Optimal. Leaf size=45 \[ \frac{a (c+d x)^2}{2 d}-\frac{b (c+d x) \cos (e+f x)}{f}+\frac{b d \sin (e+f x)}{f^2} \]
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Rubi [A] time = 0.0423611, antiderivative size = 45, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.188, Rules used = {3317, 3296, 2637} \[ \frac{a (c+d x)^2}{2 d}-\frac{b (c+d x) \cos (e+f x)}{f}+\frac{b d \sin (e+f x)}{f^2} \]
Antiderivative was successfully verified.
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Rule 3317
Rule 3296
Rule 2637
Rubi steps
\begin{align*} \int (c+d x) (a+b \sin (e+f x)) \, dx &=\int (a (c+d x)+b (c+d x) \sin (e+f x)) \, dx\\ &=\frac{a (c+d x)^2}{2 d}+b \int (c+d x) \sin (e+f x) \, dx\\ &=\frac{a (c+d x)^2}{2 d}-\frac{b (c+d x) \cos (e+f x)}{f}+\frac{(b d) \int \cos (e+f x) \, dx}{f}\\ &=\frac{a (c+d x)^2}{2 d}-\frac{b (c+d x) \cos (e+f x)}{f}+\frac{b d \sin (e+f x)}{f^2}\\ \end{align*}
Mathematica [A] time = 0.109109, size = 43, normalized size = 0.96 \[ \frac{1}{2} a x (2 c+d x)-\frac{b (c+d x) \cos (e+f x)}{f}+\frac{b d \sin (e+f x)}{f^2} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.006, size = 90, normalized size = 2. \begin{align*}{\frac{1}{f} \left ({\frac{da \left ( fx+e \right ) ^{2}}{2\,f}}+ac \left ( fx+e \right ) -{\frac{ade \left ( fx+e \right ) }{f}}+{\frac{bd \left ( \sin \left ( fx+e \right ) - \left ( fx+e \right ) \cos \left ( fx+e \right ) \right ) }{f}}-cb\cos \left ( fx+e \right ) +{\frac{bde\cos \left ( fx+e \right ) }{f}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 0.972822, size = 126, normalized size = 2.8 \begin{align*} \frac{2 \,{\left (f x + e\right )} a c + \frac{{\left (f x + e\right )}^{2} a d}{f} - \frac{2 \,{\left (f x + e\right )} a d e}{f} - 2 \, b c \cos \left (f x + e\right ) + \frac{2 \, b d e \cos \left (f x + e\right )}{f} - \frac{2 \,{\left ({\left (f x + e\right )} \cos \left (f x + e\right ) - \sin \left (f x + e\right )\right )} b d}{f}}{2 \, f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.62692, size = 126, normalized size = 2.8 \begin{align*} \frac{a d f^{2} x^{2} + 2 \, a c f^{2} x + 2 \, b d \sin \left (f x + e\right ) - 2 \,{\left (b d f x + b c f\right )} \cos \left (f x + e\right )}{2 \, f^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.342548, size = 68, normalized size = 1.51 \begin{align*} \begin{cases} a c x + \frac{a d x^{2}}{2} - \frac{b c \cos{\left (e + f x \right )}}{f} - \frac{b d x \cos{\left (e + f x \right )}}{f} + \frac{b d \sin{\left (e + f x \right )}}{f^{2}} & \text{for}\: f \neq 0 \\\left (a + b \sin{\left (e \right )}\right ) \left (c x + \frac{d x^{2}}{2}\right ) & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.75076, size = 63, normalized size = 1.4 \begin{align*} \frac{1}{2} \, a d x^{2} + a c x + \frac{b d \sin \left (f x + e\right )}{f^{2}} - \frac{{\left (b d f x + b c f\right )} \cos \left (f x + e\right )}{f^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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