Optimal. Leaf size=49 \[ \frac{a A c \sin (e+f x) \cos (e+f x)}{2 f}+\frac{1}{2} a A c x-\frac{a B c \cos ^3(e+f x)}{3 f} \]
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Rubi [A] time = 0.0826031, antiderivative size = 49, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {2967, 2669, 2635, 8} \[ \frac{a A c \sin (e+f x) \cos (e+f x)}{2 f}+\frac{1}{2} a A c x-\frac{a B c \cos ^3(e+f x)}{3 f} \]
Antiderivative was successfully verified.
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Rule 2967
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)) \, dx &=(a c) \int \cos ^2(e+f x) (A+B \sin (e+f x)) \, dx\\ &=-\frac{a B c \cos ^3(e+f x)}{3 f}+(a A c) \int \cos ^2(e+f x) \, dx\\ &=-\frac{a B c \cos ^3(e+f x)}{3 f}+\frac{a A c \cos (e+f x) \sin (e+f x)}{2 f}+\frac{1}{2} (a A c) \int 1 \, dx\\ &=\frac{1}{2} a A c x-\frac{a B c \cos ^3(e+f x)}{3 f}+\frac{a A c \cos (e+f x) \sin (e+f x)}{2 f}\\ \end{align*}
Mathematica [A] time = 0.15372, size = 48, normalized size = 0.98 \[ -\frac{a c (-3 A (\sin (2 (e+f x))-2 e+2 f x)+3 B \cos (e+f x)+B \cos (3 (e+f x)))}{12 f} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.023, size = 74, normalized size = 1.5 \begin{align*}{\frac{1}{f} \left ({\frac{Bac \left ( 2+ \left ( \sin \left ( fx+e \right ) \right ) ^{2} \right ) \cos \left ( fx+e \right ) }{3}}-Aac \left ( -{\frac{\sin \left ( fx+e \right ) \cos \left ( fx+e \right ) }{2}}+{\frac{fx}{2}}+{\frac{e}{2}} \right ) -Bac\cos \left ( fx+e \right ) +Aac \left ( fx+e \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.967706, size = 99, normalized size = 2.02 \begin{align*} -\frac{3 \,{\left (2 \, f x + 2 \, e - \sin \left (2 \, f x + 2 \, e\right )\right )} A a c - 12 \,{\left (f x + e\right )} A a c + 4 \,{\left (\cos \left (f x + e\right )^{3} - 3 \, \cos \left (f x + e\right )\right )} B a c + 12 \, B a c \cos \left (f x + e\right )}{12 \, f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.30224, size = 112, normalized size = 2.29 \begin{align*} -\frac{2 \, B a c \cos \left (f x + e\right )^{3} - 3 \, A a c f x - 3 \, A a c \cos \left (f x + e\right ) \sin \left (f x + e\right )}{6 \, f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.22869, size = 138, normalized size = 2.82 \begin{align*} \begin{cases} - \frac{A a c x \sin ^{2}{\left (e + f x \right )}}{2} - \frac{A a c x \cos ^{2}{\left (e + f x \right )}}{2} + A a c x + \frac{A a c \sin{\left (e + f x \right )} \cos{\left (e + f x \right )}}{2 f} + \frac{B a c \sin ^{2}{\left (e + f x \right )} \cos{\left (e + f x \right )}}{f} + \frac{2 B a c \cos ^{3}{\left (e + f x \right )}}{3 f} - \frac{B a 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 ) \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 = 1.14198, size = 78, normalized size = 1.59 \begin{align*} \frac{1}{2} \, A a c x - \frac{B a c \cos \left (3 \, f x + 3 \, e\right )}{12 \, f} - \frac{B a c \cos \left (f x + e\right )}{4 \, f} + \frac{A a c \sin \left (2 \, f x + 2 \, e\right )}{4 \, f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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