Integrand size = 38, antiderivative size = 210 \[ \int (a+a \sin (e+f x))^2 (A+B \sin (e+f x)) (c-c \sin (e+f x))^{7/2} \, dx=\frac {256 a^2 (13 A-3 B) c^6 \cos ^5(e+f x)}{15015 f (c-c \sin (e+f x))^{5/2}}+\frac {64 a^2 (13 A-3 B) c^5 \cos ^5(e+f x)}{3003 f (c-c \sin (e+f x))^{3/2}}+\frac {8 a^2 (13 A-3 B) c^4 \cos ^5(e+f x)}{429 f \sqrt {c-c \sin (e+f x)}}+\frac {2 a^2 (13 A-3 B) c^3 \cos ^5(e+f x) \sqrt {c-c \sin (e+f x)}}{143 f}-\frac {2 a^2 B c^2 \cos ^5(e+f x) (c-c \sin (e+f x))^{3/2}}{13 f} \]
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Time = 0.39 (sec) , antiderivative size = 210, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {3046, 2935, 2753, 2752} \[ \int (a+a \sin (e+f x))^2 (A+B \sin (e+f x)) (c-c \sin (e+f x))^{7/2} \, dx=\frac {256 a^2 c^6 (13 A-3 B) \cos ^5(e+f x)}{15015 f (c-c \sin (e+f x))^{5/2}}+\frac {64 a^2 c^5 (13 A-3 B) \cos ^5(e+f x)}{3003 f (c-c \sin (e+f x))^{3/2}}+\frac {8 a^2 c^4 (13 A-3 B) \cos ^5(e+f x)}{429 f \sqrt {c-c \sin (e+f x)}}+\frac {2 a^2 c^3 (13 A-3 B) \cos ^5(e+f x) \sqrt {c-c \sin (e+f x)}}{143 f}-\frac {2 a^2 B c^2 \cos ^5(e+f x) (c-c \sin (e+f x))^{3/2}}{13 f} \]
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Rule 2752
Rule 2753
Rule 2935
Rule 3046
Rubi steps \begin{align*} \text {integral}& = \left (a^2 c^2\right ) \int \cos ^4(e+f x) (A+B \sin (e+f x)) (c-c \sin (e+f x))^{3/2} \, dx \\ & = -\frac {2 a^2 B c^2 \cos ^5(e+f x) (c-c \sin (e+f x))^{3/2}}{13 f}+\frac {1}{13} \left (a^2 (13 A-3 B) c^2\right ) \int \cos ^4(e+f x) (c-c \sin (e+f x))^{3/2} \, dx \\ & = \frac {2 a^2 (13 A-3 B) c^3 \cos ^5(e+f x) \sqrt {c-c \sin (e+f x)}}{143 f}-\frac {2 a^2 B c^2 \cos ^5(e+f x) (c-c \sin (e+f x))^{3/2}}{13 f}+\frac {1}{143} \left (12 a^2 (13 A-3 B) c^3\right ) \int \cos ^4(e+f x) \sqrt {c-c \sin (e+f x)} \, dx \\ & = \frac {8 a^2 (13 A-3 B) c^4 \cos ^5(e+f x)}{429 f \sqrt {c-c \sin (e+f x)}}+\frac {2 a^2 (13 A-3 B) c^3 \cos ^5(e+f x) \sqrt {c-c \sin (e+f x)}}{143 f}-\frac {2 a^2 B c^2 \cos ^5(e+f x) (c-c \sin (e+f x))^{3/2}}{13 f}+\frac {1}{429} \left (32 a^2 (13 A-3 B) c^4\right ) \int \frac {\cos ^4(e+f x)}{\sqrt {c-c \sin (e+f x)}} \, dx \\ & = \frac {64 a^2 (13 A-3 B) c^5 \cos ^5(e+f x)}{3003 f (c-c \sin (e+f x))^{3/2}}+\frac {8 a^2 (13 A-3 B) c^4 \cos ^5(e+f x)}{429 f \sqrt {c-c \sin (e+f x)}}+\frac {2 a^2 (13 A-3 B) c^3 \cos ^5(e+f x) \sqrt {c-c \sin (e+f x)}}{143 f}-\frac {2 a^2 B c^2 \cos ^5(e+f x) (c-c \sin (e+f x))^{3/2}}{13 f}+\frac {\left (128 a^2 (13 A-3 B) c^5\right ) \int \frac {\cos ^4(e+f x)}{(c-c \sin (e+f x))^{3/2}} \, dx}{3003} \\ & = \frac {256 a^2 (13 A-3 B) c^6 \cos ^5(e+f x)}{15015 f (c-c \sin (e+f x))^{5/2}}+\frac {64 a^2 (13 A-3 B) c^5 \cos ^5(e+f x)}{3003 f (c-c \sin (e+f x))^{3/2}}+\frac {8 a^2 (13 A-3 B) c^4 \cos ^5(e+f x)}{429 f \sqrt {c-c \sin (e+f x)}}+\frac {2 a^2 (13 A-3 B) c^3 \cos ^5(e+f x) \sqrt {c-c \sin (e+f x)}}{143 f}-\frac {2 a^2 B c^2 \cos ^5(e+f x) (c-c \sin (e+f x))^{3/2}}{13 f} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(1355\) vs. \(2(210)=420\).
Time = 10.72 (sec) , antiderivative size = 1355, normalized size of antiderivative = 6.45 \[ \int (a+a \sin (e+f x))^2 (A+B \sin (e+f x)) (c-c \sin (e+f x))^{7/2} \, dx=\frac {(7 A-2 B) \cos \left (\frac {1}{2} (e+f x)\right ) (a+a \sin (e+f x))^2 (c-c \sin (e+f x))^{7/2}}{8 f \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^7 \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^4}-\frac {(4 A+B) \cos \left (\frac {3}{2} (e+f x)\right ) (a+a \sin (e+f x))^2 (c-c \sin (e+f x))^{7/2}}{32 f \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^7 \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^4}+\frac {(22 A-7 B) \cos \left (\frac {5}{2} (e+f x)\right ) (a+a \sin (e+f x))^2 (c-c \sin (e+f x))^{7/2}}{160 f \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^7 \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^4}+\frac {(A-4 B) \cos \left (\frac {7}{2} (e+f x)\right ) (a+a \sin (e+f x))^2 (c-c \sin (e+f x))^{7/2}}{112 f \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^7 \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^4}+\frac {A \cos \left (\frac {9}{2} (e+f x)\right ) (a+a \sin (e+f x))^2 (c-c \sin (e+f x))^{7/2}}{48 f \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^7 \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^4}+\frac {(2 A-3 B) \cos \left (\frac {11}{2} (e+f x)\right ) (a+a \sin (e+f x))^2 (c-c \sin (e+f x))^{7/2}}{352 f \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^7 \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^4}+\frac {B \cos \left (\frac {13}{2} (e+f x)\right ) (a+a \sin (e+f x))^2 (c-c \sin (e+f x))^{7/2}}{416 f \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^7 \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^4}+\frac {(7 A-2 B) \sin \left (\frac {1}{2} (e+f x)\right ) (a+a \sin (e+f x))^2 (c-c \sin (e+f x))^{7/2}}{8 f \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^7 \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^4}+\frac {(4 A+B) (a+a \sin (e+f x))^2 (c-c \sin (e+f x))^{7/2} \sin \left (\frac {3}{2} (e+f x)\right )}{32 f \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^7 \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^4}+\frac {(22 A-7 B) (a+a \sin (e+f x))^2 (c-c \sin (e+f x))^{7/2} \sin \left (\frac {5}{2} (e+f x)\right )}{160 f \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^7 \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^4}-\frac {(A-4 B) (a+a \sin (e+f x))^2 (c-c \sin (e+f x))^{7/2} \sin \left (\frac {7}{2} (e+f x)\right )}{112 f \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^7 \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^4}+\frac {A (a+a \sin (e+f x))^2 (c-c \sin (e+f x))^{7/2} \sin \left (\frac {9}{2} (e+f x)\right )}{48 f \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^7 \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^4}-\frac {(2 A-3 B) (a+a \sin (e+f x))^2 (c-c \sin (e+f x))^{7/2} \sin \left (\frac {11}{2} (e+f x)\right )}{352 f \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^7 \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^4}+\frac {B (a+a \sin (e+f x))^2 (c-c \sin (e+f x))^{7/2} \sin \left (\frac {13}{2} (e+f x)\right )}{416 f \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^7 \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^4} \]
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Time = 36.97 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.58
method | result | size |
default | \(\frac {2 \left (\sin \left (f x +e \right )-1\right ) c^{4} \left (1+\sin \left (f x +e \right )\right )^{3} a^{2} \left (1155 B \left (\cos ^{4}\left (f x +e \right )\right )+\left (-1365 A +4935 B \right ) \left (\cos ^{2}\left (f x +e \right )\right ) \sin \left (f x +e \right )+\left (5915 A -10605 B \right ) \left (\cos ^{2}\left (f x +e \right )\right )+\left (11180 A -11820 B \right ) \sin \left (f x +e \right )-12844 A +12204 B \right )}{15015 \cos \left (f x +e \right ) \sqrt {c -c \sin \left (f x +e \right )}\, f}\) | \(121\) |
parts | \(\frac {2 A \,a^{2} \left (\sin \left (f x +e \right )-1\right ) c^{4} \left (1+\sin \left (f x +e \right )\right ) \left (5 \left (\sin ^{3}\left (f x +e \right )\right )-27 \left (\sin ^{2}\left (f x +e \right )\right )+71 \sin \left (f x +e \right )-177\right )}{35 \cos \left (f x +e \right ) \sqrt {c -c \sin \left (f x +e \right )}\, f}+\frac {2 B \,a^{2} \left (\sin \left (f x +e \right )-1\right ) c^{4} \left (1+\sin \left (f x +e \right )\right ) \left (165 \left (\sin ^{6}\left (f x +e \right )\right )-765 \left (\sin ^{5}\left (f x +e \right )\right )+1565 \left (\sin ^{4}\left (f x +e \right )\right )-2095 \left (\sin ^{3}\left (f x +e \right )\right )+2514 \left (\sin ^{2}\left (f x +e \right )\right )-3352 \sin \left (f x +e \right )+6704\right )}{2145 \cos \left (f x +e \right ) \sqrt {c -c \sin \left (f x +e \right )}\, f}+\frac {2 a^{2} \left (A +2 B \right ) \left (\sin \left (f x +e \right )-1\right ) c^{4} \left (1+\sin \left (f x +e \right )\right ) \left (315 \left (\sin ^{5}\left (f x +e \right )\right )-1505 \left (\sin ^{4}\left (f x +e \right )\right )+3205 \left (\sin ^{3}\left (f x +e \right )\right )-4539 \left (\sin ^{2}\left (f x +e \right )\right )+6052 \sin \left (f x +e \right )-12104\right )}{3465 \cos \left (f x +e \right ) \sqrt {c -c \sin \left (f x +e \right )}\, f}+\frac {2 a^{2} \left (2 A +B \right ) \left (\sin \left (f x +e \right )-1\right ) c^{4} \left (1+\sin \left (f x +e \right )\right ) \left (5 \left (\sin ^{4}\left (f x +e \right )\right )-25 \left (\sin ^{3}\left (f x +e \right )\right )+57 \left (\sin ^{2}\left (f x +e \right )\right )-91 \sin \left (f x +e \right )+182\right )}{45 \cos \left (f x +e \right ) \sqrt {c -c \sin \left (f x +e \right )}\, f}\) | \(386\) |
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Time = 0.27 (sec) , antiderivative size = 358, normalized size of antiderivative = 1.70 \[ \int (a+a \sin (e+f x))^2 (A+B \sin (e+f x)) (c-c \sin (e+f x))^{7/2} \, dx=\frac {2 \, {\left (1155 \, B a^{2} c^{3} \cos \left (f x + e\right )^{7} + 105 \, {\left (13 \, A - 14 \, B\right )} a^{2} c^{3} \cos \left (f x + e\right )^{6} + 35 \, {\left (91 \, A - 87 \, B\right )} a^{2} c^{3} \cos \left (f x + e\right )^{5} - 20 \, {\left (13 \, A - 3 \, B\right )} a^{2} c^{3} \cos \left (f x + e\right )^{4} + 32 \, {\left (13 \, A - 3 \, B\right )} a^{2} c^{3} \cos \left (f x + e\right )^{3} - 64 \, {\left (13 \, A - 3 \, B\right )} a^{2} c^{3} \cos \left (f x + e\right )^{2} + 256 \, {\left (13 \, A - 3 \, B\right )} a^{2} c^{3} \cos \left (f x + e\right ) + 512 \, {\left (13 \, A - 3 \, B\right )} a^{2} c^{3} + {\left (1155 \, B a^{2} c^{3} \cos \left (f x + e\right )^{6} - 105 \, {\left (13 \, A - 25 \, B\right )} a^{2} c^{3} \cos \left (f x + e\right )^{5} + 140 \, {\left (13 \, A - 3 \, B\right )} a^{2} c^{3} \cos \left (f x + e\right )^{4} + 160 \, {\left (13 \, A - 3 \, B\right )} a^{2} c^{3} \cos \left (f x + e\right )^{3} + 192 \, {\left (13 \, A - 3 \, B\right )} a^{2} c^{3} \cos \left (f x + e\right )^{2} + 256 \, {\left (13 \, A - 3 \, B\right )} a^{2} c^{3} \cos \left (f x + e\right ) + 512 \, {\left (13 \, A - 3 \, B\right )} a^{2} c^{3}\right )} \sin \left (f x + e\right )\right )} \sqrt {-c \sin \left (f x + e\right ) + c}}{15015 \, {\left (f \cos \left (f x + e\right ) - f \sin \left (f x + e\right ) + f\right )}} \]
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Timed out. \[ \int (a+a \sin (e+f x))^2 (A+B \sin (e+f x)) (c-c \sin (e+f x))^{7/2} \, dx=\text {Timed out} \]
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\[ \int (a+a \sin (e+f x))^2 (A+B \sin (e+f x)) (c-c \sin (e+f x))^{7/2} \, dx=\int { {\left (B \sin \left (f x + e\right ) + A\right )} {\left (a \sin \left (f x + e\right ) + a\right )}^{2} {\left (-c \sin \left (f x + e\right ) + c\right )}^{\frac {7}{2}} \,d x } \]
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Time = 0.53 (sec) , antiderivative size = 373, normalized size of antiderivative = 1.78 \[ \int (a+a \sin (e+f x))^2 (A+B \sin (e+f x)) (c-c \sin (e+f x))^{7/2} \, dx=-\frac {\sqrt {2} {\left (10010 \, A a^{2} c^{3} \cos \left (-\frac {9}{4} \, \pi + \frac {9}{2} \, f x + \frac {9}{2} \, e\right ) \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) - 1155 \, B a^{2} c^{3} \cos \left (-\frac {13}{4} \, \pi + \frac {13}{2} \, f x + \frac {13}{2} \, e\right ) \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) + 60060 \, {\left (7 \, A a^{2} c^{3} \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) - 2 \, B a^{2} c^{3} \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )\right )} \cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 15015 \, {\left (4 \, A a^{2} c^{3} \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) + B a^{2} c^{3} \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )\right )} \cos \left (-\frac {3}{4} \, \pi + \frac {3}{2} \, f x + \frac {3}{2} \, e\right ) - 3003 \, {\left (22 \, A a^{2} c^{3} \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) - 7 \, B a^{2} c^{3} \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )\right )} \cos \left (-\frac {5}{4} \, \pi + \frac {5}{2} \, f x + \frac {5}{2} \, e\right ) + 4290 \, {\left (A a^{2} c^{3} \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) - 4 \, B a^{2} c^{3} \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )\right )} \cos \left (-\frac {7}{4} \, \pi + \frac {7}{2} \, f x + \frac {7}{2} \, e\right ) - 1365 \, {\left (2 \, A a^{2} c^{3} \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) - 3 \, B a^{2} c^{3} \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )\right )} \cos \left (-\frac {11}{4} \, \pi + \frac {11}{2} \, f x + \frac {11}{2} \, e\right )\right )} \sqrt {c}}{480480 \, f} \]
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Timed out. \[ \int (a+a \sin (e+f x))^2 (A+B \sin (e+f x)) (c-c \sin (e+f x))^{7/2} \, dx=\int \left (A+B\,\sin \left (e+f\,x\right )\right )\,{\left (a+a\,\sin \left (e+f\,x\right )\right )}^2\,{\left (c-c\,\sin \left (e+f\,x\right )\right )}^{7/2} \,d x \]
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