Optimal. Leaf size=117 \[ \frac {a^3 A c^3 \sin (e+f x) \cos ^5(e+f x)}{6 f}+\frac {5 a^3 A c^3 \sin (e+f x) \cos ^3(e+f x)}{24 f}+\frac {5 a^3 A c^3 \sin (e+f x) \cos (e+f x)}{16 f}+\frac {5}{16} a^3 A c^3 x-\frac {a^3 B c^3 \cos ^7(e+f x)}{7 f} \]
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Rubi [A] time = 0.15, antiderivative size = 117, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 36, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {2967, 2669, 2635, 8} \[ \frac {a^3 A c^3 \sin (e+f x) \cos ^5(e+f x)}{6 f}+\frac {5 a^3 A c^3 \sin (e+f x) \cos ^3(e+f x)}{24 f}+\frac {5 a^3 A c^3 \sin (e+f x) \cos (e+f x)}{16 f}+\frac {5}{16} a^3 A c^3 x-\frac {a^3 B c^3 \cos ^7(e+f x)}{7 f} \]
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))^3 (A+B \sin (e+f x)) (c-c \sin (e+f x))^3 \, dx &=\left (a^3 c^3\right ) \int \cos ^6(e+f x) (A+B \sin (e+f x)) \, dx\\ &=-\frac {a^3 B c^3 \cos ^7(e+f x)}{7 f}+\left (a^3 A c^3\right ) \int \cos ^6(e+f x) \, dx\\ &=-\frac {a^3 B c^3 \cos ^7(e+f x)}{7 f}+\frac {a^3 A c^3 \cos ^5(e+f x) \sin (e+f x)}{6 f}+\frac {1}{6} \left (5 a^3 A c^3\right ) \int \cos ^4(e+f x) \, dx\\ &=-\frac {a^3 B c^3 \cos ^7(e+f x)}{7 f}+\frac {5 a^3 A c^3 \cos ^3(e+f x) \sin (e+f x)}{24 f}+\frac {a^3 A c^3 \cos ^5(e+f x) \sin (e+f x)}{6 f}+\frac {1}{8} \left (5 a^3 A c^3\right ) \int \cos ^2(e+f x) \, dx\\ &=-\frac {a^3 B c^3 \cos ^7(e+f x)}{7 f}+\frac {5 a^3 A c^3 \cos (e+f x) \sin (e+f x)}{16 f}+\frac {5 a^3 A c^3 \cos ^3(e+f x) \sin (e+f x)}{24 f}+\frac {a^3 A c^3 \cos ^5(e+f x) \sin (e+f x)}{6 f}+\frac {1}{16} \left (5 a^3 A c^3\right ) \int 1 \, dx\\ &=\frac {5}{16} a^3 A c^3 x-\frac {a^3 B c^3 \cos ^7(e+f x)}{7 f}+\frac {5 a^3 A c^3 \cos (e+f x) \sin (e+f x)}{16 f}+\frac {5 a^3 A c^3 \cos ^3(e+f x) \sin (e+f x)}{24 f}+\frac {a^3 A c^3 \cos ^5(e+f x) \sin (e+f x)}{6 f}\\ \end {align*}
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Mathematica [A] time = 0.23, size = 64, normalized size = 0.55 \[ \frac {a^3 c^3 \left (7 A (45 \sin (2 (e+f x))+9 \sin (4 (e+f x))+\sin (6 (e+f x))+60 e+60 f x)-192 B \cos ^7(e+f x)\right )}{1344 f} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 92, normalized size = 0.79 \[ -\frac {48 \, B a^{3} c^{3} \cos \left (f x + e\right )^{7} - 105 \, A a^{3} c^{3} f x - 7 \, {\left (8 \, A a^{3} c^{3} \cos \left (f x + e\right )^{5} + 10 \, A a^{3} c^{3} \cos \left (f x + e\right )^{3} + 15 \, A a^{3} c^{3} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{336 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.19, size = 162, normalized size = 1.38 \[ \frac {5}{16} \, A a^{3} c^{3} x - \frac {B a^{3} c^{3} \cos \left (7 \, f x + 7 \, e\right )}{448 \, f} - \frac {B a^{3} c^{3} \cos \left (5 \, f x + 5 \, e\right )}{64 \, f} - \frac {3 \, B a^{3} c^{3} \cos \left (3 \, f x + 3 \, e\right )}{64 \, f} - \frac {5 \, B a^{3} c^{3} \cos \left (f x + e\right )}{64 \, f} + \frac {A a^{3} c^{3} \sin \left (6 \, f x + 6 \, e\right )}{192 \, f} + \frac {3 \, A a^{3} c^{3} \sin \left (4 \, f x + 4 \, e\right )}{64 \, f} + \frac {15 \, A a^{3} c^{3} \sin \left (2 \, f x + 2 \, e\right )}{64 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.67, size = 263, normalized size = 2.25 \[ \frac {\frac {B \,a^{3} c^{3} \left (\frac {16}{5}+\sin ^{6}\left (f x +e \right )+\frac {6 \left (\sin ^{4}\left (f x +e \right )\right )}{5}+\frac {8 \left (\sin ^{2}\left (f x +e \right )\right )}{5}\right ) \cos \left (f x +e \right )}{7}-a^{3} A \,c^{3} \left (-\frac {\left (\sin ^{5}\left (f x +e \right )+\frac {5 \left (\sin ^{3}\left (f x +e \right )\right )}{4}+\frac {15 \sin \left (f x +e \right )}{8}\right ) \cos \left (f x +e \right )}{6}+\frac {5 f x}{16}+\frac {5 e}{16}\right )-\frac {3 B \,a^{3} c^{3} \left (\frac {8}{3}+\sin ^{4}\left (f x +e \right )+\frac {4 \left (\sin ^{2}\left (f x +e \right )\right )}{3}\right ) \cos \left (f x +e \right )}{5}+3 a^{3} A \,c^{3} \left (-\frac {\left (\sin ^{3}\left (f x +e \right )+\frac {3 \sin \left (f x +e \right )}{2}\right ) \cos \left (f x +e \right )}{4}+\frac {3 f x}{8}+\frac {3 e}{8}\right )+B \,a^{3} c^{3} \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )-3 a^{3} A \,c^{3} \left (-\frac {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )-B \,a^{3} c^{3} \cos \left (f x +e \right )+a^{3} A \,c^{3} \left (f x +e \right )}{f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.38, size = 264, normalized size = 2.26 \[ -\frac {35 \, {\left (4 \, \sin \left (2 \, f x + 2 \, e\right )^{3} + 60 \, f x + 60 \, e + 9 \, \sin \left (4 \, f x + 4 \, e\right ) - 48 \, \sin \left (2 \, f x + 2 \, e\right )\right )} A a^{3} c^{3} - 630 \, {\left (12 \, f x + 12 \, e + \sin \left (4 \, f x + 4 \, e\right ) - 8 \, \sin \left (2 \, f x + 2 \, e\right )\right )} A a^{3} c^{3} + 5040 \, {\left (2 \, f x + 2 \, e - \sin \left (2 \, f x + 2 \, e\right )\right )} A a^{3} c^{3} - 6720 \, {\left (f x + e\right )} A a^{3} c^{3} + 192 \, {\left (5 \, \cos \left (f x + e\right )^{7} - 21 \, \cos \left (f x + e\right )^{5} + 35 \, \cos \left (f x + e\right )^{3} - 35 \, \cos \left (f x + e\right )\right )} B a^{3} c^{3} + 1344 \, {\left (3 \, \cos \left (f x + e\right )^{5} - 10 \, \cos \left (f x + e\right )^{3} + 15 \, \cos \left (f x + e\right )\right )} B a^{3} c^{3} + 6720 \, {\left (\cos \left (f x + e\right )^{3} - 3 \, \cos \left (f x + e\right )\right )} B a^{3} c^{3} + 6720 \, B a^{3} c^{3} \cos \left (f x + e\right )}{6720 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 14.29, size = 325, normalized size = 2.78 \[ \frac {5\,A\,a^3\,c^3\,x}{16}-\frac {{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{12}\,\left (\frac {a^3\,c^3\,\left (672\,B-735\,A\,\left (e+f\,x\right )\right )}{336}+\frac {35\,A\,a^3\,c^3\,\left (e+f\,x\right )}{16}\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4\,\left (\frac {a^3\,c^3\,\left (2016\,B-2205\,A\,\left (e+f\,x\right )\right )}{336}+\frac {105\,A\,a^3\,c^3\,\left (e+f\,x\right )}{16}\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^8\,\left (\frac {a^3\,c^3\,\left (3360\,B-3675\,A\,\left (e+f\,x\right )\right )}{336}+\frac {175\,A\,a^3\,c^3\,\left (e+f\,x\right )}{16}\right )+\frac {a^3\,c^3\,\left (96\,B-105\,A\,\left (e+f\,x\right )\right )}{336}+\frac {5\,A\,a^3\,c^3\,\left (e+f\,x\right )}{16}-\frac {7\,A\,a^3\,c^3\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3}{6}-\frac {85\,A\,a^3\,c^3\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^5}{24}+\frac {85\,A\,a^3\,c^3\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^9}{24}+\frac {7\,A\,a^3\,c^3\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{11}}{6}+\frac {11\,A\,a^3\,c^3\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{13}}{8}-\frac {11\,A\,a^3\,c^3\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}{8}}{f\,{\left ({\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2+1\right )}^7} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 10.35, size = 682, normalized size = 5.83 \[ \begin {cases} - \frac {5 A a^{3} c^{3} x \sin ^{6}{\left (e + f x \right )}}{16} - \frac {15 A a^{3} c^{3} x \sin ^{4}{\left (e + f x \right )} \cos ^{2}{\left (e + f x \right )}}{16} + \frac {9 A a^{3} c^{3} x \sin ^{4}{\left (e + f x \right )}}{8} - \frac {15 A a^{3} c^{3} x \sin ^{2}{\left (e + f x \right )} \cos ^{4}{\left (e + f x \right )}}{16} + \frac {9 A a^{3} c^{3} x \sin ^{2}{\left (e + f x \right )} \cos ^{2}{\left (e + f x \right )}}{4} - \frac {3 A a^{3} c^{3} x \sin ^{2}{\left (e + f x \right )}}{2} - \frac {5 A a^{3} c^{3} x \cos ^{6}{\left (e + f x \right )}}{16} + \frac {9 A a^{3} c^{3} x \cos ^{4}{\left (e + f x \right )}}{8} - \frac {3 A a^{3} c^{3} x \cos ^{2}{\left (e + f x \right )}}{2} + A a^{3} c^{3} x + \frac {11 A a^{3} c^{3} \sin ^{5}{\left (e + f x \right )} \cos {\left (e + f x \right )}}{16 f} + \frac {5 A a^{3} c^{3} \sin ^{3}{\left (e + f x \right )} \cos ^{3}{\left (e + f x \right )}}{6 f} - \frac {15 A a^{3} c^{3} \sin ^{3}{\left (e + f x \right )} \cos {\left (e + f x \right )}}{8 f} + \frac {5 A a^{3} c^{3} \sin {\left (e + f x \right )} \cos ^{5}{\left (e + f x \right )}}{16 f} - \frac {9 A a^{3} c^{3} \sin {\left (e + f x \right )} \cos ^{3}{\left (e + f x \right )}}{8 f} + \frac {3 A a^{3} c^{3} \sin {\left (e + f x \right )} \cos {\left (e + f x \right )}}{2 f} + \frac {B a^{3} c^{3} \sin ^{6}{\left (e + f x \right )} \cos {\left (e + f x \right )}}{f} + \frac {2 B a^{3} c^{3} \sin ^{4}{\left (e + f x \right )} \cos ^{3}{\left (e + f x \right )}}{f} - \frac {3 B a^{3} c^{3} \sin ^{4}{\left (e + f x \right )} \cos {\left (e + f x \right )}}{f} + \frac {8 B a^{3} c^{3} \sin ^{2}{\left (e + f x \right )} \cos ^{5}{\left (e + f x \right )}}{5 f} - \frac {4 B a^{3} c^{3} \sin ^{2}{\left (e + f x \right )} \cos ^{3}{\left (e + f x \right )}}{f} + \frac {3 B a^{3} c^{3} \sin ^{2}{\left (e + f x \right )} \cos {\left (e + f x \right )}}{f} + \frac {16 B a^{3} c^{3} \cos ^{7}{\left (e + f x \right )}}{35 f} - \frac {8 B a^{3} c^{3} \cos ^{5}{\left (e + f x \right )}}{5 f} + \frac {2 B a^{3} c^{3} \cos ^{3}{\left (e + f x \right )}}{f} - \frac {B a^{3} 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 )^{3} \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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