Optimal. Leaf size=97 \[ \frac{a c^2 (A-B) \cos ^3(e+f x)}{3 f}+\frac{a c^2 (4 A-B) \sin (e+f x) \cos (e+f x)}{8 f}+\frac{1}{8} a c^2 x (4 A-B)+\frac{a B c^2 \sin (e+f x) \cos ^3(e+f x)}{4 f} \]
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Rubi [A] time = 0.186352, antiderivative size = 105, normalized size of antiderivative = 1.08, 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} \[ \frac{a c^2 (4 A-B) \cos ^3(e+f x)}{12 f}+\frac{a c^2 (4 A-B) \sin (e+f x) \cos (e+f x)}{8 f}+\frac{1}{8} a c^2 x (4 A-B)-\frac{a B \cos ^3(e+f x) \left (c^2-c^2 \sin (e+f x)\right )}{4 f} \]
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)) (A+B \sin (e+f x)) (c-c \sin (e+f x))^2 \, dx &=(a c) \int \cos ^2(e+f x) (A+B \sin (e+f x)) (c-c \sin (e+f x)) \, dx\\ &=-\frac{a B \cos ^3(e+f x) \left (c^2-c^2 \sin (e+f x)\right )}{4 f}+\frac{1}{4} (a (4 A-B) c) \int \cos ^2(e+f x) (c-c \sin (e+f x)) \, dx\\ &=\frac{a (4 A-B) c^2 \cos ^3(e+f x)}{12 f}-\frac{a B \cos ^3(e+f x) \left (c^2-c^2 \sin (e+f x)\right )}{4 f}+\frac{1}{4} \left (a (4 A-B) c^2\right ) \int \cos ^2(e+f x) \, dx\\ &=\frac{a (4 A-B) c^2 \cos ^3(e+f x)}{12 f}+\frac{a (4 A-B) c^2 \cos (e+f x) \sin (e+f x)}{8 f}-\frac{a B \cos ^3(e+f x) \left (c^2-c^2 \sin (e+f x)\right )}{4 f}+\frac{1}{8} \left (a (4 A-B) c^2\right ) \int 1 \, dx\\ &=\frac{1}{8} a (4 A-B) c^2 x+\frac{a (4 A-B) c^2 \cos ^3(e+f x)}{12 f}+\frac{a (4 A-B) c^2 \cos (e+f x) \sin (e+f x)}{8 f}-\frac{a B \cos ^3(e+f x) \left (c^2-c^2 \sin (e+f x)\right )}{4 f}\\ \end{align*}
Mathematica [A] time = 0.649647, size = 74, normalized size = 0.76 \[ \frac{a c^2 (3 (8 A \sin (2 (e+f x))+16 A f x+B \sin (4 (e+f x))-4 B f x)+24 (A-B) \cos (e+f x)+8 (A-B) \cos (3 (e+f x)))}{96 f} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.029, size = 185, normalized size = 1.9 \begin{align*}{\frac{1}{f} \left ( -{\frac{A{c}^{2}a \left ( 2+ \left ( \sin \left ( fx+e \right ) \right ) ^{2} \right ) \cos \left ( fx+e \right ) }{3}}-A{c}^{2}a \left ( -{\frac{\sin \left ( fx+e \right ) \cos \left ( fx+e \right ) }{2}}+{\frac{fx}{2}}+{\frac{e}{2}} \right ) +A{c}^{2}a\cos \left ( fx+e \right ) +B{c}^{2}a \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{c}^{2}a \left ( 2+ \left ( \sin \left ( fx+e \right ) \right ) ^{2} \right ) \cos \left ( fx+e \right ) }{3}}-B{c}^{2}a \left ( -{\frac{\sin \left ( fx+e \right ) \cos \left ( fx+e \right ) }{2}}+{\frac{fx}{2}}+{\frac{e}{2}} \right ) +A{c}^{2}a \left ( fx+e \right ) -B{c}^{2}a\cos \left ( fx+e \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 0.966047, size = 242, normalized size = 2.49 \begin{align*} \frac{32 \,{\left (\cos \left (f x + e\right )^{3} - 3 \, \cos \left (f x + e\right )\right )} A a c^{2} - 24 \,{\left (2 \, f x + 2 \, e - \sin \left (2 \, f x + 2 \, e\right )\right )} A a c^{2} + 96 \,{\left (f x + e\right )} A a c^{2} - 32 \,{\left (\cos \left (f x + e\right )^{3} - 3 \, \cos \left (f x + e\right )\right )} B a c^{2} + 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 c^{2} - 24 \,{\left (2 \, f x + 2 \, e - \sin \left (2 \, f x + 2 \, e\right )\right )} B a c^{2} + 96 \, A a c^{2} \cos \left (f x + e\right ) - 96 \, B a c^{2} \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 = 1.51681, size = 189, normalized size = 1.95 \begin{align*} \frac{8 \,{\left (A - B\right )} a c^{2} \cos \left (f x + e\right )^{3} + 3 \,{\left (4 \, A - B\right )} a c^{2} f x + 3 \,{\left (2 \, B a c^{2} \cos \left (f x + e\right )^{3} +{\left (4 \, A - B\right )} a c^{2} \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 = 2.62768, size = 396, normalized size = 4.08 \begin{align*} \begin{cases} - \frac{A a c^{2} x \sin ^{2}{\left (e + f x \right )}}{2} - \frac{A a c^{2} x \cos ^{2}{\left (e + f x \right )}}{2} + A a c^{2} x - \frac{A a c^{2} \sin ^{2}{\left (e + f x \right )} \cos{\left (e + f x \right )}}{f} + \frac{A a c^{2} \sin{\left (e + f x \right )} \cos{\left (e + f x \right )}}{2 f} - \frac{2 A a c^{2} \cos ^{3}{\left (e + f x \right )}}{3 f} + \frac{A a c^{2} \cos{\left (e + f x \right )}}{f} + \frac{3 B a c^{2} x \sin ^{4}{\left (e + f x \right )}}{8} + \frac{3 B a c^{2} x \sin ^{2}{\left (e + f x \right )} \cos ^{2}{\left (e + f x \right )}}{4} - \frac{B a c^{2} x \sin ^{2}{\left (e + f x \right )}}{2} + \frac{3 B a c^{2} x \cos ^{4}{\left (e + f x \right )}}{8} - \frac{B a c^{2} x \cos ^{2}{\left (e + f x \right )}}{2} - \frac{5 B a c^{2} \sin ^{3}{\left (e + f x \right )} \cos{\left (e + f x \right )}}{8 f} + \frac{B a c^{2} \sin ^{2}{\left (e + f x \right )} \cos{\left (e + f x \right )}}{f} - \frac{3 B a c^{2} \sin{\left (e + f x \right )} \cos ^{3}{\left (e + f x \right )}}{8 f} + \frac{B a c^{2} \sin{\left (e + f x \right )} \cos{\left (e + f x \right )}}{2 f} + \frac{2 B a c^{2} \cos ^{3}{\left (e + f x \right )}}{3 f} - \frac{B a c^{2} \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 ) \left (- c \sin{\left (e \right )} + c\right )^{2} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.17095, size = 154, normalized size = 1.59 \begin{align*} \frac{B a c^{2} \sin \left (4 \, f x + 4 \, e\right )}{32 \, f} + \frac{A a c^{2} \sin \left (2 \, f x + 2 \, e\right )}{4 \, f} + \frac{1}{8} \,{\left (4 \, A a c^{2} - B a c^{2}\right )} x + \frac{{\left (A a c^{2} - B a c^{2}\right )} \cos \left (3 \, f x + 3 \, e\right )}{12 \, f} + \frac{{\left (A a c^{2} - B a c^{2}\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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