Optimal. Leaf size=28 \[ \frac{d \sin (a+b x)}{b^2}-\frac{(c+d x) \cos (a+b x)}{b} \]
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Rubi [A] time = 0.0162482, 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} \[ \frac{d \sin (a+b x)}{b^2}-\frac{(c+d x) \cos (a+b x)}{b} \]
Antiderivative was successfully verified.
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Rule 3296
Rule 2637
Rubi steps
\begin{align*} \int (c+d x) \sin (a+b x) \, dx &=-\frac{(c+d x) \cos (a+b x)}{b}+\frac{d \int \cos (a+b x) \, dx}{b}\\ &=-\frac{(c+d x) \cos (a+b x)}{b}+\frac{d \sin (a+b x)}{b^2}\\ \end{align*}
Mathematica [A] time = 0.0703575, size = 27, normalized size = 0.96 \[ \frac{d \sin (a+b x)-b (c+d x) \cos (a+b x)}{b^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 52, normalized size = 1.9 \begin{align*}{\frac{1}{b} \left ({\frac{d \left ( \sin \left ( bx+a \right ) - \left ( bx+a \right ) \cos \left ( bx+a \right ) \right ) }{b}}+{\frac{da\cos \left ( bx+a \right ) }{b}}-c\cos \left ( bx+a \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.01222, size = 72, normalized size = 2.57 \begin{align*} -\frac{c \cos \left (b x + a\right ) - \frac{a d \cos \left (b x + a\right )}{b} + \frac{{\left ({\left (b x + a\right )} \cos \left (b x + a\right ) - \sin \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 = 1.66359, size = 70, normalized size = 2.5 \begin{align*} -\frac{{\left (b d x + b c\right )} \cos \left (b x + a\right ) - d \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 = 0.235433, size = 46, normalized size = 1.64 \begin{align*} \begin{cases} - \frac{c \cos{\left (a + b x \right )}}{b} - \frac{d x \cos{\left (a + b x \right )}}{b} + \frac{d \sin{\left (a + b x \right )}}{b^{2}} & \text{for}\: b \neq 0 \\\left (c x + \frac{d x^{2}}{2}\right ) \sin{\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 = 1.1022, size = 42, normalized size = 1.5 \begin{align*} -\frac{{\left (b d x + b c\right )} \cos \left (b x + a\right )}{b^{2}} + \frac{d \sin \left (b x + a\right )}{b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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