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 = 0.03, antiderivative size = 50, normalized size of antiderivative = 1.00, 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 2554
Rule 2637
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*}
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Mathematica [A] time = 0.01, 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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fricas [A] time = 0.49, size = 65, normalized size = 1.30 \[ \frac {2 \, {\left (\cos \left (\frac {1}{2} \, b x + \frac {1}{2} \, a\right ) \log \left (\cos \left (\frac {1}{2} \, b x + \frac {1}{2} \, a\right ) \sin \left (\frac {1}{2} \, b x + \frac {1}{2} \, a\right )\right ) \sin \left (\frac {1}{2} \, b x + \frac {1}{2} \, a\right ) - \cos \left (\frac {1}{2} \, b x + \frac {1}{2} \, a\right ) \sin \left (\frac {1}{2} \, b x + \frac {1}{2} \, a\right )\right )}}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.93, size = 42, normalized size = 0.84 \[ \frac {\log \left (\cos \left (\frac {1}{2} \, b x + \frac {1}{2} \, a\right ) \sin \left (\frac {1}{2} \, b x + \frac {1}{2} \, a\right )\right ) \sin \left (b x + a\right )}{b} - \frac {\sin \left (b x + a\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.74, size = 32, normalized size = 0.64 \[ \frac {\ln \left (\frac {\sin \left (b x +a \right )}{2}\right ) \sin \left (b x +a \right )}{b}-\frac {\sin \left (b x +a \right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.54, size = 42, normalized size = 0.84 \[ \frac {\log \left (\cos \left (\frac {1}{2} \, b x + \frac {1}{2} \, a\right ) \sin \left (\frac {1}{2} \, b x + \frac {1}{2} \, a\right )\right ) \sin \left (b x + a\right )}{b} - \frac {\sin \left (b x + a\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.52, size = 29, normalized size = 0.58 \[ -\frac {\sin \left (a+b\,x\right )-\ln \left (\frac {\sin \left (a+b\,x\right )}{2}\right )\,\sin \left (a+b\,x\right )}{b} \]
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 \[ \int \log {\left (\sin {\left (\frac {a}{2} + \frac {b x}{2} \right )} \cos {\left (\frac {a}{2} + \frac {b x}{2} \right )} \right )} \cos {\left (a + b x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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