Optimal. Leaf size=27 \[ \frac{d \cos (a+b x)}{b^2}+\frac{(c+d x) \sin (a+b x)}{b} \]
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Rubi [A] time = 0.0164785, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {3296, 2638} \[ \frac{d \cos (a+b x)}{b^2}+\frac{(c+d x) \sin (a+b x)}{b} \]
Antiderivative was successfully verified.
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Rule 3296
Rule 2638
Rubi steps
\begin{align*} \int (c+d x) \cos (a+b x) \, dx &=\frac{(c+d x) \sin (a+b x)}{b}-\frac{d \int \sin (a+b x) \, dx}{b}\\ &=\frac{d \cos (a+b x)}{b^2}+\frac{(c+d x) \sin (a+b x)}{b}\\ \end{align*}
Mathematica [A] time = 0.0561993, size = 26, normalized size = 0.96 \[ \frac{b (c+d x) \sin (a+b x)+d \cos (a+b x)}{b^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.029, size = 51, normalized size = 1.9 \begin{align*}{\frac{1}{b} \left ({\frac{d \left ( \cos \left ( bx+a \right ) + \left ( bx+a \right ) \sin \left ( bx+a \right ) \right ) }{b}}-{\frac{da\sin \left ( bx+a \right ) }{b}}+c\sin \left ( bx+a \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.959584, size = 68, normalized size = 2.52 \begin{align*} \frac{c \sin \left (b x + a\right ) - \frac{a d \sin \left (b x + a\right )}{b} + \frac{{\left ({\left (b x + a\right )} \sin \left (b x + a\right ) + \cos \left (b x + a\right )\right )} d}{b}}{b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.05773, size = 69, normalized size = 2.56 \begin{align*} \frac{d \cos \left (b x + a\right ) +{\left (b d x + b c\right )} \sin \left (b x + a\right )}{b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.233663, size = 46, normalized size = 1.7 \begin{align*} \begin{cases} \frac{c \sin{\left (a + b x \right )}}{b} + \frac{d x \sin{\left (a + b x \right )}}{b} + \frac{d \cos{\left (a + b x \right )}}{b^{2}} & \text{for}\: b \neq 0 \\\left (c x + \frac{d x^{2}}{2}\right ) \cos{\left (a \right )} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.09986, size = 41, normalized size = 1.52 \begin{align*} \frac{d \cos \left (b x + a\right )}{b^{2}} + \frac{{\left (b d x + b c\right )} \sin \left (b x + a\right )}{b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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