Optimal. Leaf size=111 \[ -\frac{a (3 A (c+d)+B (3 c+d)) \cos (e+f x)}{3 f}-\frac{a (3 A d+3 B c-B d) \sin (e+f x) \cos (e+f x)}{6 f}+\frac{1}{2} a x (A (2 c+d)+B (c+d))-\frac{B d \cos (e+f x) (a \sin (e+f x)+a)^2}{3 a f} \]
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Rubi [A] time = 0.15674, antiderivative size = 111, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.097, Rules used = {2968, 3023, 2734} \[ -\frac{a (3 A (c+d)+B (3 c+d)) \cos (e+f x)}{3 f}-\frac{a (3 A d+3 B c-B d) \sin (e+f x) \cos (e+f x)}{6 f}+\frac{1}{2} a x (A (2 c+d)+B (c+d))-\frac{B d \cos (e+f x) (a \sin (e+f x)+a)^2}{3 a f} \]
Antiderivative was successfully verified.
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Rule 2968
Rule 3023
Rule 2734
Rubi steps
\begin{align*} \int (a+a \sin (e+f x)) (A+B \sin (e+f x)) (c+d \sin (e+f x)) \, dx &=\int (a+a \sin (e+f x)) \left (A c+(B c+A d) \sin (e+f x)+B d \sin ^2(e+f x)\right ) \, dx\\ &=-\frac{B d \cos (e+f x) (a+a \sin (e+f x))^2}{3 a f}+\frac{\int (a+a \sin (e+f x)) (a (3 A c+2 B d)+a (3 B c+3 A d-B d) \sin (e+f x)) \, dx}{3 a}\\ &=\frac{1}{2} a (B (c+d)+A (2 c+d)) x-\frac{a (3 A (c+d)+B (3 c+d)) \cos (e+f x)}{3 f}-\frac{a (3 B c+3 A d-B d) \cos (e+f x) \sin (e+f x)}{6 f}-\frac{B d \cos (e+f x) (a+a \sin (e+f x))^2}{3 a f}\\ \end{align*}
Mathematica [A] time = 0.426629, size = 104, normalized size = 0.94 \[ \frac{a (-3 (4 A (c+d)+B (4 c+3 d)) \cos (e+f x)+12 A c f x-3 A d \sin (2 (e+f x))+6 A d f x-3 B c \sin (2 (e+f x))+6 B c f x-3 B d \sin (2 (e+f x))+B d \cos (3 (e+f x))+6 B d f x)}{12 f} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.047, size = 147, normalized size = 1.3 \begin{align*}{\frac{1}{f} \left ( -{\frac{Bad \left ( 2+ \left ( \sin \left ( fx+e \right ) \right ) ^{2} \right ) \cos \left ( fx+e \right ) }{3}}+Aad \left ( -{\frac{\sin \left ( fx+e \right ) \cos \left ( fx+e \right ) }{2}}+{\frac{fx}{2}}+{\frac{e}{2}} \right ) +Bac \left ( -{\frac{\sin \left ( fx+e \right ) \cos \left ( fx+e \right ) }{2}}+{\frac{fx}{2}}+{\frac{e}{2}} \right ) +Bad \left ( -{\frac{\sin \left ( fx+e \right ) \cos \left ( fx+e \right ) }{2}}+{\frac{fx}{2}}+{\frac{e}{2}} \right ) -Aac\cos \left ( fx+e \right ) -Aad\cos \left ( fx+e \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 = 0.957769, size = 193, normalized size = 1.74 \begin{align*} \frac{12 \,{\left (f x + e\right )} A a c + 3 \,{\left (2 \, f x + 2 \, e - \sin \left (2 \, f x + 2 \, e\right )\right )} B a c + 3 \,{\left (2 \, f x + 2 \, e - \sin \left (2 \, f x + 2 \, e\right )\right )} A a d + 4 \,{\left (\cos \left (f x + e\right )^{3} - 3 \, \cos \left (f x + e\right )\right )} B a d + 3 \,{\left (2 \, f x + 2 \, e - \sin \left (2 \, f x + 2 \, e\right )\right )} B a d - 12 \, A a c \cos \left (f x + e\right ) - 12 \, B a c \cos \left (f x + e\right ) - 12 \, A a d \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 = 1.94864, size = 225, normalized size = 2.03 \begin{align*} \frac{2 \, B a d \cos \left (f x + e\right )^{3} + 3 \,{\left ({\left (2 \, A + B\right )} a c +{\left (A + B\right )} a d\right )} f x - 3 \,{\left (B a c +{\left (A + B\right )} a d\right )} \cos \left (f x + e\right ) \sin \left (f x + e\right ) - 6 \,{\left ({\left (A + B\right )} a c +{\left (A + B\right )} a d\right )} \cos \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 = 1.35109, size = 277, normalized size = 2.5 \begin{align*} \begin{cases} A a c x - \frac{A a c \cos{\left (e + f x \right )}}{f} + \frac{A a d x \sin ^{2}{\left (e + f x \right )}}{2} + \frac{A a d x \cos ^{2}{\left (e + f x \right )}}{2} - \frac{A a d \sin{\left (e + f x \right )} \cos{\left (e + f x \right )}}{2 f} - \frac{A a d \cos{\left (e + f x \right )}}{f} + \frac{B a c x \sin ^{2}{\left (e + f x \right )}}{2} + \frac{B a c x \cos ^{2}{\left (e + f x \right )}}{2} - \frac{B a c \sin{\left (e + f x \right )} \cos{\left (e + f x \right )}}{2 f} - \frac{B a c \cos{\left (e + f x \right )}}{f} + \frac{B a d x \sin ^{2}{\left (e + f x \right )}}{2} + \frac{B a d x \cos ^{2}{\left (e + f x \right )}}{2} - \frac{B a d \sin ^{2}{\left (e + f x \right )} \cos{\left (e + f x \right )}}{f} - \frac{B a d \sin{\left (e + f x \right )} \cos{\left (e + f x \right )}}{2 f} - \frac{2 B a d \cos ^{3}{\left (e + f x \right )}}{3 f} & \text{for}\: f \neq 0 \\x \left (A + B \sin{\left (e \right )}\right ) \left (c + d \sin{\left (e \right )}\right ) \left (a \sin{\left (e \right )} + a\right ) & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.21923, size = 136, normalized size = 1.23 \begin{align*} \frac{B a d \cos \left (3 \, f x + 3 \, e\right )}{12 \, f} + \frac{1}{2} \,{\left (2 \, A a c + B a c + A a d + B a d\right )} x - \frac{{\left (4 \, A a c + 4 \, B a c + 4 \, A a d + 3 \, B a d\right )} \cos \left (f x + e\right )}{4 \, f} - \frac{{\left (B a c + A a d + B a d\right )} \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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