Optimal. Leaf size=27 \[ \frac{1}{2} x \cos (a-c)-\frac{\sin (a+2 b x+c)}{4 b} \]
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Rubi [A] time = 0.0243191, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {4569, 2637} \[ \frac{1}{2} x \cos (a-c)-\frac{\sin (a+2 b x+c)}{4 b} \]
Antiderivative was successfully verified.
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Rule 4569
Rule 2637
Rubi steps
\begin{align*} \int \sin (a+b x) \sin (c+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 = 0.0476423, 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 = 0.011, 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 = 0.999916, 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 [B] time = 2.4508, size = 127, normalized size = 4.7 \begin{align*} \frac{b x \cos \left (-a + c\right ) - \cos \left (b x + c\right ) \cos \left (-a + c\right ) \sin \left (b x + c\right ) + \cos \left (b x + c\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 = 1.39209, size = 58, normalized size = 2.15 \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 = 1.13145, 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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