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 = 0.02, antiderivative size = 27, normalized size of antiderivative = 1.00, 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 2637
Rule 4569
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*}
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Mathematica [A] time = 0.05, 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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fricas [B] time = 0.62, size = 50, normalized size = 1.85 \[ \frac {b x \cos \left (-a + c\right ) - \cos \left (b x + c\right ) \cos \left (-a + c\right ) \sin \left (b x + c\right ) + \cos \left (b x + c\right )^{2} \sin \left (-a + c\right )}{2 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.12, size = 23, normalized size = 0.85 \[ \frac {1}{2} \, x \cos \left (a - c\right ) - \frac {\sin \left (2 \, b x + a + c\right )}{4 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 24, normalized size = 0.89 \[ \frac {x \cos \left (a -c \right )}{2}-\frac {\sin \left (2 b x +a +c \right )}{4 b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 23, normalized size = 0.85 \[ \frac {1}{2} \, x \cos \left (-a + c\right ) - \frac {\sin \left (2 \, b x + a + c\right )}{4 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.48, size = 36, normalized size = 1.33 \[ \left \{\begin {array}{cl} x\,\sin \relax (a)\,\sin \relax (c) & \text {\ if\ \ }b=0\\ \frac {x\,\cos \left (a-c\right )}{2}-\frac {\sin \left (a+c+2\,b\,x\right )}{4\,b} & \text {\ if\ \ }b\neq 0 \end {array}\right . \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.74, size = 58, normalized size = 2.15 \[ \begin {cases} \frac {x \sin {\left (a + b x \right )} \sin {\left (b x + c \right )}}{2} + \frac {x \cos {\left (a + b x \right )} \cos {\left (b x + c \right )}}{2} - \frac {\sin {\left (b x + c \right )} \cos {\left (a + b x \right )}}{2 b} & \text {for}\: b \neq 0 \\x \sin {\relax (a )} \sin {\relax (c )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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