Optimal. Leaf size=79 \[ -\frac{a (c+d) \cos ^3(e+f x)}{3 f}+\frac{a (4 c+d) \sin (e+f x) \cos (e+f x)}{8 f}+\frac{1}{8} a x (4 c+d)-\frac{a d \sin (e+f x) \cos ^3(e+f x)}{4 f} \]
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Rubi [A] time = 0.0918696, antiderivative size = 84, normalized size of antiderivative = 1.06, number of steps used = 4, number of rules used = 4, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.138, Rules used = {2860, 2669, 2635, 8} \[ -\frac{a (4 c+d) \cos ^3(e+f x)}{12 f}+\frac{a (4 c+d) \sin (e+f x) \cos (e+f x)}{8 f}+\frac{1}{8} a x (4 c+d)-\frac{d \cos ^3(e+f x) (a \sin (e+f x)+a)}{4 f} \]
Antiderivative was successfully verified.
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Rule 2860
Rule 2669
Rule 2635
Rule 8
Rubi steps
\begin{align*} \int \cos ^2(e+f x) (a+a \sin (e+f x)) (c+d \sin (e+f x)) \, dx &=-\frac{d \cos ^3(e+f x) (a+a \sin (e+f x))}{4 f}+\frac{1}{4} (4 c+d) \int \cos ^2(e+f x) (a+a \sin (e+f x)) \, dx\\ &=-\frac{a (4 c+d) \cos ^3(e+f x)}{12 f}-\frac{d \cos ^3(e+f x) (a+a \sin (e+f x))}{4 f}+\frac{1}{4} (a (4 c+d)) \int \cos ^2(e+f x) \, dx\\ &=-\frac{a (4 c+d) \cos ^3(e+f x)}{12 f}+\frac{a (4 c+d) \cos (e+f x) \sin (e+f x)}{8 f}-\frac{d \cos ^3(e+f x) (a+a \sin (e+f x))}{4 f}+\frac{1}{8} (a (4 c+d)) \int 1 \, dx\\ &=\frac{1}{8} a (4 c+d) x-\frac{a (4 c+d) \cos ^3(e+f x)}{12 f}+\frac{a (4 c+d) \cos (e+f x) \sin (e+f x)}{8 f}-\frac{d \cos ^3(e+f x) (a+a \sin (e+f x))}{4 f}\\ \end{align*}
Mathematica [A] time = 0.608047, size = 64, normalized size = 0.81 \[ -\frac{a (24 (c+d) \cos (e+f x)+8 (c+d) \cos (3 (e+f x))-12 f x (4 c+d)-24 c \sin (2 (e+f x))+3 d \sin (4 (e+f x)))}{96 f} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.053, size = 96, normalized size = 1.2 \begin{align*}{\frac{1}{f} \left ( da \left ( -{\frac{\sin \left ( fx+e \right ) \left ( \cos \left ( fx+e \right ) \right ) ^{3}}{4}}+{\frac{\sin \left ( fx+e \right ) \cos \left ( fx+e \right ) }{8}}+{\frac{fx}{8}}+{\frac{e}{8}} \right ) -{\frac{ \left ( \cos \left ( fx+e \right ) \right ) ^{3}ac}{3}}-{\frac{da \left ( \cos \left ( fx+e \right ) \right ) ^{3}}{3}}+ca \left ({\frac{\sin \left ( fx+e \right ) \cos \left ( fx+e \right ) }{2}}+{\frac{fx}{2}}+{\frac{e}{2}} \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.0376, size = 100, normalized size = 1.27 \begin{align*} -\frac{32 \, a c \cos \left (f x + e\right )^{3} + 32 \, a d \cos \left (f x + e\right )^{3} - 24 \,{\left (2 \, f x + 2 \, e + \sin \left (2 \, f x + 2 \, e\right )\right )} a c - 3 \,{\left (4 \, f x + 4 \, e - \sin \left (4 \, f x + 4 \, e\right )\right )} a d}{96 \, f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.7148, size = 177, normalized size = 2.24 \begin{align*} -\frac{8 \,{\left (a c + a d\right )} \cos \left (f x + e\right )^{3} - 3 \,{\left (4 \, a c + a d\right )} f x + 3 \,{\left (2 \, a d \cos \left (f x + e\right )^{3} -{\left (4 \, a c + a d\right )} \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 = 1.31873, size = 199, normalized size = 2.52 \begin{align*} \begin{cases} \frac{a c x \sin ^{2}{\left (e + f x \right )}}{2} + \frac{a c x \cos ^{2}{\left (e + f x \right )}}{2} + \frac{a c \sin{\left (e + f x \right )} \cos{\left (e + f x \right )}}{2 f} - \frac{a c \cos ^{3}{\left (e + f x \right )}}{3 f} + \frac{a d x \sin ^{4}{\left (e + f x \right )}}{8} + \frac{a d x \sin ^{2}{\left (e + f x \right )} \cos ^{2}{\left (e + f x \right )}}{4} + \frac{a d x \cos ^{4}{\left (e + f x \right )}}{8} + \frac{a d \sin ^{3}{\left (e + f x \right )} \cos{\left (e + f x \right )}}{8 f} - \frac{a d \sin{\left (e + f x \right )} \cos ^{3}{\left (e + f x \right )}}{8 f} - \frac{a d \cos ^{3}{\left (e + f x \right )}}{3 f} & \text{for}\: f \neq 0 \\x \left (c + d \sin{\left (e \right )}\right ) \left (a \sin{\left (e \right )} + a\right ) \cos ^{2}{\left (e \right )} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.27094, size = 117, normalized size = 1.48 \begin{align*} \frac{1}{8} \,{\left (4 \, a c + a d\right )} x - \frac{a d \sin \left (4 \, f x + 4 \, e\right )}{32 \, f} + \frac{a c \sin \left (2 \, f x + 2 \, e\right )}{4 \, f} - \frac{{\left (a c + a d\right )} \cos \left (3 \, f x + 3 \, e\right )}{12 \, f} - \frac{{\left (a c + a d\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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