Optimal. Leaf size=27 \[ \frac{\sin (a+2 b x-c)}{4 b}-\frac{1}{2} x \cos (a+c) \]
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Rubi [A] time = 0.025399, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {4569, 2637} \[ \frac{\sin (a+2 b x-c)}{4 b}-\frac{1}{2} x \cos (a+c) \]
Antiderivative was successfully verified.
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Rule 4569
Rule 2637
Rubi steps
\begin{align*} \int \sin (c-b x) \sin (a+b x) \, dx &=\int \left (-\frac{1}{2} \cos (a+c)+\frac{1}{2} \cos (a-c+2 b x)\right ) \, dx\\ &=-\frac{1}{2} x \cos (a+c)+\frac{1}{2} \int \cos (a-c+2 b x) \, dx\\ &=-\frac{1}{2} x \cos (a+c)+\frac{\sin (a-c+2 b x)}{4 b}\\ \end{align*}
Mathematica [A] time = 0.0338703, size = 26, normalized size = 0.96 \[ \frac{\sin (a+2 b x-c)-2 b x \cos (a+c)}{4 b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.01, size = 24, normalized size = 0.9 \begin{align*} -{\frac{x\cos \left ( a+c \right ) }{2}}+{\frac{\sin \left ( 2\,bx+a-c \right ) }{4\,b}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.00554, size = 31, normalized size = 1.15 \begin{align*} -\frac{1}{2} \, x \cos \left (a + c\right ) + \frac{\sin \left (2 \, b x + a - c\right )}{4 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.4156, size = 124, normalized size = 4.59 \begin{align*} -\frac{b x \cos \left (a + c\right ) - \cos \left (b x + a\right ) \cos \left (a + c\right ) \sin \left (b x + a\right ) + \cos \left (b x + a\right )^{2} \sin \left (a + c\right )}{2 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.21596, size = 61, normalized size = 2.26 \begin{align*} - \begin{cases} \frac{x \sin{\left (a + b x \right )} \sin{\left (b x - c \right )}}{2} + \frac{x \cos{\left (a + b x \right )} \cos{\left (b x - c \right )}}{2} - \frac{\sin{\left (b x - c \right )} \cos{\left (a + b x \right )}}{2 b} & \text{for}\: b \neq 0 \\- x \sin{\left (a \right )} \sin{\left (c \right )} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.11759, size = 31, normalized size = 1.15 \begin{align*} -\frac{1}{2} \, x \cos \left (a + c\right ) + \frac{\sin \left (2 \, b x + a - c\right )}{4 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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