Optimal. Leaf size=142 \[ \frac {a c^3 (5 A-2 B) \cos ^3(e+f x)}{12 f}+\frac {a (5 A-2 B) \cos ^3(e+f x) \left (c^3-c^3 \sin (e+f x)\right )}{20 f}+\frac {a c^3 (5 A-2 B) \sin (e+f x) \cos (e+f x)}{8 f}+\frac {1}{8} a c^3 x (5 A-2 B)-\frac {a B c \cos ^3(e+f x) (c-c \sin (e+f x))^2}{5 f} \]
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Rubi [A] time = 0.25, antiderivative size = 142, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 34, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {2967, 2860, 2678, 2669, 2635, 8} \[ \frac {a c^3 (5 A-2 B) \cos ^3(e+f x)}{12 f}+\frac {a (5 A-2 B) \cos ^3(e+f x) \left (c^3-c^3 \sin (e+f x)\right )}{20 f}+\frac {a c^3 (5 A-2 B) \sin (e+f x) \cos (e+f x)}{8 f}+\frac {1}{8} a c^3 x (5 A-2 B)-\frac {a B c \cos ^3(e+f x) (c-c \sin (e+f x))^2}{5 f} \]
Antiderivative was successfully verified.
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Rule 8
Rule 2635
Rule 2669
Rule 2678
Rule 2860
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))^3 \, dx &=(a c) \int \cos ^2(e+f x) (A+B \sin (e+f x)) (c-c \sin (e+f x))^2 \, dx\\ &=-\frac {a B c \cos ^3(e+f x) (c-c \sin (e+f x))^2}{5 f}+\frac {1}{5} (a (5 A-2 B) c) \int \cos ^2(e+f x) (c-c \sin (e+f x))^2 \, dx\\ &=-\frac {a B c \cos ^3(e+f x) (c-c \sin (e+f x))^2}{5 f}+\frac {a (5 A-2 B) \cos ^3(e+f x) \left (c^3-c^3 \sin (e+f x)\right )}{20 f}+\frac {1}{4} \left (a (5 A-2 B) c^2\right ) \int \cos ^2(e+f x) (c-c \sin (e+f x)) \, dx\\ &=\frac {a (5 A-2 B) c^3 \cos ^3(e+f x)}{12 f}-\frac {a B c \cos ^3(e+f x) (c-c \sin (e+f x))^2}{5 f}+\frac {a (5 A-2 B) \cos ^3(e+f x) \left (c^3-c^3 \sin (e+f x)\right )}{20 f}+\frac {1}{4} \left (a (5 A-2 B) c^3\right ) \int \cos ^2(e+f x) \, dx\\ &=\frac {a (5 A-2 B) c^3 \cos ^3(e+f x)}{12 f}+\frac {a (5 A-2 B) c^3 \cos (e+f x) \sin (e+f x)}{8 f}-\frac {a B c \cos ^3(e+f x) (c-c \sin (e+f x))^2}{5 f}+\frac {a (5 A-2 B) \cos ^3(e+f x) \left (c^3-c^3 \sin (e+f x)\right )}{20 f}+\frac {1}{8} \left (a (5 A-2 B) c^3\right ) \int 1 \, dx\\ &=\frac {1}{8} a (5 A-2 B) c^3 x+\frac {a (5 A-2 B) c^3 \cos ^3(e+f x)}{12 f}+\frac {a (5 A-2 B) c^3 \cos (e+f x) \sin (e+f x)}{8 f}-\frac {a B c \cos ^3(e+f x) (c-c \sin (e+f x))^2}{5 f}+\frac {a (5 A-2 B) \cos ^3(e+f x) \left (c^3-c^3 \sin (e+f x)\right )}{20 f}\\ \end {align*}
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Mathematica [A] time = 0.84, size = 95, normalized size = 0.67 \[ \frac {a c^3 (15 (-(A-2 B) \sin (4 (e+f x))+4 f x (5 A-2 B)+8 A \sin (2 (e+f x)))+60 (4 A-3 B) \cos (e+f x)+10 (8 A-5 B) \cos (3 (e+f x))+6 B \cos (5 (e+f x)))}{480 f} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 102, normalized size = 0.72 \[ \frac {24 \, B a c^{3} \cos \left (f x + e\right )^{5} + 80 \, {\left (A - B\right )} a c^{3} \cos \left (f x + e\right )^{3} + 15 \, {\left (5 \, A - 2 \, B\right )} a c^{3} f x - 15 \, {\left (2 \, {\left (A - 2 \, B\right )} a c^{3} \cos \left (f x + e\right )^{3} - {\left (5 \, A - 2 \, B\right )} a c^{3} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{120 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.18, size = 145, normalized size = 1.02 \[ \frac {B a c^{3} \cos \left (5 \, f x + 5 \, e\right )}{80 \, f} + \frac {A a c^{3} \sin \left (2 \, f x + 2 \, e\right )}{4 \, f} + \frac {1}{8} \, {\left (5 \, A a c^{3} - 2 \, B a c^{3}\right )} x + \frac {{\left (8 \, A a c^{3} - 5 \, B a c^{3}\right )} \cos \left (3 \, f x + 3 \, e\right )}{48 \, f} + \frac {{\left (4 \, A a c^{3} - 3 \, B a c^{3}\right )} \cos \left (f x + e\right )}{8 \, f} - \frac {{\left (A a c^{3} - 2 \, B a c^{3}\right )} \sin \left (4 \, f x + 4 \, e\right )}{32 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.48, size = 208, normalized size = 1.46 \[ \frac {-A \,c^{3} a \left (-\frac {\left (\sin ^{3}\left (f x +e \right )+\frac {3 \sin \left (f x +e \right )}{2}\right ) \cos \left (f x +e \right )}{4}+\frac {3 f x}{8}+\frac {3 e}{8}\right )-\frac {2 A \,c^{3} a \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )}{3}+2 A \,c^{3} a \cos \left (f x +e \right )+\frac {B \,c^{3} a \left (\frac {8}{3}+\sin ^{4}\left (f x +e \right )+\frac {4 \left (\sin ^{2}\left (f x +e \right )\right )}{3}\right ) \cos \left (f x +e \right )}{5}+2 B \,c^{3} a \left (-\frac {\left (\sin ^{3}\left (f x +e \right )+\frac {3 \sin \left (f x +e \right )}{2}\right ) \cos \left (f x +e \right )}{4}+\frac {3 f x}{8}+\frac {3 e}{8}\right )-2 B \,c^{3} a \left (-\frac {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )+A \,c^{3} a \left (f x +e \right )-B \,c^{3} a \cos \left (f x +e \right )}{f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.33, size = 200, normalized size = 1.41 \[ \frac {320 \, {\left (\cos \left (f x + e\right )^{3} - 3 \, \cos \left (f x + e\right )\right )} A a c^{3} - 15 \, {\left (12 \, f x + 12 \, e + \sin \left (4 \, f x + 4 \, e\right ) - 8 \, \sin \left (2 \, f x + 2 \, e\right )\right )} A a c^{3} + 480 \, {\left (f x + e\right )} A a c^{3} + 32 \, {\left (3 \, \cos \left (f x + e\right )^{5} - 10 \, \cos \left (f x + e\right )^{3} + 15 \, \cos \left (f x + e\right )\right )} B a c^{3} + 30 \, {\left (12 \, f x + 12 \, e + \sin \left (4 \, f x + 4 \, e\right ) - 8 \, \sin \left (2 \, f x + 2 \, e\right )\right )} B a c^{3} - 240 \, {\left (2 \, f x + 2 \, e - \sin \left (2 \, f x + 2 \, e\right )\right )} B a c^{3} + 960 \, A a c^{3} \cos \left (f x + e\right ) - 480 \, B a c^{3} \cos \left (f x + e\right )}{480 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 13.80, size = 389, normalized size = 2.74 \[ \frac {\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (\frac {3\,A\,a\,c^3}{4}+\frac {B\,a\,c^3}{2}\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^8\,\left (4\,A\,a\,c^3-2\,B\,a\,c^3\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3\,\left (\frac {7\,A\,a\,c^3}{2}-3\,B\,a\,c^3\right )-{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^7\,\left (\frac {7\,A\,a\,c^3}{2}-3\,B\,a\,c^3\right )-{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^9\,\left (\frac {3\,A\,a\,c^3}{4}+\frac {B\,a\,c^3}{2}\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6\,\left (8\,A\,a\,c^3-8\,B\,a\,c^3\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2\,\left (\frac {8\,A\,a\,c^3}{3}-\frac {8\,B\,a\,c^3}{3}\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4\,\left (\frac {16\,A\,a\,c^3}{3}-\frac {4\,B\,a\,c^3}{3}\right )+\frac {4\,A\,a\,c^3}{3}-\frac {14\,B\,a\,c^3}{15}}{f\,\left ({\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{10}+5\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^8+10\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6+10\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4+5\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2+1\right )}+\frac {a\,c^3\,\mathrm {atan}\left (\frac {a\,c^3\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (5\,A-2\,B\right )}{4\,\left (\frac {5\,A\,a\,c^3}{4}-\frac {B\,a\,c^3}{2}\right )}\right )\,\left (5\,A-2\,B\right )}{4\,f}-\frac {a\,c^3\,\left (5\,A-2\,B\right )\,\left (\mathrm {atan}\left (\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\right )-\frac {f\,x}{2}\right )}{4\,f} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 4.09, size = 486, normalized size = 3.42 \[ \begin {cases} - \frac {3 A a c^{3} x \sin ^{4}{\left (e + f x \right )}}{8} - \frac {3 A a c^{3} x \sin ^{2}{\left (e + f x \right )} \cos ^{2}{\left (e + f x \right )}}{4} - \frac {3 A a c^{3} x \cos ^{4}{\left (e + f x \right )}}{8} + A a c^{3} x + \frac {5 A a c^{3} \sin ^{3}{\left (e + f x \right )} \cos {\left (e + f x \right )}}{8 f} - \frac {2 A a c^{3} \sin ^{2}{\left (e + f x \right )} \cos {\left (e + f x \right )}}{f} + \frac {3 A a c^{3} \sin {\left (e + f x \right )} \cos ^{3}{\left (e + f x \right )}}{8 f} - \frac {4 A a c^{3} \cos ^{3}{\left (e + f x \right )}}{3 f} + \frac {2 A a c^{3} \cos {\left (e + f x \right )}}{f} + \frac {3 B a c^{3} x \sin ^{4}{\left (e + f x \right )}}{4} + \frac {3 B a c^{3} x \sin ^{2}{\left (e + f x \right )} \cos ^{2}{\left (e + f x \right )}}{2} - B a c^{3} x \sin ^{2}{\left (e + f x \right )} + \frac {3 B a c^{3} x \cos ^{4}{\left (e + f x \right )}}{4} - B a c^{3} x \cos ^{2}{\left (e + f x \right )} + \frac {B a c^{3} \sin ^{4}{\left (e + f x \right )} \cos {\left (e + f x \right )}}{f} - \frac {5 B a c^{3} \sin ^{3}{\left (e + f x \right )} \cos {\left (e + f x \right )}}{4 f} + \frac {4 B a c^{3} \sin ^{2}{\left (e + f x \right )} \cos ^{3}{\left (e + f x \right )}}{3 f} - \frac {3 B a c^{3} \sin {\left (e + f x \right )} \cos ^{3}{\left (e + f x \right )}}{4 f} + \frac {B a c^{3} \sin {\left (e + f x \right )} \cos {\left (e + f x \right )}}{f} + \frac {8 B a c^{3} \cos ^{5}{\left (e + f x \right )}}{15 f} - \frac {B a c^{3} \cos {\left (e + f x \right )}}{f} & \text {for}\: f \neq 0 \\x \left (A + B \sin {\relax (e )}\right ) \left (a \sin {\relax (e )} + a\right ) \left (- c \sin {\relax (e )} + c\right )^{3} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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