Optimal. Leaf size=118 \[ -\frac{2 a^2 (c+d x) \cos (e+f x)}{f}-\frac{a^2 (c+d x) \sin (e+f x) \cos (e+f x)}{2 f}+\frac{a^2 (c+d x)^2}{2 d}+\frac{1}{2} a^2 c x+\frac{a^2 d \sin ^2(e+f x)}{4 f^2}+\frac{2 a^2 d \sin (e+f x)}{f^2}+\frac{1}{4} a^2 d x^2 \]
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Rubi [A] time = 0.10374, antiderivative size = 118, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {3317, 3296, 2637, 3310} \[ -\frac{2 a^2 (c+d x) \cos (e+f x)}{f}-\frac{a^2 (c+d x) \sin (e+f x) \cos (e+f x)}{2 f}+\frac{a^2 (c+d x)^2}{2 d}+\frac{1}{2} a^2 c x+\frac{a^2 d \sin ^2(e+f x)}{4 f^2}+\frac{2 a^2 d \sin (e+f x)}{f^2}+\frac{1}{4} a^2 d x^2 \]
Antiderivative was successfully verified.
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Rule 3317
Rule 3296
Rule 2637
Rule 3310
Rubi steps
\begin{align*} \int (c+d x) (a+a \sin (e+f x))^2 \, dx &=\int \left (a^2 (c+d x)+2 a^2 (c+d x) \sin (e+f x)+a^2 (c+d x) \sin ^2(e+f x)\right ) \, dx\\ &=\frac{a^2 (c+d x)^2}{2 d}+a^2 \int (c+d x) \sin ^2(e+f x) \, dx+\left (2 a^2\right ) \int (c+d x) \sin (e+f x) \, dx\\ &=\frac{a^2 (c+d x)^2}{2 d}-\frac{2 a^2 (c+d x) \cos (e+f x)}{f}-\frac{a^2 (c+d x) \cos (e+f x) \sin (e+f x)}{2 f}+\frac{a^2 d \sin ^2(e+f x)}{4 f^2}+\frac{1}{2} a^2 \int (c+d x) \, dx+\frac{\left (2 a^2 d\right ) \int \cos (e+f x) \, dx}{f}\\ &=\frac{1}{2} a^2 c x+\frac{1}{4} a^2 d x^2+\frac{a^2 (c+d x)^2}{2 d}-\frac{2 a^2 (c+d x) \cos (e+f x)}{f}+\frac{2 a^2 d \sin (e+f x)}{f^2}-\frac{a^2 (c+d x) \cos (e+f x) \sin (e+f x)}{2 f}+\frac{a^2 d \sin ^2(e+f x)}{4 f^2}\\ \end{align*}
Mathematica [A] time = 1.04758, size = 80, normalized size = 0.68 \[ -\frac{a^2 (6 (e+f x) (d (e-f x)-2 c f)+2 f (c+d x) \sin (2 (e+f x))+16 f (c+d x) \cos (e+f x)-16 d \sin (e+f x)+d \cos (2 (e+f x)))}{8 f^2} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.02, size = 219, normalized size = 1.9 \begin{align*}{\frac{1}{f} \left ({\frac{{a}^{2}d}{f} \left ( \left ( fx+e \right ) \left ( -{\frac{\sin \left ( fx+e \right ) \cos \left ( fx+e \right ) }{2}}+{\frac{fx}{2}}+{\frac{e}{2}} \right ) -{\frac{ \left ( fx+e \right ) ^{2}}{4}}+{\frac{ \left ( \sin \left ( fx+e \right ) \right ) ^{2}}{4}} \right ) }+{a}^{2}c \left ( -{\frac{\sin \left ( fx+e \right ) \cos \left ( fx+e \right ) }{2}}+{\frac{fx}{2}}+{\frac{e}{2}} \right ) -{\frac{{a}^{2}de}{f} \left ( -{\frac{\sin \left ( fx+e \right ) \cos \left ( fx+e \right ) }{2}}+{\frac{fx}{2}}+{\frac{e}{2}} \right ) }+2\,{\frac{{a}^{2}d \left ( \sin \left ( fx+e \right ) - \left ( fx+e \right ) \cos \left ( fx+e \right ) \right ) }{f}}-2\,{a}^{2}c\cos \left ( fx+e \right ) +2\,{\frac{{a}^{2}de\cos \left ( fx+e \right ) }{f}}+{\frac{{a}^{2}d \left ( fx+e \right ) ^{2}}{2\,f}}+{a}^{2}c \left ( fx+e \right ) -{\frac{{a}^{2}de \left ( fx+e \right ) }{f}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.99571, size = 277, normalized size = 2.35 \begin{align*} \frac{2 \,{\left (2 \, f x + 2 \, e - \sin \left (2 \, f x + 2 \, e\right )\right )} a^{2} c + 8 \,{\left (f x + e\right )} a^{2} c + \frac{4 \,{\left (f x + e\right )}^{2} a^{2} d}{f} - \frac{2 \,{\left (2 \, f x + 2 \, e - \sin \left (2 \, f x + 2 \, e\right )\right )} a^{2} d e}{f} - \frac{8 \,{\left (f x + e\right )} a^{2} d e}{f} - 16 \, a^{2} c \cos \left (f x + e\right ) + \frac{16 \, a^{2} d e \cos \left (f x + e\right )}{f} + \frac{{\left (2 \,{\left (f x + e\right )}^{2} - 2 \,{\left (f x + e\right )} \sin \left (2 \, f x + 2 \, e\right ) - \cos \left (2 \, f x + 2 \, e\right )\right )} a^{2} d}{f} - \frac{16 \,{\left ({\left (f x + e\right )} \cos \left (f x + e\right ) - \sin \left (f x + e\right )\right )} a^{2} d}{f}}{8 \, f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.73688, size = 228, normalized size = 1.93 \begin{align*} \frac{3 \, a^{2} d f^{2} x^{2} + 6 \, a^{2} c f^{2} x - a^{2} d \cos \left (f x + e\right )^{2} - 8 \,{\left (a^{2} d f x + a^{2} c f\right )} \cos \left (f x + e\right ) + 2 \,{\left (4 \, a^{2} d -{\left (a^{2} d f x + a^{2} c f\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{4 \, f^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.844491, size = 219, normalized size = 1.86 \begin{align*} \begin{cases} \frac{a^{2} c x \sin ^{2}{\left (e + f x \right )}}{2} + \frac{a^{2} c x \cos ^{2}{\left (e + f x \right )}}{2} + a^{2} c x - \frac{a^{2} c \sin{\left (e + f x \right )} \cos{\left (e + f x \right )}}{2 f} - \frac{2 a^{2} c \cos{\left (e + f x \right )}}{f} + \frac{a^{2} d x^{2} \sin ^{2}{\left (e + f x \right )}}{4} + \frac{a^{2} d x^{2} \cos ^{2}{\left (e + f x \right )}}{4} + \frac{a^{2} d x^{2}}{2} - \frac{a^{2} d x \sin{\left (e + f x \right )} \cos{\left (e + f x \right )}}{2 f} - \frac{2 a^{2} d x \cos{\left (e + f x \right )}}{f} + \frac{2 a^{2} d \sin{\left (e + f x \right )}}{f^{2}} - \frac{a^{2} d \cos ^{2}{\left (e + f x \right )}}{4 f^{2}} & \text{for}\: f \neq 0 \\\left (a \sin{\left (e \right )} + a\right )^{2} \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 = 1.10989, size = 144, normalized size = 1.22 \begin{align*} \frac{3}{4} \, a^{2} d x^{2} + \frac{3}{2} \, a^{2} c x - \frac{a^{2} d \cos \left (2 \, f x + 2 \, e\right )}{8 \, f^{2}} + \frac{2 \, a^{2} d \sin \left (f x + e\right )}{f^{2}} - \frac{2 \,{\left (a^{2} d f x + a^{2} c f\right )} \cos \left (f x + e\right )}{f^{2}} - \frac{{\left (a^{2} d f x + a^{2} c f\right )} \sin \left (2 \, f x + 2 \, e\right )}{4 \, f^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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