Optimal. Leaf size=77 \[ -\frac {a^2 c \cos ^3(e+f x)}{3 f}+\frac {a^2 c \sin ^3(e+f x) \cos (e+f x)}{4 f}-\frac {a^2 c \sin (e+f x) \cos (e+f x)}{8 f}+\frac {1}{8} a^2 c x \]
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Rubi [A] time = 0.10, antiderivative size = 77, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 5, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {2966, 2638, 2635, 8, 2633} \[ -\frac {a^2 c \cos ^3(e+f x)}{3 f}+\frac {a^2 c \sin ^3(e+f x) \cos (e+f x)}{4 f}-\frac {a^2 c \sin (e+f x) \cos (e+f x)}{8 f}+\frac {1}{8} a^2 c x \]
Antiderivative was successfully verified.
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Rule 8
Rule 2633
Rule 2635
Rule 2638
Rule 2966
Rubi steps
\begin {align*} \int \sin (e+f x) (a+a \sin (e+f x))^2 (c-c \sin (e+f x)) \, dx &=\int \left (a^2 c \sin (e+f x)+a^2 c \sin ^2(e+f x)-a^2 c \sin ^3(e+f x)-a^2 c \sin ^4(e+f x)\right ) \, dx\\ &=\left (a^2 c\right ) \int \sin (e+f x) \, dx+\left (a^2 c\right ) \int \sin ^2(e+f x) \, dx-\left (a^2 c\right ) \int \sin ^3(e+f x) \, dx-\left (a^2 c\right ) \int \sin ^4(e+f x) \, dx\\ &=-\frac {a^2 c \cos (e+f x)}{f}-\frac {a^2 c \cos (e+f x) \sin (e+f x)}{2 f}+\frac {a^2 c \cos (e+f x) \sin ^3(e+f x)}{4 f}+\frac {1}{2} \left (a^2 c\right ) \int 1 \, dx-\frac {1}{4} \left (3 a^2 c\right ) \int \sin ^2(e+f x) \, dx+\frac {\left (a^2 c\right ) \operatorname {Subst}\left (\int \left (1-x^2\right ) \, dx,x,\cos (e+f x)\right )}{f}\\ &=\frac {1}{2} a^2 c x-\frac {a^2 c \cos ^3(e+f x)}{3 f}-\frac {a^2 c \cos (e+f x) \sin (e+f x)}{8 f}+\frac {a^2 c \cos (e+f x) \sin ^3(e+f x)}{4 f}-\frac {1}{8} \left (3 a^2 c\right ) \int 1 \, dx\\ &=\frac {1}{8} a^2 c x-\frac {a^2 c \cos ^3(e+f x)}{3 f}-\frac {a^2 c \cos (e+f x) \sin (e+f x)}{8 f}+\frac {a^2 c \cos (e+f x) \sin ^3(e+f x)}{4 f}\\ \end {align*}
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Mathematica [A] time = 0.11, size = 47, normalized size = 0.61 \[ \frac {a^2 c (-3 \sin (4 (e+f x))-24 \cos (e+f x)-8 \cos (3 (e+f x))+12 e+12 f x)}{96 f} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 63, normalized size = 0.82 \[ -\frac {8 \, a^{2} c \cos \left (f x + e\right )^{3} - 3 \, a^{2} c f x + 3 \, {\left (2 \, a^{2} c \cos \left (f x + e\right )^{3} - a^{2} c \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{24 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 62, normalized size = 0.81 \[ \frac {1}{8} \, a^{2} c x - \frac {a^{2} c \cos \left (3 \, f x + 3 \, e\right )}{12 \, f} - \frac {a^{2} c \cos \left (f x + e\right )}{4 \, f} - \frac {a^{2} c \sin \left (4 \, f x + 4 \, e\right )}{32 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.38, size = 106, normalized size = 1.38 \[ \frac {-a^{2} c \left (-\frac {\left (\sin ^{3}\left (f x +e \right )+\frac {3 \sin \left (f x +e \right )}{2}\right ) \cos \left (f x +e \right )}{4}+\frac {3 f x}{8}+\frac {3 e}{8}\right )+\frac {a^{2} c \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )}{3}+a^{2} c \left (-\frac {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )-a^{2} c \cos \left (f x +e \right )}{f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.33, size = 100, normalized size = 1.30 \[ -\frac {32 \, {\left (\cos \left (f x + e\right )^{3} - 3 \, \cos \left (f x + e\right )\right )} a^{2} c + 3 \, {\left (12 \, f x + 12 \, e + \sin \left (4 \, f x + 4 \, e\right ) - 8 \, \sin \left (2 \, f x + 2 \, e\right )\right )} a^{2} c - 24 \, {\left (2 \, f x + 2 \, e - \sin \left (2 \, f x + 2 \, e\right )\right )} a^{2} c + 96 \, a^{2} c \cos \left (f x + e\right )}{96 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 14.12, size = 250, normalized size = 3.25 \[ \frac {a^2\,c\,x}{8}-\frac {{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2\,\left (\frac {a^2\,c\,\left (3\,e+3\,f\,x\right )}{6}-\frac {a^2\,c\,\left (12\,e+12\,f\,x-16\right )}{24}\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6\,\left (\frac {a^2\,c\,\left (3\,e+3\,f\,x\right )}{6}-\frac {a^2\,c\,\left (12\,e+12\,f\,x-48\right )}{24}\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4\,\left (\frac {a^2\,c\,\left (3\,e+3\,f\,x\right )}{4}-\frac {a^2\,c\,\left (18\,e+18\,f\,x-48\right )}{24}\right )+\frac {a^2\,c\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}{4}-\frac {7\,a^2\,c\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3}{4}+\frac {7\,a^2\,c\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^5}{4}-\frac {a^2\,c\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^7}{4}+\frac {a^2\,c\,\left (3\,e+3\,f\,x\right )}{24}-\frac {a^2\,c\,\left (3\,e+3\,f\,x-16\right )}{24}}{f\,{\left ({\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2+1\right )}^4} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.36, size = 245, normalized size = 3.18 \[ \begin {cases} - \frac {3 a^{2} c x \sin ^{4}{\left (e + f x \right )}}{8} - \frac {3 a^{2} c x \sin ^{2}{\left (e + f x \right )} \cos ^{2}{\left (e + f x \right )}}{4} + \frac {a^{2} c x \sin ^{2}{\left (e + f x \right )}}{2} - \frac {3 a^{2} c x \cos ^{4}{\left (e + f x \right )}}{8} + \frac {a^{2} c x \cos ^{2}{\left (e + f x \right )}}{2} + \frac {5 a^{2} c \sin ^{3}{\left (e + f x \right )} \cos {\left (e + f x \right )}}{8 f} + \frac {a^{2} c \sin ^{2}{\left (e + f x \right )} \cos {\left (e + f x \right )}}{f} + \frac {3 a^{2} c \sin {\left (e + f x \right )} \cos ^{3}{\left (e + f x \right )}}{8 f} - \frac {a^{2} c \sin {\left (e + f x \right )} \cos {\left (e + f x \right )}}{2 f} + \frac {2 a^{2} c \cos ^{3}{\left (e + f x \right )}}{3 f} - \frac {a^{2} c \cos {\left (e + f x \right )}}{f} & \text {for}\: f \neq 0 \\x \left (a \sin {\relax (e )} + a\right )^{2} \left (- c \sin {\relax (e )} + c\right ) \sin {\relax (e )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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