Optimal. Leaf size=50 \[ \frac{\sin (a+b x) \log \left (\sin \left (\frac{a}{2}+\frac{b x}{2}\right ) \cos \left (\frac{a}{2}+\frac{b x}{2}\right )\right )}{b}-\frac{\sin (a+b x)}{b} \]
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Rubi [A] time = 0.026015, antiderivative size = 50, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.057, Rules used = {2637, 2554} \[ \frac{\sin (a+b x) \log \left (\sin \left (\frac{a}{2}+\frac{b x}{2}\right ) \cos \left (\frac{a}{2}+\frac{b x}{2}\right )\right )}{b}-\frac{\sin (a+b x)}{b} \]
Antiderivative was successfully verified.
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Rule 2637
Rule 2554
Rubi steps
\begin{align*} \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{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*}
Mathematica [A] time = 0.0101918, size = 33, normalized size = 0.66 \[ \frac{\sin (a+b x) \log \left (\frac{1}{2} \sin (a+b x)\right )}{b}-\frac{\sin (a+b x)}{b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.031, size = 32, normalized size = 0.6 \begin{align*}{\frac{\sin \left ( bx+a \right ) }{b}\ln \left ({\frac{\sin \left ( bx+a \right ) }{2}} \right ) }-{\frac{\sin \left ( bx+a \right ) }{b}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.07812, size = 57, normalized size = 1.14 \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.60892, size = 189, normalized size = 3.78 \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.47756, size = 57, normalized size = 1.14 \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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