Optimal. Leaf size=49 \[ \frac {a A c \sin (e+f x) \cos (e+f x)}{2 f}+\frac {1}{2} a A c x-\frac {a B c \cos ^3(e+f x)}{3 f} \]
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Rubi [A] time = 0.08, antiderivative size = 49, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {2967, 2669, 2635, 8} \[ \frac {a A c \sin (e+f x) \cos (e+f x)}{2 f}+\frac {1}{2} a A c x-\frac {a B c \cos ^3(e+f x)}{3 f} \]
Antiderivative was successfully verified.
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Rule 8
Rule 2635
Rule 2669
Rule 2967
Rubi steps
\begin {align*} \int (a+a \sin (e+f x)) (A+B \sin (e+f x)) (c-c \sin (e+f x)) \, dx &=(a c) \int \cos ^2(e+f x) (A+B \sin (e+f x)) \, dx\\ &=-\frac {a B c \cos ^3(e+f x)}{3 f}+(a A c) \int \cos ^2(e+f x) \, dx\\ &=-\frac {a B c \cos ^3(e+f x)}{3 f}+\frac {a A c \cos (e+f x) \sin (e+f x)}{2 f}+\frac {1}{2} (a A c) \int 1 \, dx\\ &=\frac {1}{2} a A c x-\frac {a B c \cos ^3(e+f x)}{3 f}+\frac {a A c \cos (e+f x) \sin (e+f x)}{2 f}\\ \end {align*}
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Mathematica [A] time = 0.16, size = 48, normalized size = 0.98 \[ -\frac {a c (-3 A (\sin (2 (e+f x))-2 e+2 f x)+3 B \cos (e+f x)+B \cos (3 (e+f x)))}{12 f} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 43, normalized size = 0.88 \[ -\frac {2 \, B a c \cos \left (f x + e\right )^{3} - 3 \, A a c f x - 3 \, A a c \cos \left (f x + e\right ) \sin \left (f x + e\right )}{6 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.14, size = 58, normalized size = 1.18 \[ \frac {1}{2} \, A a c x - \frac {B a c \cos \left (3 \, f x + 3 \, e\right )}{12 \, f} - \frac {B a c \cos \left (f x + e\right )}{4 \, f} + \frac {A a c \sin \left (2 \, f x + 2 \, e\right )}{4 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.29, size = 74, normalized size = 1.51 \[ \frac {\frac {B a c \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )}{3}-A a c \left (-\frac {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )-B a c \cos \left (f x +e \right )+A a c \left (f x +e \right )}{f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.38, size = 73, normalized size = 1.49 \[ -\frac {3 \, {\left (2 \, f x + 2 \, e - \sin \left (2 \, f x + 2 \, e\right )\right )} A a c - 12 \, {\left (f x + e\right )} A a c + 4 \, {\left (\cos \left (f x + e\right )^{3} - 3 \, \cos \left (f x + e\right )\right )} B a c + 12 \, B a c \cos \left (f x + e\right )}{12 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 14.33, size = 122, normalized size = 2.49 \[ \frac {A\,a\,c\,x}{2}-\frac {A\,a\,c\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^5+\left (\frac {a\,c\,\left (12\,B-9\,A\,\left (e+f\,x\right )\right )}{6}+\frac {3\,A\,a\,c\,\left (e+f\,x\right )}{2}\right )\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4-A\,a\,c\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )+\frac {a\,c\,\left (4\,B-3\,A\,\left (e+f\,x\right )\right )}{6}+\frac {A\,a\,c\,\left (e+f\,x\right )}{2}}{f\,{\left ({\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2+1\right )}^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.79, size = 138, normalized size = 2.82 \[ \begin {cases} - \frac {A a c x \sin ^{2}{\left (e + f x \right )}}{2} - \frac {A a c x \cos ^{2}{\left (e + f x \right )}}{2} + A a c x + \frac {A a c \sin {\left (e + f x \right )} \cos {\left (e + f x \right )}}{2 f} + \frac {B a c \sin ^{2}{\left (e + f x \right )} \cos {\left (e + f x \right )}}{f} + \frac {2 B a c \cos ^{3}{\left (e + f x \right )}}{3 f} - \frac {B a c \cos {\left (e + f x \right )}}{f} & \text {for}\: f \neq 0 \\x \left (A + B \sin {\relax (e )}\right ) \left (a \sin {\relax (e )} + a\right ) \left (- c \sin {\relax (e )} + c\right ) & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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