Integrand size = 35, antiderivative size = 50 \[ \int \cos (a+b x) \log \left (\cos \left (\frac {a}{2}+\frac {b x}{2}\right ) \sin \left (\frac {a}{2}+\frac {b x}{2}\right )\right ) \, dx=-\frac {\sin (a+b x)}{b}+\frac {\log \left (\cos \left (\frac {a}{2}+\frac {b x}{2}\right ) \sin \left (\frac {a}{2}+\frac {b x}{2}\right )\right ) \sin (a+b x)}{b} \]
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Time = 0.02 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.057, Rules used = {2717, 2634} \[ \int \cos (a+b x) \log \left (\cos \left (\frac {a}{2}+\frac {b x}{2}\right ) \sin \left (\frac {a}{2}+\frac {b x}{2}\right )\right ) \, dx=\frac {\sin (a+b x) \log \left (\sin \left (\frac {a}{2}+\frac {b x}{2}\right ) \cos \left (\frac {a}{2}+\frac {b x}{2}\right )\right )}{b}-\frac {\sin (a+b x)}{b} \]
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Rule 2634
Rule 2717
Rubi steps \begin{align*} \text {integral}& = \frac {\log \left (\cos \left (\frac {a}{2}+\frac {b x}{2}\right ) \sin \left (\frac {a}{2}+\frac {b x}{2}\right )\right ) \sin (a+b x)}{b}-\int \cos (a+b x) \, dx \\ & = -\frac {\sin (a+b x)}{b}+\frac {\log \left (\cos \left (\frac {a}{2}+\frac {b x}{2}\right ) \sin \left (\frac {a}{2}+\frac {b x}{2}\right )\right ) \sin (a+b x)}{b} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.66 \[ \int \cos (a+b x) \log \left (\cos \left (\frac {a}{2}+\frac {b x}{2}\right ) \sin \left (\frac {a}{2}+\frac {b x}{2}\right )\right ) \, dx=-\frac {\sin (a+b x)}{b}+\frac {\log \left (\frac {1}{2} \sin (a+b x)\right ) \sin (a+b x)}{b} \]
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Time = 7.66 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.60
method | result | size |
default | \(\frac {\ln \left (\frac {\sin \left (b x +a \right )}{2}\right ) \sin \left (b x +a \right )-\sin \left (b x +a \right )}{b}\) | \(30\) |
risch | \(\text {Expression too large to display}\) | \(1389\) |
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Time = 0.36 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.30 \[ \int \cos (a+b x) \log \left (\cos \left (\frac {a}{2}+\frac {b x}{2}\right ) \sin \left (\frac {a}{2}+\frac {b x}{2}\right )\right ) \, dx=\frac {2 \, {\left (\cos \left (\frac {1}{2} \, b x + \frac {1}{2} \, a\right ) \log \left (\cos \left (\frac {1}{2} \, b x + \frac {1}{2} \, a\right ) \sin \left (\frac {1}{2} \, b x + \frac {1}{2} \, a\right )\right ) \sin \left (\frac {1}{2} \, b x + \frac {1}{2} \, a\right ) - \cos \left (\frac {1}{2} \, b x + \frac {1}{2} \, a\right ) \sin \left (\frac {1}{2} \, b x + \frac {1}{2} \, a\right )\right )}}{b} \]
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\[ \int \cos (a+b x) \log \left (\cos \left (\frac {a}{2}+\frac {b x}{2}\right ) \sin \left (\frac {a}{2}+\frac {b x}{2}\right )\right ) \, dx=\int \log {\left (\sin {\left (\frac {a}{2} + \frac {b x}{2} \right )} \cos {\left (\frac {a}{2} + \frac {b x}{2} \right )} \right )} \cos {\left (a + b x \right )}\, dx \]
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Time = 0.21 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.84 \[ \int \cos (a+b x) \log \left (\cos \left (\frac {a}{2}+\frac {b x}{2}\right ) \sin \left (\frac {a}{2}+\frac {b x}{2}\right )\right ) \, dx=\frac {\log \left (\cos \left (\frac {1}{2} \, b x + \frac {1}{2} \, a\right ) \sin \left (\frac {1}{2} \, b x + \frac {1}{2} \, a\right )\right ) \sin \left (b x + a\right )}{b} - \frac {\sin \left (b x + a\right )}{b} \]
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Exception generated. \[ \int \cos (a+b x) \log \left (\cos \left (\frac {a}{2}+\frac {b x}{2}\right ) \sin \left (\frac {a}{2}+\frac {b x}{2}\right )\right ) \, dx=\text {Exception raised: TypeError} \]
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Time = 1.70 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.58 \[ \int \cos (a+b x) \log \left (\cos \left (\frac {a}{2}+\frac {b x}{2}\right ) \sin \left (\frac {a}{2}+\frac {b x}{2}\right )\right ) \, dx=-\frac {\sin \left (a+b\,x\right )-\ln \left (\frac {\sin \left (a+b\,x\right )}{2}\right )\,\sin \left (a+b\,x\right )}{b} \]
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