Optimal. Leaf size=64 \[ \frac{a^2 c^2 \sin (e+f x) \cos ^3(e+f x)}{4 f}+\frac{3 a^2 c^2 \sin (e+f x) \cos (e+f x)}{8 f}+\frac{3}{8} a^2 c^2 x \]
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Rubi [A] time = 0.0687816, antiderivative size = 64, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115, Rules used = {2736, 2635, 8} \[ \frac{a^2 c^2 \sin (e+f x) \cos ^3(e+f x)}{4 f}+\frac{3 a^2 c^2 \sin (e+f x) \cos (e+f x)}{8 f}+\frac{3}{8} a^2 c^2 x \]
Antiderivative was successfully verified.
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Rule 2736
Rule 2635
Rule 8
Rubi steps
\begin{align*} \int (a+a \sin (e+f x))^2 (c-c \sin (e+f x))^2 \, dx &=\left (a^2 c^2\right ) \int \cos ^4(e+f x) \, dx\\ &=\frac{a^2 c^2 \cos ^3(e+f x) \sin (e+f x)}{4 f}+\frac{1}{4} \left (3 a^2 c^2\right ) \int \cos ^2(e+f x) \, dx\\ &=\frac{3 a^2 c^2 \cos (e+f x) \sin (e+f x)}{8 f}+\frac{a^2 c^2 \cos ^3(e+f x) \sin (e+f x)}{4 f}+\frac{1}{8} \left (3 a^2 c^2\right ) \int 1 \, dx\\ &=\frac{3}{8} a^2 c^2 x+\frac{3 a^2 c^2 \cos (e+f x) \sin (e+f x)}{8 f}+\frac{a^2 c^2 \cos ^3(e+f x) \sin (e+f x)}{4 f}\\ \end{align*}
Mathematica [A] time = 0.0484535, size = 39, normalized size = 0.61 \[ \frac{a^2 c^2 (12 (e+f x)+8 \sin (2 (e+f x))+\sin (4 (e+f x)))}{32 f} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.014, size = 88, normalized size = 1.4 \begin{align*}{\frac{1}{f} \left ({c}^{2}{a}^{2} \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 ) -2\,{c}^{2}{a}^{2} \left ( -1/2\,\sin \left ( fx+e \right ) \cos \left ( fx+e \right ) +1/2\,fx+e/2 \right ) +{c}^{2}{a}^{2} \left ( fx+e \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.11695, size = 109, normalized size = 1.7 \begin{align*} \frac{{\left (12 \, f x + 12 \, e + \sin \left (4 \, f x + 4 \, e\right ) - 8 \, \sin \left (2 \, f x + 2 \, e\right )\right )} a^{2} c^{2} - 16 \,{\left (2 \, f x + 2 \, e - \sin \left (2 \, f x + 2 \, e\right )\right )} a^{2} c^{2} + 32 \,{\left (f x + e\right )} a^{2} c^{2}}{32 \, f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.42943, size = 122, normalized size = 1.91 \begin{align*} \frac{3 \, a^{2} c^{2} f x +{\left (2 \, a^{2} c^{2} \cos \left (f x + e\right )^{3} + 3 \, a^{2} c^{2} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{8 \, f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 2.32217, size = 206, normalized size = 3.22 \begin{align*} \begin{cases} \frac{3 a^{2} c^{2} x \sin ^{4}{\left (e + f x \right )}}{8} + \frac{3 a^{2} c^{2} x \sin ^{2}{\left (e + f x \right )} \cos ^{2}{\left (e + f x \right )}}{4} - a^{2} c^{2} x \sin ^{2}{\left (e + f x \right )} + \frac{3 a^{2} c^{2} x \cos ^{4}{\left (e + f x \right )}}{8} - a^{2} c^{2} x \cos ^{2}{\left (e + f x \right )} + a^{2} c^{2} x - \frac{5 a^{2} c^{2} \sin ^{3}{\left (e + f x \right )} \cos{\left (e + f x \right )}}{8 f} - \frac{3 a^{2} c^{2} \sin{\left (e + f x \right )} \cos ^{3}{\left (e + f x \right )}}{8 f} + \frac{a^{2} c^{2} \sin{\left (e + f x \right )} \cos{\left (e + f x \right )}}{f} & \text{for}\: f \neq 0 \\x \left (a \sin{\left (e \right )} + a\right )^{2} \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 = 1.97903, size = 70, normalized size = 1.09 \begin{align*} \frac{3}{8} \, a^{2} c^{2} x + \frac{a^{2} c^{2} \sin \left (4 \, f x + 4 \, e\right )}{32 \, f} + \frac{a^{2} c^{2} \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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