Optimal. Leaf size=107 \[ -\frac{2 \left (a^2 d+3 a b c+b^2 d\right ) \cos (e+f x)}{3 f}+\frac{1}{2} x \left (2 a^2 c+2 a b d+b^2 c\right )-\frac{b (2 a d+3 b c) \sin (e+f x) \cos (e+f x)}{6 f}-\frac{d \cos (e+f x) (a+b \sin (e+f x))^2}{3 f} \]
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Rubi [A] time = 0.0936561, antiderivative size = 107, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.087, Rules used = {2753, 2734} \[ -\frac{2 \left (a^2 d+3 a b c+b^2 d\right ) \cos (e+f x)}{3 f}+\frac{1}{2} x \left (2 a^2 c+2 a b d+b^2 c\right )-\frac{b (2 a d+3 b c) \sin (e+f x) \cos (e+f x)}{6 f}-\frac{d \cos (e+f x) (a+b \sin (e+f x))^2}{3 f} \]
Antiderivative was successfully verified.
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Rule 2753
Rule 2734
Rubi steps
\begin{align*} \int (a+b \sin (e+f x))^2 (c+d \sin (e+f x)) \, dx &=-\frac{d \cos (e+f x) (a+b \sin (e+f x))^2}{3 f}+\frac{1}{3} \int (a+b \sin (e+f x)) (3 a c+2 b d+(3 b c+2 a d) \sin (e+f x)) \, dx\\ &=\frac{1}{2} \left (2 a^2 c+b^2 c+2 a b d\right ) x-\frac{2 \left (3 a b c+a^2 d+b^2 d\right ) \cos (e+f x)}{3 f}-\frac{b (3 b c+2 a d) \cos (e+f x) \sin (e+f x)}{6 f}-\frac{d \cos (e+f x) (a+b \sin (e+f x))^2}{3 f}\\ \end{align*}
Mathematica [A] time = 0.294564, size = 90, normalized size = 0.84 \[ \frac{6 (e+f x) \left (2 a^2 c+2 a b d+b^2 c\right )-3 \left (4 a^2 d+8 a b c+3 b^2 d\right ) \cos (e+f x)-3 b (2 a d+b c) \sin (2 (e+f x))+b^2 d \cos (3 (e+f x))}{12 f} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.026, size = 115, normalized size = 1.1 \begin{align*}{\frac{1}{f} \left ({a}^{2}c \left ( fx+e \right ) -{a}^{2}d\cos \left ( fx+e \right ) -2\,abc\cos \left ( fx+e \right ) +2\,abd \left ( -1/2\,\sin \left ( fx+e \right ) \cos \left ( fx+e \right ) +1/2\,fx+e/2 \right ) +{b}^{2}c \left ( -{\frac{\sin \left ( fx+e \right ) \cos \left ( fx+e \right ) }{2}}+{\frac{fx}{2}}+{\frac{e}{2}} \right ) -{\frac{{b}^{2}d \left ( 2+ \left ( \sin \left ( fx+e \right ) \right ) ^{2} \right ) \cos \left ( fx+e \right ) }{3}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.12328, size = 151, normalized size = 1.41 \begin{align*} \frac{12 \,{\left (f x + e\right )} a^{2} c + 3 \,{\left (2 \, f x + 2 \, e - \sin \left (2 \, f x + 2 \, e\right )\right )} b^{2} c + 6 \,{\left (2 \, f x + 2 \, e - \sin \left (2 \, f x + 2 \, e\right )\right )} a b d + 4 \,{\left (\cos \left (f x + e\right )^{3} - 3 \, \cos \left (f x + e\right )\right )} b^{2} d - 24 \, a b c \cos \left (f x + e\right ) - 12 \, a^{2} 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 = 1.59181, size = 215, normalized size = 2.01 \begin{align*} \frac{2 \, b^{2} d \cos \left (f x + e\right )^{3} + 3 \,{\left (2 \, a b d +{\left (2 \, a^{2} + b^{2}\right )} c\right )} f x - 3 \,{\left (b^{2} c + 2 \, a b d\right )} \cos \left (f x + e\right ) \sin \left (f x + e\right ) - 6 \,{\left (2 \, a b c +{\left (a^{2} + b^{2}\right )} 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 = 0.786323, size = 199, normalized size = 1.86 \begin{align*} \begin{cases} a^{2} c x - \frac{a^{2} d \cos{\left (e + f x \right )}}{f} - \frac{2 a b c \cos{\left (e + f x \right )}}{f} + a b d x \sin ^{2}{\left (e + f x \right )} + a b d x \cos ^{2}{\left (e + f x \right )} - \frac{a b d \sin{\left (e + f x \right )} \cos{\left (e + f x \right )}}{f} + \frac{b^{2} c x \sin ^{2}{\left (e + f x \right )}}{2} + \frac{b^{2} c x \cos ^{2}{\left (e + f x \right )}}{2} - \frac{b^{2} c \sin{\left (e + f x \right )} \cos{\left (e + f x \right )}}{2 f} - \frac{b^{2} d \sin ^{2}{\left (e + f x \right )} \cos{\left (e + f x \right )}}{f} - \frac{2 b^{2} d \cos ^{3}{\left (e + f x \right )}}{3 f} & \text{for}\: f \neq 0 \\x \left (a + b \sin{\left (e \right )}\right )^{2} \left (c + d \sin{\left (e \right )}\right ) & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.33933, size = 130, normalized size = 1.21 \begin{align*} \frac{b^{2} d \cos \left (3 \, f x + 3 \, e\right )}{12 \, f} + \frac{1}{2} \,{\left (2 \, a^{2} c + b^{2} c + 2 \, a b d\right )} x - \frac{{\left (8 \, a b c + 4 \, a^{2} d + 3 \, b^{2} d\right )} \cos \left (f x + e\right )}{4 \, f} - \frac{{\left (b^{2} c + 2 \, a b 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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