Optimal. Leaf size=189 \[ \frac {a^2 c^4 (7 A-2 B) \cos ^5(e+f x)}{30 f}+\frac {a^2 (7 A-2 B) \cos ^5(e+f x) \left (c^4-c^4 \sin (e+f x)\right )}{42 f}+\frac {a^2 c^4 (7 A-2 B) \sin (e+f x) \cos ^3(e+f x)}{24 f}+\frac {a^2 c^4 (7 A-2 B) \sin (e+f x) \cos (e+f x)}{16 f}+\frac {1}{16} a^2 c^4 x (7 A-2 B)-\frac {a^2 B \cos ^5(e+f x) \left (c^2-c^2 \sin (e+f x)\right )^2}{7 f} \]
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Rubi [A] time = 0.30, antiderivative size = 189, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 36, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {2967, 2860, 2678, 2669, 2635, 8} \[ \frac {a^2 c^4 (7 A-2 B) \cos ^5(e+f x)}{30 f}+\frac {a^2 (7 A-2 B) \cos ^5(e+f x) \left (c^4-c^4 \sin (e+f x)\right )}{42 f}+\frac {a^2 c^4 (7 A-2 B) \sin (e+f x) \cos ^3(e+f x)}{24 f}+\frac {a^2 c^4 (7 A-2 B) \sin (e+f x) \cos (e+f x)}{16 f}+\frac {1}{16} a^2 c^4 x (7 A-2 B)-\frac {a^2 B \cos ^5(e+f x) \left (c^2-c^2 \sin (e+f x)\right )^2}{7 f} \]
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))^2 (A+B \sin (e+f x)) (c-c \sin (e+f x))^4 \, dx &=\left (a^2 c^2\right ) \int \cos ^4(e+f x) (A+B \sin (e+f x)) (c-c \sin (e+f x))^2 \, dx\\ &=-\frac {a^2 B \cos ^5(e+f x) \left (c^2-c^2 \sin (e+f x)\right )^2}{7 f}+\frac {1}{7} \left (a^2 (7 A-2 B) c^2\right ) \int \cos ^4(e+f x) (c-c \sin (e+f x))^2 \, dx\\ &=-\frac {a^2 B \cos ^5(e+f x) \left (c^2-c^2 \sin (e+f x)\right )^2}{7 f}+\frac {a^2 (7 A-2 B) \cos ^5(e+f x) \left (c^4-c^4 \sin (e+f x)\right )}{42 f}+\frac {1}{6} \left (a^2 (7 A-2 B) c^3\right ) \int \cos ^4(e+f x) (c-c \sin (e+f x)) \, dx\\ &=\frac {a^2 (7 A-2 B) c^4 \cos ^5(e+f x)}{30 f}-\frac {a^2 B \cos ^5(e+f x) \left (c^2-c^2 \sin (e+f x)\right )^2}{7 f}+\frac {a^2 (7 A-2 B) \cos ^5(e+f x) \left (c^4-c^4 \sin (e+f x)\right )}{42 f}+\frac {1}{6} \left (a^2 (7 A-2 B) c^4\right ) \int \cos ^4(e+f x) \, dx\\ &=\frac {a^2 (7 A-2 B) c^4 \cos ^5(e+f x)}{30 f}+\frac {a^2 (7 A-2 B) c^4 \cos ^3(e+f x) \sin (e+f x)}{24 f}-\frac {a^2 B \cos ^5(e+f x) \left (c^2-c^2 \sin (e+f x)\right )^2}{7 f}+\frac {a^2 (7 A-2 B) \cos ^5(e+f x) \left (c^4-c^4 \sin (e+f x)\right )}{42 f}+\frac {1}{8} \left (a^2 (7 A-2 B) c^4\right ) \int \cos ^2(e+f x) \, dx\\ &=\frac {a^2 (7 A-2 B) c^4 \cos ^5(e+f x)}{30 f}+\frac {a^2 (7 A-2 B) c^4 \cos (e+f x) \sin (e+f x)}{16 f}+\frac {a^2 (7 A-2 B) c^4 \cos ^3(e+f x) \sin (e+f x)}{24 f}-\frac {a^2 B \cos ^5(e+f x) \left (c^2-c^2 \sin (e+f x)\right )^2}{7 f}+\frac {a^2 (7 A-2 B) \cos ^5(e+f x) \left (c^4-c^4 \sin (e+f x)\right )}{42 f}+\frac {1}{16} \left (a^2 (7 A-2 B) c^4\right ) \int 1 \, dx\\ &=\frac {1}{16} a^2 (7 A-2 B) c^4 x+\frac {a^2 (7 A-2 B) c^4 \cos ^5(e+f x)}{30 f}+\frac {a^2 (7 A-2 B) c^4 \cos (e+f x) \sin (e+f x)}{16 f}+\frac {a^2 (7 A-2 B) c^4 \cos ^3(e+f x) \sin (e+f x)}{24 f}-\frac {a^2 B \cos ^5(e+f x) \left (c^2-c^2 \sin (e+f x)\right )^2}{7 f}+\frac {a^2 (7 A-2 B) \cos ^5(e+f x) \left (c^4-c^4 \sin (e+f x)\right )}{42 f}\\ \end {align*}
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Mathematica [A] time = 1.59, size = 163, normalized size = 0.86 \[ \frac {a^2 c^4 (105 (16 A-11 B) \cos (e+f x)+105 (8 A-5 B) \cos (3 (e+f x))+1785 A \sin (2 (e+f x))+105 A \sin (4 (e+f x))-35 A \sin (6 (e+f x))+168 A \cos (5 (e+f x))+2940 A e+2940 A f x-210 B \sin (2 (e+f x))+210 B \sin (4 (e+f x))+70 B \sin (6 (e+f x))-63 B \cos (5 (e+f x))+15 B \cos (7 (e+f x))-840 B e-840 B f x)}{6720 f} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 135, normalized size = 0.71 \[ \frac {240 \, B a^{2} c^{4} \cos \left (f x + e\right )^{7} + 672 \, {\left (A - B\right )} a^{2} c^{4} \cos \left (f x + e\right )^{5} + 105 \, {\left (7 \, A - 2 \, B\right )} a^{2} c^{4} f x - 35 \, {\left (8 \, {\left (A - 2 \, B\right )} a^{2} c^{4} \cos \left (f x + e\right )^{5} - 2 \, {\left (7 \, A - 2 \, B\right )} a^{2} c^{4} \cos \left (f x + e\right )^{3} - 3 \, {\left (7 \, A - 2 \, B\right )} a^{2} c^{4} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{1680 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.21, size = 244, normalized size = 1.29 \[ \frac {B a^{2} c^{4} \cos \left (7 \, f x + 7 \, e\right )}{448 \, f} + \frac {1}{16} \, {\left (7 \, A a^{2} c^{4} - 2 \, B a^{2} c^{4}\right )} x + \frac {{\left (8 \, A a^{2} c^{4} - 3 \, B a^{2} c^{4}\right )} \cos \left (5 \, f x + 5 \, e\right )}{320 \, f} + \frac {{\left (8 \, A a^{2} c^{4} - 5 \, B a^{2} c^{4}\right )} \cos \left (3 \, f x + 3 \, e\right )}{64 \, f} + \frac {{\left (16 \, A a^{2} c^{4} - 11 \, B a^{2} c^{4}\right )} \cos \left (f x + e\right )}{64 \, f} - \frac {{\left (A a^{2} c^{4} - 2 \, B a^{2} c^{4}\right )} \sin \left (6 \, f x + 6 \, e\right )}{192 \, f} + \frac {{\left (A a^{2} c^{4} + 2 \, B a^{2} c^{4}\right )} \sin \left (4 \, f x + 4 \, e\right )}{64 \, f} + \frac {{\left (17 \, A a^{2} c^{4} - 2 \, B a^{2} c^{4}\right )} \sin \left (2 \, f x + 2 \, e\right )}{64 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.57, size = 463, normalized size = 2.45 \[ \frac {a^{2} A \,c^{4} \left (f x +e \right )+4 B \,a^{2} c^{4} \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 {B \,a^{2} c^{4} \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )}{3}+2 a^{2} A \,c^{4} \cos \left (f x +e \right )-2 B \,a^{2} c^{4} \left (-\frac {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )-\frac {4 a^{2} A \,c^{4} \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )}{3}-\frac {B \,a^{2} c^{4} \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}-2 B \,a^{2} c^{4} \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^{2} c^{4} \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^{2} A \,c^{4} \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 )+a^{2} A \,c^{4} \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 {2 a^{2} A \,c^{4} \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^{2} A \,c^{4} \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^{2} c^{4} \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 = 460, normalized size = 2.43 \[ \frac {896 \, {\left (3 \, \cos \left (f x + e\right )^{5} - 10 \, \cos \left (f x + e\right )^{3} + 15 \, \cos \left (f x + e\right )\right )} A a^{2} c^{4} + 8960 \, {\left (\cos \left (f x + e\right )^{3} - 3 \, \cos \left (f x + e\right )\right )} A a^{2} c^{4} + 35 \, {\left (4 \, \sin \left (2 \, f x + 2 \, e\right )^{3} + 60 \, f x + 60 \, e + 9 \, \sin \left (4 \, f x + 4 \, e\right ) - 48 \, \sin \left (2 \, f x + 2 \, e\right )\right )} A a^{2} c^{4} - 210 \, {\left (12 \, f x + 12 \, e + \sin \left (4 \, f x + 4 \, e\right ) - 8 \, \sin \left (2 \, f x + 2 \, e\right )\right )} A a^{2} c^{4} - 1680 \, {\left (2 \, f x + 2 \, e - \sin \left (2 \, f x + 2 \, e\right )\right )} A a^{2} c^{4} + 6720 \, {\left (f x + e\right )} A a^{2} c^{4} + 192 \, {\left (5 \, \cos \left (f x + e\right )^{7} - 21 \, \cos \left (f x + e\right )^{5} + 35 \, \cos \left (f x + e\right )^{3} - 35 \, \cos \left (f x + e\right )\right )} B a^{2} c^{4} + 448 \, {\left (3 \, \cos \left (f x + e\right )^{5} - 10 \, \cos \left (f x + e\right )^{3} + 15 \, \cos \left (f x + e\right )\right )} B a^{2} c^{4} - 2240 \, {\left (\cos \left (f x + e\right )^{3} - 3 \, \cos \left (f x + e\right )\right )} B a^{2} c^{4} - 70 \, {\left (4 \, \sin \left (2 \, f x + 2 \, e\right )^{3} + 60 \, f x + 60 \, e + 9 \, \sin \left (4 \, f x + 4 \, e\right ) - 48 \, \sin \left (2 \, f x + 2 \, e\right )\right )} B a^{2} c^{4} + 840 \, {\left (12 \, f x + 12 \, e + \sin \left (4 \, f x + 4 \, e\right ) - 8 \, \sin \left (2 \, f x + 2 \, e\right )\right )} B a^{2} c^{4} - 3360 \, {\left (2 \, f x + 2 \, e - \sin \left (2 \, f x + 2 \, e\right )\right )} B a^{2} c^{4} + 13440 \, A a^{2} c^{4} \cos \left (f x + e\right ) - 6720 \, B a^{2} c^{4} \cos \left (f x + e\right )}{6720 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 14.89, size = 553, normalized size = 2.93 \[ \frac {{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{12}\,\left (4\,A\,a^2\,c^4-2\,B\,a^2\,c^4\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^8\,\left (12\,A\,a^2\,c^4-2\,B\,a^2\,c^4\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{10}\,\left (8\,A\,a^2\,c^4-8\,B\,a^2\,c^4\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2\,\left (\frac {8\,A\,a^2\,c^4}{5}-\frac {8\,B\,a^2\,c^4}{5}\right )-{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{13}\,\left (\frac {9\,A\,a^2\,c^4}{8}+\frac {B\,a^2\,c^4}{4}\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6\,\left (16\,A\,a^2\,c^4-16\,B\,a^2\,c^4\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3\,\left (\frac {29\,A\,a^2\,c^4}{6}-\frac {11\,B\,a^2\,c^4}{3}\right )-{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{11}\,\left (\frac {29\,A\,a^2\,c^4}{6}-\frac {11\,B\,a^2\,c^4}{3}\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4\,\left (\frac {44\,A\,a^2\,c^4}{5}-\frac {14\,B\,a^2\,c^4}{5}\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^5\,\left (\frac {23\,A\,a^2\,c^4}{24}+\frac {31\,B\,a^2\,c^4}{12}\right )-{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^9\,\left (\frac {23\,A\,a^2\,c^4}{24}+\frac {31\,B\,a^2\,c^4}{12}\right )+\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (\frac {9\,A\,a^2\,c^4}{8}+\frac {B\,a^2\,c^4}{4}\right )+\frac {4\,A\,a^2\,c^4}{5}-\frac {18\,B\,a^2\,c^4}{35}}{f\,\left ({\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{14}+7\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{12}+21\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{10}+35\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^8+35\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6+21\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4+7\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2+1\right )}+\frac {a^2\,c^4\,\mathrm {atan}\left (\frac {a^2\,c^4\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (7\,A-2\,B\right )}{8\,\left (\frac {7\,A\,a^2\,c^4}{8}-\frac {B\,a^2\,c^4}{4}\right )}\right )\,\left (7\,A-2\,B\right )}{8\,f} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 15.02, size = 1210, normalized size = 6.40 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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