Optimal. Leaf size=43 \[ -\frac{\cos (a+x (b-d)-c)}{2 (b-d)}-\frac{\cos (a+x (b+d)+c)}{2 (b+d)} \]
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Rubi [A] time = 0.0357136, antiderivative size = 43, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {4574, 2638} \[ -\frac{\cos (a+x (b-d)-c)}{2 (b-d)}-\frac{\cos (a+x (b+d)+c)}{2 (b+d)} \]
Antiderivative was successfully verified.
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Rule 4574
Rule 2638
Rubi steps
\begin{align*} \int \cos (c+d x) \sin (a+b x) \, dx &=\int \left (\frac{1}{2} \sin (a-c+(b-d) x)+\frac{1}{2} \sin (a+c+(b+d) x)\right ) \, dx\\ &=\frac{1}{2} \int \sin (a-c+(b-d) x) \, dx+\frac{1}{2} \int \sin (a+c+(b+d) x) \, dx\\ &=-\frac{\cos (a-c+(b-d) x)}{2 (b-d)}-\frac{\cos (a+c+(b+d) x)}{2 (b+d)}\\ \end{align*}
Mathematica [A] time = 0.195882, size = 43, normalized size = 1. \[ -\frac{\cos (a+x (b-d)-c)}{2 (b-d)}-\frac{\cos (a+x (b+d)+c)}{2 (b+d)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.014, size = 40, normalized size = 0.9 \begin{align*} -{\frac{\cos \left ( a-c+ \left ( b-d \right ) x \right ) }{2\,b-2\,d}}-{\frac{\cos \left ( a+c+ \left ( b+d \right ) x \right ) }{2\,b+2\,d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.17808, size = 54, normalized size = 1.26 \begin{align*} -\frac{\cos \left (b x + d x + a + c\right )}{2 \,{\left (b + d\right )}} - \frac{\cos \left (-b x + d x - a + c\right )}{2 \,{\left (b - d\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.473851, size = 100, normalized size = 2.33 \begin{align*} -\frac{b \cos \left (b x + a\right ) \cos \left (d x + c\right ) + d \sin \left (b x + a\right ) \sin \left (d x + c\right )}{b^{2} - d^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 2.87205, size = 155, normalized size = 3.6 \begin{align*} \begin{cases} x \sin{\left (a \right )} \cos{\left (c \right )} & \text{for}\: b = 0 \wedge d = 0 \\\frac{x \sin{\left (a - d x \right )} \cos{\left (c + d x \right )}}{2} + \frac{x \sin{\left (c + d x \right )} \cos{\left (a - d x \right )}}{2} + \frac{\sin{\left (a - d x \right )} \sin{\left (c + d x \right )}}{2 d} & \text{for}\: b = - d \\\frac{x \sin{\left (a + d x \right )} \cos{\left (c + d x \right )}}{2} - \frac{x \sin{\left (c + d x \right )} \cos{\left (a + d x \right )}}{2} - \frac{\cos{\left (a + d x \right )} \cos{\left (c + d x \right )}}{2 d} & \text{for}\: b = d \\- \frac{b \cos{\left (a + b x \right )} \cos{\left (c + d x \right )}}{b^{2} - d^{2}} - \frac{d \sin{\left (a + b x \right )} \sin{\left (c + d x \right )}}{b^{2} - d^{2}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.13548, size = 54, normalized size = 1.26 \begin{align*} -\frac{\cos \left (b x + d x + a + c\right )}{2 \,{\left (b + d\right )}} - \frac{\cos \left (b x - d x + a - c\right )}{2 \,{\left (b - d\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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