2 b x) \sin (2 a)}{4 b}-\frac {\cos (a+b x) \log (x) \sin (a+b x)}{2 b}+\frac {\cos (2 a) \text {Si}(2 b x)}{4 b} \]
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Time = 0.08 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.545, Rules used = {2715, 8, 2634, 3384, 3380, 3383} \[ \int \log (x) \sin ^2(a+b x) \, dx=\frac {\sin (2 a) \operatorname {CosIntegral}(2 b x)}{4 b}+\frac {\cos (2 a) \text {Si}(2 b x)}{4 b}-\frac {\log (x) \sin (a+b x) \cos (a+b x)}{2 b}-\frac {x}{2}+\frac {1}{2} x \log (x) \]
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Rule 8
Rule 2634
Rule 2715
Rule 3380
Rule 3383
Rule 3384
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} x \log (x)-\frac {\cos (a+b x) \log (x) \sin (a+b x)}{2 b}-\int \left (\frac {1}{2}-\frac {\sin (2 a+2 b x)}{4 b x}\right ) \, dx \\ & = -\frac {x}{2}+\frac {1}{2} x \log (x)-\frac {\cos (a+b x) \log (x) \sin (a+b x)}{2 b}+\frac {\int \frac {\sin (2 a+2 b x)}{x} \, dx}{4 b} \\ & = -\frac {x}{2}+\frac {1}{2} x \log (x)-\frac {\cos (a+b x) \log (x) \sin (a+b x)}{2 b}+\frac {\cos (2 a) \int \frac {\sin (2 b x)}{x} \, dx}{4 b}+\frac {\sin (2 a) \int \frac {\cos (2 b x)}{x} \, dx}{4 b} \\ & = -\frac {x}{2}+\frac {1}{2} x \log (x)+\frac {\text {Ci}(2 b x) \sin (2 a)}{4 b}-\frac {\cos (a+b x) \log (x) \sin (a+b x)}{2 b}+\frac {\cos (2 a) \text {Si}(2 b x)}{4 b} \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.76 \[ \int \log (x) \sin ^2(a+b x) \, dx=\frac {-2 b x+2 b x \log (x)+\operatorname {CosIntegral}(2 b x) \sin (2 a)-\log (x) \sin (2 (a+b x))+\cos (2 a) \text {Si}(2 b x)}{4 b} \]
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 1.60 (sec) , antiderivative size = 132, normalized size of antiderivative = 2.00
method | result | size |
risch | \(\frac {\ln \left (x \right ) x}{2}-\frac {\sin \left (2 b x +2 a \right ) \ln \left (x \right )}{4 b}-\frac {{\mathrm e}^{-2 i a} \pi \,\operatorname {csgn}\left (b x \right )}{8 b}+\frac {{\mathrm e}^{-2 i a} \operatorname {Si}\left (2 b x \right )}{4 b}-\frac {i {\mathrm e}^{-2 i a} \operatorname {Ei}_{1}\left (-2 i b x \right )}{8 b}+\frac {a \ln \left (i b x \right )}{2 b}-\frac {\ln \left (a +i \left (i b x +i a \right )\right ) a}{2 b}-\frac {x}{2}-\frac {a}{2 b}+\frac {i {\mathrm e}^{2 i a} \operatorname {Ei}_{1}\left (-2 i b x \right )}{8 b}\) | \(132\) |
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none
Time = 0.35 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.79 \[ \int \log (x) \sin ^2(a+b x) \, dx=\frac {2 \, b x \log \left (x\right ) - 2 \, \cos \left (b x + a\right ) \log \left (x\right ) \sin \left (b x + a\right ) - 2 \, b x + \operatorname {Ci}\left (2 \, b x\right ) \sin \left (2 \, a\right ) + \cos \left (2 \, a\right ) \operatorname {Si}\left (2 \, b x\right )}{4 \, b} \]
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\[ \int \log (x) \sin ^2(a+b x) \, dx=\int \log {\left (x \right )} \sin ^{2}{\left (a + b x \right )}\, dx \]
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Result contains complex when optimal does not.
Time = 0.28 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.20 \[ \int \log (x) \sin ^2(a+b x) \, dx=\frac {{\left (2 \, b x + 2 \, a - \sin \left (2 \, b x + 2 \, a\right )\right )} \log \left (x\right )}{4 \, b} - \frac {4 \, b x + {\left (i \, {\rm Ei}\left (2 i \, b x\right ) - i \, {\rm Ei}\left (-2 i \, b x\right )\right )} \cos \left (2 \, a\right ) + 4 \, a \log \left (x\right ) - {\left ({\rm Ei}\left (2 i \, b x\right ) + {\rm Ei}\left (-2 i \, b x\right )\right )} \sin \left (2 \, a\right )}{8 \, b} \]
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.34 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.86 \[ \int \log (x) \sin ^2(a+b x) \, dx=\frac {1}{4} \, {\left (2 \, x - \frac {\sin \left (2 \, b x + 2 \, a\right )}{b}\right )} \log \left (x\right ) - \frac {4 \, b x \tan \left (a\right )^{2} + \Im \left ( \operatorname {Ci}\left (2 \, b x\right ) \right ) \tan \left (a\right )^{2} - \Im \left ( \operatorname {Ci}\left (-2 \, b x\right ) \right ) \tan \left (a\right )^{2} + 2 \, \operatorname {Si}\left (2 \, b x\right ) \tan \left (a\right )^{2} + 4 \, b x - 2 \, \Re \left ( \operatorname {Ci}\left (2 \, b x\right ) \right ) \tan \left (a\right ) - 2 \, \Re \left ( \operatorname {Ci}\left (-2 \, b x\right ) \right ) \tan \left (a\right ) - \Im \left ( \operatorname {Ci}\left (2 \, b x\right ) \right ) + \Im \left ( \operatorname {Ci}\left (-2 \, b x\right ) \right ) - 2 \, \operatorname {Si}\left (2 \, b x\right )}{8 \, {\left (b \tan \left (a\right )^{2} + b\right )}} \]
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Timed out. \[ \int \log (x) \sin ^2(a+b x) \, dx=\int {\sin \left (a+b\,x\right )}^2\,\ln \left (x\right ) \,d x \]
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