Optimal. Leaf size=98 \[ -\frac{a^2 c (4 A+B) \cos ^3(e+f x)}{12 f}+\frac{a^2 c (4 A+B) \sin (e+f x) \cos (e+f x)}{8 f}+\frac{1}{8} a^2 c x (4 A+B)-\frac{B c \cos ^3(e+f x) \left (a^2 \sin (e+f x)+a^2\right )}{4 f} \]
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Rubi [A] time = 0.148919, antiderivative size = 98, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 34, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.147, Rules used = {2967, 2860, 2669, 2635, 8} \[ -\frac{a^2 c (4 A+B) \cos ^3(e+f x)}{12 f}+\frac{a^2 c (4 A+B) \sin (e+f x) \cos (e+f x)}{8 f}+\frac{1}{8} a^2 c x (4 A+B)-\frac{B c \cos ^3(e+f x) \left (a^2 \sin (e+f x)+a^2\right )}{4 f} \]
Antiderivative was successfully verified.
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Rule 2967
Rule 2860
Rule 2669
Rule 2635
Rule 8
Rubi steps
\begin{align*} \int (a+a \sin (e+f x))^2 (A+B \sin (e+f x)) (c-c \sin (e+f x)) \, dx &=(a c) \int \cos ^2(e+f x) (a+a \sin (e+f x)) (A+B \sin (e+f x)) \, dx\\ &=-\frac{B c \cos ^3(e+f x) \left (a^2+a^2 \sin (e+f x)\right )}{4 f}+\frac{1}{4} (a (4 A+B) c) \int \cos ^2(e+f x) (a+a \sin (e+f x)) \, dx\\ &=-\frac{a^2 (4 A+B) c \cos ^3(e+f x)}{12 f}-\frac{B c \cos ^3(e+f x) \left (a^2+a^2 \sin (e+f x)\right )}{4 f}+\frac{1}{4} \left (a^2 (4 A+B) c\right ) \int \cos ^2(e+f x) \, dx\\ &=-\frac{a^2 (4 A+B) c \cos ^3(e+f x)}{12 f}+\frac{a^2 (4 A+B) c \cos (e+f x) \sin (e+f x)}{8 f}-\frac{B c \cos ^3(e+f x) \left (a^2+a^2 \sin (e+f x)\right )}{4 f}+\frac{1}{8} \left (a^2 (4 A+B) c\right ) \int 1 \, dx\\ &=\frac{1}{8} a^2 (4 A+B) c x-\frac{a^2 (4 A+B) c \cos ^3(e+f x)}{12 f}+\frac{a^2 (4 A+B) c \cos (e+f x) \sin (e+f x)}{8 f}-\frac{B c \cos ^3(e+f x) \left (a^2+a^2 \sin (e+f x)\right )}{4 f}\\ \end{align*}
Mathematica [A] time = 0.786035, size = 67, normalized size = 0.68 \[ -\frac{a^2 c (24 (A+B) \cos (e+f x)+8 (A+B) \cos (3 (e+f x))-12 f x (4 A+B)-24 A \sin (2 (e+f x))+3 B \sin (4 (e+f x)))}{96 f} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.027, size = 186, normalized size = 1.9 \begin{align*}{\frac{1}{f} \left ({\frac{A{a}^{2}c \left ( 2+ \left ( \sin \left ( fx+e \right ) \right ) ^{2} \right ) \cos \left ( fx+e \right ) }{3}}-A{a}^{2}c \left ( -{\frac{\sin \left ( fx+e \right ) \cos \left ( fx+e \right ) }{2}}+{\frac{fx}{2}}+{\frac{e}{2}} \right ) -B{a}^{2}c \left ( -{\frac{\cos \left ( fx+e \right ) }{4} \left ( \left ( \sin \left ( fx+e \right ) \right ) ^{3}+{\frac{3\,\sin \left ( fx+e \right ) }{2}} \right ) }+{\frac{3\,fx}{8}}+{\frac{3\,e}{8}} \right ) +{\frac{B{a}^{2}c \left ( 2+ \left ( \sin \left ( fx+e \right ) \right ) ^{2} \right ) \cos \left ( fx+e \right ) }{3}}-A{a}^{2}c\cos \left ( fx+e \right ) +B{a}^{2}c \left ( -{\frac{\sin \left ( fx+e \right ) \cos \left ( fx+e \right ) }{2}}+{\frac{fx}{2}}+{\frac{e}{2}} \right ) +A{a}^{2}c \left ( fx+e \right ) -B{a}^{2}c\cos \left ( fx+e \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.971595, size = 242, normalized size = 2.47 \begin{align*} -\frac{32 \,{\left (\cos \left (f x + e\right )^{3} - 3 \, \cos \left (f x + e\right )\right )} A a^{2} c + 24 \,{\left (2 \, f x + 2 \, e - \sin \left (2 \, f x + 2 \, e\right )\right )} A a^{2} c - 96 \,{\left (f x + e\right )} A a^{2} c + 32 \,{\left (\cos \left (f x + e\right )^{3} - 3 \, \cos \left (f x + e\right )\right )} B a^{2} c + 3 \,{\left (12 \, f x + 12 \, e + \sin \left (4 \, f x + 4 \, e\right ) - 8 \, \sin \left (2 \, f x + 2 \, e\right )\right )} B a^{2} c - 24 \,{\left (2 \, f x + 2 \, e - \sin \left (2 \, f x + 2 \, e\right )\right )} B a^{2} c + 96 \, A a^{2} c \cos \left (f x + e\right ) + 96 \, B a^{2} c \cos \left (f x + e\right )}{96 \, f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.40412, size = 190, normalized size = 1.94 \begin{align*} -\frac{8 \,{\left (A + B\right )} a^{2} c \cos \left (f x + e\right )^{3} - 3 \,{\left (4 \, A + B\right )} a^{2} c f x + 3 \,{\left (2 \, B a^{2} c \cos \left (f x + e\right )^{3} -{\left (4 \, A + B\right )} a^{2} c \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{24 \, f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.79887, size = 396, normalized size = 4.04 \begin{align*} \begin{cases} - \frac{A a^{2} c x \sin ^{2}{\left (e + f x \right )}}{2} - \frac{A a^{2} c x \cos ^{2}{\left (e + f x \right )}}{2} + A a^{2} c x + \frac{A a^{2} c \sin ^{2}{\left (e + f x \right )} \cos{\left (e + f x \right )}}{f} + \frac{A a^{2} c \sin{\left (e + f x \right )} \cos{\left (e + f x \right )}}{2 f} + \frac{2 A a^{2} c \cos ^{3}{\left (e + f x \right )}}{3 f} - \frac{A a^{2} c \cos{\left (e + f x \right )}}{f} - \frac{3 B a^{2} c x \sin ^{4}{\left (e + f x \right )}}{8} - \frac{3 B a^{2} c x \sin ^{2}{\left (e + f x \right )} \cos ^{2}{\left (e + f x \right )}}{4} + \frac{B a^{2} c x \sin ^{2}{\left (e + f x \right )}}{2} - \frac{3 B a^{2} c x \cos ^{4}{\left (e + f x \right )}}{8} + \frac{B a^{2} c x \cos ^{2}{\left (e + f x \right )}}{2} + \frac{5 B a^{2} c \sin ^{3}{\left (e + f x \right )} \cos{\left (e + f x \right )}}{8 f} + \frac{B a^{2} c \sin ^{2}{\left (e + f x \right )} \cos{\left (e + f x \right )}}{f} + \frac{3 B a^{2} c \sin{\left (e + f x \right )} \cos ^{3}{\left (e + f x \right )}}{8 f} - \frac{B a^{2} c \sin{\left (e + f x \right )} \cos{\left (e + f x \right )}}{2 f} + \frac{2 B a^{2} c \cos ^{3}{\left (e + f x \right )}}{3 f} - \frac{B a^{2} c \cos{\left (e + f x \right )}}{f} & \text{for}\: f \neq 0 \\x \left (A + B \sin{\left (e \right )}\right ) \left (a \sin{\left (e \right )} + a\right )^{2} \left (- c \sin{\left (e \right )} + 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 = 1.16806, size = 150, normalized size = 1.53 \begin{align*} -\frac{B a^{2} c \sin \left (4 \, f x + 4 \, e\right )}{32 \, f} + \frac{A a^{2} c \sin \left (2 \, f x + 2 \, e\right )}{4 \, f} + \frac{1}{8} \,{\left (4 \, A a^{2} c + B a^{2} c\right )} x - \frac{{\left (A a^{2} c + B a^{2} c\right )} \cos \left (3 \, f x + 3 \, e\right )}{12 \, f} - \frac{{\left (A a^{2} c + B a^{2} c\right )} \cos \left (f x + e\right )}{4 \, f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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