Optimal. Leaf size=138 \[ -\frac {a^3 c^2 (6 A+B) \cos ^5(e+f x)}{30 f}+\frac {a^3 c^2 (6 A+B) \sin (e+f x) \cos ^3(e+f x)}{24 f}+\frac {a^3 c^2 (6 A+B) \sin (e+f x) \cos (e+f x)}{16 f}+\frac {1}{16} a^3 c^2 x (6 A+B)-\frac {B c^2 \cos ^5(e+f x) \left (a^3 \sin (e+f x)+a^3\right )}{6 f} \]
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Rubi [A] time = 0.20, antiderivative size = 138, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 36, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.139, Rules used = {2967, 2860, 2669, 2635, 8} \[ -\frac {a^3 c^2 (6 A+B) \cos ^5(e+f x)}{30 f}+\frac {a^3 c^2 (6 A+B) \sin (e+f x) \cos ^3(e+f x)}{24 f}+\frac {a^3 c^2 (6 A+B) \sin (e+f x) \cos (e+f x)}{16 f}+\frac {1}{16} a^3 c^2 x (6 A+B)-\frac {B c^2 \cos ^5(e+f x) \left (a^3 \sin (e+f x)+a^3\right )}{6 f} \]
Antiderivative was successfully verified.
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Rule 8
Rule 2635
Rule 2669
Rule 2860
Rule 2967
Rubi steps
\begin {align*} \int (a+a \sin (e+f x))^3 (A+B \sin (e+f x)) (c-c \sin (e+f x))^2 \, dx &=\left (a^2 c^2\right ) \int \cos ^4(e+f x) (a+a \sin (e+f x)) (A+B \sin (e+f x)) \, dx\\ &=-\frac {B c^2 \cos ^5(e+f x) \left (a^3+a^3 \sin (e+f x)\right )}{6 f}+\frac {1}{6} \left (a^2 (6 A+B) c^2\right ) \int \cos ^4(e+f x) (a+a \sin (e+f x)) \, dx\\ &=-\frac {a^3 (6 A+B) c^2 \cos ^5(e+f x)}{30 f}-\frac {B c^2 \cos ^5(e+f x) \left (a^3+a^3 \sin (e+f x)\right )}{6 f}+\frac {1}{6} \left (a^3 (6 A+B) c^2\right ) \int \cos ^4(e+f x) \, dx\\ &=-\frac {a^3 (6 A+B) c^2 \cos ^5(e+f x)}{30 f}+\frac {a^3 (6 A+B) c^2 \cos ^3(e+f x) \sin (e+f x)}{24 f}-\frac {B c^2 \cos ^5(e+f x) \left (a^3+a^3 \sin (e+f x)\right )}{6 f}+\frac {1}{8} \left (a^3 (6 A+B) c^2\right ) \int \cos ^2(e+f x) \, dx\\ &=-\frac {a^3 (6 A+B) c^2 \cos ^5(e+f x)}{30 f}+\frac {a^3 (6 A+B) c^2 \cos (e+f x) \sin (e+f x)}{16 f}+\frac {a^3 (6 A+B) c^2 \cos ^3(e+f x) \sin (e+f x)}{24 f}-\frac {B c^2 \cos ^5(e+f x) \left (a^3+a^3 \sin (e+f x)\right )}{6 f}+\frac {1}{16} \left (a^3 (6 A+B) c^2\right ) \int 1 \, dx\\ &=\frac {1}{16} a^3 (6 A+B) c^2 x-\frac {a^3 (6 A+B) c^2 \cos ^5(e+f x)}{30 f}+\frac {a^3 (6 A+B) c^2 \cos (e+f x) \sin (e+f x)}{16 f}+\frac {a^3 (6 A+B) c^2 \cos ^3(e+f x) \sin (e+f x)}{24 f}-\frac {B c^2 \cos ^5(e+f x) \left (a^3+a^3 \sin (e+f x)\right )}{6 f}\\ \end {align*}
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Mathematica [A] time = 1.09, size = 133, normalized size = 0.96 \[ \frac {a^3 c^2 (-120 (A+B) \cos (e+f x)-60 (A+B) \cos (3 (e+f x))+240 A \sin (2 (e+f x))+30 A \sin (4 (e+f x))-12 A \cos (5 (e+f x))+360 A e+360 A f x+15 B \sin (2 (e+f x))-15 B \sin (4 (e+f x))-5 B \sin (6 (e+f x))-12 B \cos (5 (e+f x))+60 B e+60 B f x)}{960 f} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 106, normalized size = 0.77 \[ -\frac {48 \, {\left (A + B\right )} a^{3} c^{2} \cos \left (f x + e\right )^{5} - 15 \, {\left (6 \, A + B\right )} a^{3} c^{2} f x + 5 \, {\left (8 \, B a^{3} c^{2} \cos \left (f x + e\right )^{5} - 2 \, {\left (6 \, A + B\right )} a^{3} c^{2} \cos \left (f x + e\right )^{3} - 3 \, {\left (6 \, A + B\right )} a^{3} c^{2} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{240 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.19, size = 204, normalized size = 1.48 \[ -\frac {B a^{3} c^{2} \sin \left (6 \, f x + 6 \, e\right )}{192 \, f} + \frac {1}{16} \, {\left (6 \, A a^{3} c^{2} + B a^{3} c^{2}\right )} x - \frac {{\left (A a^{3} c^{2} + B a^{3} c^{2}\right )} \cos \left (5 \, f x + 5 \, e\right )}{80 \, f} - \frac {{\left (A a^{3} c^{2} + B a^{3} c^{2}\right )} \cos \left (3 \, f x + 3 \, e\right )}{16 \, f} - \frac {{\left (A a^{3} c^{2} + B a^{3} c^{2}\right )} \cos \left (f x + e\right )}{8 \, f} + \frac {{\left (2 \, A a^{3} c^{2} - B a^{3} c^{2}\right )} \sin \left (4 \, f x + 4 \, e\right )}{64 \, f} + \frac {{\left (16 \, A a^{3} c^{2} + B a^{3} c^{2}\right )} \sin \left (2 \, f x + 2 \, e\right )}{64 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.58, size = 364, normalized size = 2.64 \[ \frac {-\frac {a^{3} A \,c^{2} \left (\frac {8}{3}+\sin ^{4}\left (f x +e \right )+\frac {4 \left (\sin ^{2}\left (f x +e \right )\right )}{3}\right ) \cos \left (f x +e \right )}{5}+a^{3} A \,c^{2} \left (-\frac {\left (\sin ^{3}\left (f x +e \right )+\frac {3 \sin \left (f x +e \right )}{2}\right ) \cos \left (f x +e \right )}{4}+\frac {3 f x}{8}+\frac {3 e}{8}\right )+\frac {2 a^{3} A \,c^{2} \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )}{3}+B \,a^{3} c^{2} \left (-\frac {\left (\sin ^{5}\left (f x +e \right )+\frac {5 \left (\sin ^{3}\left (f x +e \right )\right )}{4}+\frac {15 \sin \left (f x +e \right )}{8}\right ) \cos \left (f x +e \right )}{6}+\frac {5 f x}{16}+\frac {5 e}{16}\right )-\frac {B \,a^{3} c^{2} \left (\frac {8}{3}+\sin ^{4}\left (f x +e \right )+\frac {4 \left (\sin ^{2}\left (f x +e \right )\right )}{3}\right ) \cos \left (f x +e \right )}{5}-2 B \,a^{3} c^{2} \left (-\frac {\left (\sin ^{3}\left (f x +e \right )+\frac {3 \sin \left (f x +e \right )}{2}\right ) \cos \left (f x +e \right )}{4}+\frac {3 f x}{8}+\frac {3 e}{8}\right )-2 a^{3} A \,c^{2} \left (-\frac {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )+\frac {2 B \,a^{3} c^{2} \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )}{3}-a^{3} A \,c^{2} \cos \left (f x +e \right )+B \,a^{3} c^{2} \left (-\frac {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )+a^{3} A \,c^{2} \left (f x +e \right )-B \,a^{3} c^{2} \cos \left (f x +e \right )}{f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.34, size = 360, normalized size = 2.61 \[ -\frac {64 \, {\left (3 \, \cos \left (f x + e\right )^{5} - 10 \, \cos \left (f x + e\right )^{3} + 15 \, \cos \left (f x + e\right )\right )} A a^{3} c^{2} + 640 \, {\left (\cos \left (f x + e\right )^{3} - 3 \, \cos \left (f x + e\right )\right )} A a^{3} c^{2} - 30 \, {\left (12 \, f x + 12 \, e + \sin \left (4 \, f x + 4 \, e\right ) - 8 \, \sin \left (2 \, f x + 2 \, e\right )\right )} A a^{3} c^{2} + 480 \, {\left (2 \, f x + 2 \, e - \sin \left (2 \, f x + 2 \, e\right )\right )} A a^{3} c^{2} - 960 \, {\left (f x + e\right )} A a^{3} c^{2} + 64 \, {\left (3 \, \cos \left (f x + e\right )^{5} - 10 \, \cos \left (f x + e\right )^{3} + 15 \, \cos \left (f x + e\right )\right )} B a^{3} c^{2} + 640 \, {\left (\cos \left (f x + e\right )^{3} - 3 \, \cos \left (f x + e\right )\right )} B a^{3} c^{2} - 5 \, {\left (4 \, \sin \left (2 \, f x + 2 \, e\right )^{3} + 60 \, f x + 60 \, e + 9 \, \sin \left (4 \, f x + 4 \, e\right ) - 48 \, \sin \left (2 \, f x + 2 \, e\right )\right )} B a^{3} c^{2} + 60 \, {\left (12 \, f x + 12 \, e + \sin \left (4 \, f x + 4 \, e\right ) - 8 \, \sin \left (2 \, f x + 2 \, e\right )\right )} B a^{3} c^{2} - 240 \, {\left (2 \, f x + 2 \, e - \sin \left (2 \, f x + 2 \, e\right )\right )} B a^{3} c^{2} + 960 \, A a^{3} c^{2} \cos \left (f x + e\right ) + 960 \, B a^{3} c^{2} \cos \left (f x + e\right )}{960 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 14.17, size = 536, normalized size = 3.88 \[ \frac {a^3\,c^2\,\mathrm {atan}\left (\frac {a^3\,c^2\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (6\,A+B\right )}{8\,\left (\frac {3\,A\,a^3\,c^2}{4}+\frac {B\,a^3\,c^2}{8}\right )}\right )\,\left (6\,A+B\right )}{8\,f}-\frac {{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4\,\left (4\,A\,a^3\,c^2+4\,B\,a^3\,c^2\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^8\,\left (2\,A\,a^3\,c^2+2\,B\,a^3\,c^2\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6\,\left (4\,A\,a^3\,c^2+4\,B\,a^3\,c^2\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{10}\,\left (2\,A\,a^3\,c^2+2\,B\,a^3\,c^2\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2\,\left (\frac {2\,A\,a^3\,c^2}{5}+\frac {2\,B\,a^3\,c^2}{5}\right )-{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^5\,\left (\frac {A\,a^3\,c^2}{2}-\frac {13\,B\,a^3\,c^2}{4}\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^7\,\left (\frac {A\,a^3\,c^2}{2}-\frac {13\,B\,a^3\,c^2}{4}\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{11}\,\left (\frac {5\,A\,a^3\,c^2}{4}-\frac {B\,a^3\,c^2}{8}\right )-{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3\,\left (\frac {7\,A\,a^3\,c^2}{4}+\frac {47\,B\,a^3\,c^2}{24}\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^9\,\left (\frac {7\,A\,a^3\,c^2}{4}+\frac {47\,B\,a^3\,c^2}{24}\right )-\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (\frac {5\,A\,a^3\,c^2}{4}-\frac {B\,a^3\,c^2}{8}\right )+\frac {2\,A\,a^3\,c^2}{5}+\frac {2\,B\,a^3\,c^2}{5}}{f\,\left ({\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{12}+6\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{10}+15\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^8+20\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6+15\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4+6\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2+1\right )}-\frac {a^3\,c^2\,\left (6\,A+B\right )\,\left (\mathrm {atan}\left (\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\right )-\frac {f\,x}{2}\right )}{8\,f} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 8.10, size = 910, normalized size = 6.59 \[ \begin {cases} \frac {3 A a^{3} c^{2} x \sin ^{4}{\left (e + f x \right )}}{8} + \frac {3 A a^{3} c^{2} x \sin ^{2}{\left (e + f x \right )} \cos ^{2}{\left (e + f x \right )}}{4} - A a^{3} c^{2} x \sin ^{2}{\left (e + f x \right )} + \frac {3 A a^{3} c^{2} x \cos ^{4}{\left (e + f x \right )}}{8} - A a^{3} c^{2} x \cos ^{2}{\left (e + f x \right )} + A a^{3} c^{2} x - \frac {A a^{3} c^{2} \sin ^{4}{\left (e + f x \right )} \cos {\left (e + f x \right )}}{f} - \frac {5 A a^{3} c^{2} \sin ^{3}{\left (e + f x \right )} \cos {\left (e + f x \right )}}{8 f} - \frac {4 A a^{3} c^{2} \sin ^{2}{\left (e + f x \right )} \cos ^{3}{\left (e + f x \right )}}{3 f} + \frac {2 A a^{3} c^{2} \sin ^{2}{\left (e + f x \right )} \cos {\left (e + f x \right )}}{f} - \frac {3 A a^{3} c^{2} \sin {\left (e + f x \right )} \cos ^{3}{\left (e + f x \right )}}{8 f} + \frac {A a^{3} c^{2} \sin {\left (e + f x \right )} \cos {\left (e + f x \right )}}{f} - \frac {8 A a^{3} c^{2} \cos ^{5}{\left (e + f x \right )}}{15 f} + \frac {4 A a^{3} c^{2} \cos ^{3}{\left (e + f x \right )}}{3 f} - \frac {A a^{3} c^{2} \cos {\left (e + f x \right )}}{f} + \frac {5 B a^{3} c^{2} x \sin ^{6}{\left (e + f x \right )}}{16} + \frac {15 B a^{3} c^{2} x \sin ^{4}{\left (e + f x \right )} \cos ^{2}{\left (e + f x \right )}}{16} - \frac {3 B a^{3} c^{2} x \sin ^{4}{\left (e + f x \right )}}{4} + \frac {15 B a^{3} c^{2} x \sin ^{2}{\left (e + f x \right )} \cos ^{4}{\left (e + f x \right )}}{16} - \frac {3 B a^{3} c^{2} x \sin ^{2}{\left (e + f x \right )} \cos ^{2}{\left (e + f x \right )}}{2} + \frac {B a^{3} c^{2} x \sin ^{2}{\left (e + f x \right )}}{2} + \frac {5 B a^{3} c^{2} x \cos ^{6}{\left (e + f x \right )}}{16} - \frac {3 B a^{3} c^{2} x \cos ^{4}{\left (e + f x \right )}}{4} + \frac {B a^{3} c^{2} x \cos ^{2}{\left (e + f x \right )}}{2} - \frac {11 B a^{3} c^{2} \sin ^{5}{\left (e + f x \right )} \cos {\left (e + f x \right )}}{16 f} - \frac {B a^{3} c^{2} \sin ^{4}{\left (e + f x \right )} \cos {\left (e + f x \right )}}{f} - \frac {5 B a^{3} c^{2} \sin ^{3}{\left (e + f x \right )} \cos ^{3}{\left (e + f x \right )}}{6 f} + \frac {5 B a^{3} c^{2} \sin ^{3}{\left (e + f x \right )} \cos {\left (e + f x \right )}}{4 f} - \frac {4 B a^{3} c^{2} \sin ^{2}{\left (e + f x \right )} \cos ^{3}{\left (e + f x \right )}}{3 f} + \frac {2 B a^{3} c^{2} \sin ^{2}{\left (e + f x \right )} \cos {\left (e + f x \right )}}{f} - \frac {5 B a^{3} c^{2} \sin {\left (e + f x \right )} \cos ^{5}{\left (e + f x \right )}}{16 f} + \frac {3 B a^{3} c^{2} \sin {\left (e + f x \right )} \cos ^{3}{\left (e + f x \right )}}{4 f} - \frac {B a^{3} c^{2} \sin {\left (e + f x \right )} \cos {\left (e + f x \right )}}{2 f} - \frac {8 B a^{3} c^{2} \cos ^{5}{\left (e + f x \right )}}{15 f} + \frac {4 B a^{3} c^{2} \cos ^{3}{\left (e + f x \right )}}{3 f} - \frac {B a^{3} c^{2} \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 )^{3} \left (- c \sin {\relax (e )} + c\right )^{2} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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