Optimal. Leaf size=28 \[ \frac{b \sin (c+d x)}{d^2}-\frac{(a+b x) \cos (c+d x)}{d} \]
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Rubi [A] time = 0.0166503, antiderivative size = 28, 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, 2637} \[ \frac{b \sin (c+d x)}{d^2}-\frac{(a+b x) \cos (c+d x)}{d} \]
Antiderivative was successfully verified.
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Rule 3296
Rule 2637
Rubi steps
\begin{align*} \int (a+b x) \sin (c+d x) \, dx &=-\frac{(a+b x) \cos (c+d x)}{d}+\frac{b \int \cos (c+d x) \, dx}{d}\\ &=-\frac{(a+b x) \cos (c+d x)}{d}+\frac{b \sin (c+d x)}{d^2}\\ \end{align*}
Mathematica [A] time = 0.0747545, size = 27, normalized size = 0.96 \[ \frac{b \sin (c+d x)-d (a+b x) \cos (c+d x)}{d^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 52, normalized size = 1.9 \begin{align*}{\frac{1}{d} \left ({\frac{b \left ( \sin \left ( dx+c \right ) - \left ( dx+c \right ) \cos \left ( dx+c \right ) \right ) }{d}}-\cos \left ( dx+c \right ) a+{\frac{cb\cos \left ( dx+c \right ) }{d}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.984978, size = 72, normalized size = 2.57 \begin{align*} -\frac{a \cos \left (d x + c\right ) - \frac{b c \cos \left (d x + c\right )}{d} + \frac{{\left ({\left (d x + c\right )} \cos \left (d x + c\right ) - \sin \left (d x + c\right )\right )} b}{d}}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.65849, size = 70, normalized size = 2.5 \begin{align*} -\frac{{\left (b d x + a d\right )} \cos \left (d x + c\right ) - b \sin \left (d x + c\right )}{d^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.242294, size = 46, normalized size = 1.64 \begin{align*} \begin{cases} - \frac{a \cos{\left (c + d x \right )}}{d} - \frac{b x \cos{\left (c + d x \right )}}{d} + \frac{b \sin{\left (c + d x \right )}}{d^{2}} & \text{for}\: d \neq 0 \\\left (a x + \frac{b x^{2}}{2}\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 = 1.10727, size = 42, normalized size = 1.5 \begin{align*} -\frac{{\left (b d x + a d\right )} \cos \left (d x + c\right )}{d^{2}} + \frac{b \sin \left (d x + c\right )}{d^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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