Integrand size = 40, antiderivative size = 384 \[ \int (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{5/2} \left (A+C \sin ^2(e+f x)\right ) \, dx=\frac {64 c^3 \left (C \left (39-16 m+4 m^2\right )+A \left (63+32 m+4 m^2\right )\right ) \cos (e+f x) (a+a \sin (e+f x))^m}{f (5+2 m) (7+2 m) (9+2 m) \left (3+8 m+4 m^2\right ) \sqrt {c-c \sin (e+f x)}}+\frac {16 c^2 \left (C \left (39-16 m+4 m^2\right )+A \left (63+32 m+4 m^2\right )\right ) \cos (e+f x) (a+a \sin (e+f x))^m \sqrt {c-c \sin (e+f x)}}{f (7+2 m) (9+2 m) \left (15+16 m+4 m^2\right )}+\frac {2 c \left (C \left (39-16 m+4 m^2\right )+A \left (63+32 m+4 m^2\right )\right ) \cos (e+f x) (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{3/2}}{f (5+2 m) (7+2 m) (9+2 m)}-\frac {4 C (1+2 m) \cos (e+f x) (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{5/2}}{f (7+2 m) (9+2 m)}+\frac {2 C \cos (e+f x) (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{7/2}}{c f (9+2 m)} \]
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Time = 0.60 (sec) , antiderivative size = 384, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {3119, 3052, 2819, 2817} \[ \int (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{5/2} \left (A+C \sin ^2(e+f x)\right ) \, dx=\frac {64 c^3 \left (A \left (4 m^2+32 m+63\right )+C \left (4 m^2-16 m+39\right )\right ) \cos (e+f x) (a \sin (e+f x)+a)^m}{f (2 m+5) (2 m+7) (2 m+9) \left (4 m^2+8 m+3\right ) \sqrt {c-c \sin (e+f x)}}+\frac {16 c^2 \left (A \left (4 m^2+32 m+63\right )+C \left (4 m^2-16 m+39\right )\right ) \cos (e+f x) \sqrt {c-c \sin (e+f x)} (a \sin (e+f x)+a)^m}{f (2 m+7) (2 m+9) \left (4 m^2+16 m+15\right )}+\frac {2 c \left (A \left (4 m^2+32 m+63\right )+C \left (4 m^2-16 m+39\right )\right ) \cos (e+f x) (c-c \sin (e+f x))^{3/2} (a \sin (e+f x)+a)^m}{f (2 m+5) (2 m+7) (2 m+9)}+\frac {2 C \cos (e+f x) (c-c \sin (e+f x))^{7/2} (a \sin (e+f x)+a)^m}{c f (2 m+9)}-\frac {4 C (2 m+1) \cos (e+f x) (c-c \sin (e+f x))^{5/2} (a \sin (e+f x)+a)^m}{f (2 m+7) (2 m+9)} \]
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Rule 2817
Rule 2819
Rule 3052
Rule 3119
Rubi steps \begin{align*} \text {integral}& = \frac {2 C \cos (e+f x) (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{7/2}}{c f (9+2 m)}-\frac {2 \int (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{5/2} \left (-\frac {1}{2} a c (C (7-2 m)+A (9+2 m))-a c C (1+2 m) \sin (e+f x)\right ) \, dx}{a c (9+2 m)} \\ & = -\frac {4 C (1+2 m) \cos (e+f x) (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{5/2}}{f (7+2 m) (9+2 m)}+\frac {2 C \cos (e+f x) (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{7/2}}{c f (9+2 m)}+\frac {\left (C \left (39-16 m+4 m^2\right )+A \left (63+32 m+4 m^2\right )\right ) \int (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{5/2} \, dx}{(7+2 m) (9+2 m)} \\ & = \frac {2 c \left (C \left (39-16 m+4 m^2\right )+A \left (63+32 m+4 m^2\right )\right ) \cos (e+f x) (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{3/2}}{f (5+2 m) (7+2 m) (9+2 m)}-\frac {4 C (1+2 m) \cos (e+f x) (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{5/2}}{f (7+2 m) (9+2 m)}+\frac {2 C \cos (e+f x) (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{7/2}}{c f (9+2 m)}+\frac {\left (8 c \left (C \left (39-16 m+4 m^2\right )+A \left (63+32 m+4 m^2\right )\right )\right ) \int (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{3/2} \, dx}{(5+2 m) (7+2 m) (9+2 m)} \\ & = \frac {16 c^2 \left (C \left (39-16 m+4 m^2\right )+A \left (63+32 m+4 m^2\right )\right ) \cos (e+f x) (a+a \sin (e+f x))^m \sqrt {c-c \sin (e+f x)}}{f (3+2 m) (5+2 m) (7+2 m) (9+2 m)}+\frac {2 c \left (C \left (39-16 m+4 m^2\right )+A \left (63+32 m+4 m^2\right )\right ) \cos (e+f x) (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{3/2}}{f (5+2 m) (7+2 m) (9+2 m)}-\frac {4 C (1+2 m) \cos (e+f x) (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{5/2}}{f (7+2 m) (9+2 m)}+\frac {2 C \cos (e+f x) (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{7/2}}{c f (9+2 m)}+\frac {\left (32 c^2 \left (C \left (39-16 m+4 m^2\right )+A \left (63+32 m+4 m^2\right )\right )\right ) \int (a+a \sin (e+f x))^m \sqrt {c-c \sin (e+f x)} \, dx}{(3+2 m) (5+2 m) (7+2 m) (9+2 m)} \\ & = \frac {64 c^3 \left (C \left (39-16 m+4 m^2\right )+A \left (63+32 m+4 m^2\right )\right ) \cos (e+f x) (a+a \sin (e+f x))^m}{f (1+2 m) (3+2 m) (5+2 m) (7+2 m) (9+2 m) \sqrt {c-c \sin (e+f x)}}+\frac {16 c^2 \left (C \left (39-16 m+4 m^2\right )+A \left (63+32 m+4 m^2\right )\right ) \cos (e+f x) (a+a \sin (e+f x))^m \sqrt {c-c \sin (e+f x)}}{f (3+2 m) (5+2 m) (7+2 m) (9+2 m)}+\frac {2 c \left (C \left (39-16 m+4 m^2\right )+A \left (63+32 m+4 m^2\right )\right ) \cos (e+f x) (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{3/2}}{f (5+2 m) (7+2 m) (9+2 m)}-\frac {4 C (1+2 m) \cos (e+f x) (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{5/2}}{f (7+2 m) (9+2 m)}+\frac {2 C \cos (e+f x) (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{7/2}}{c f (9+2 m)} \\ \end{align*}
Result contains complex when optimal does not.
Time = 13.45 (sec) , antiderivative size = 899, normalized size of antiderivative = 2.34 \[ \int (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{5/2} \left (A+C \sin ^2(e+f x)\right ) \, dx=\frac {(a (1+\sin (e+f x)))^m (c-c \sin (e+f x))^{5/2} \left (\frac {\left (18900 A+12285 C+15648 A m+648 C m+5280 A m^2+1416 C m^2+896 A m^3+224 C m^3+64 A m^4+16 C m^4\right ) \left (\left (\frac {1}{8}+\frac {i}{8}\right ) \cos \left (\frac {1}{2} (e+f x)\right )+\left (\frac {1}{8}-\frac {i}{8}\right ) \sin \left (\frac {1}{2} (e+f x)\right )\right )}{(1+2 m) (3+2 m) (5+2 m) (7+2 m) (9+2 m)}+\frac {\left (18900 A+12285 C+15648 A m+648 C m+5280 A m^2+1416 C m^2+896 A m^3+224 C m^3+64 A m^4+16 C m^4\right ) \left (\left (\frac {1}{8}-\frac {i}{8}\right ) \cos \left (\frac {1}{2} (e+f x)\right )+\left (\frac {1}{8}+\frac {i}{8}\right ) \sin \left (\frac {1}{2} (e+f x)\right )\right )}{(1+2 m) (3+2 m) (5+2 m) (7+2 m) (9+2 m)}+\frac {\left (1575 A+1575 C+1178 A m+414 C m+292 A m^2+100 C m^2+24 A m^3+8 C m^3\right ) \left (\left (\frac {1}{4}-\frac {i}{4}\right ) \cos \left (\frac {3}{2} (e+f x)\right )-\left (\frac {1}{4}+\frac {i}{4}\right ) \sin \left (\frac {3}{2} (e+f x)\right )\right )}{(3+2 m) (5+2 m) (7+2 m) (9+2 m)}+\frac {\left (1575 A+1575 C+1178 A m+414 C m+292 A m^2+100 C m^2+24 A m^3+8 C m^3\right ) \left (\left (\frac {1}{4}+\frac {i}{4}\right ) \cos \left (\frac {3}{2} (e+f x)\right )-\left (\frac {1}{4}-\frac {i}{4}\right ) \sin \left (\frac {3}{2} (e+f x)\right )\right )}{(3+2 m) (5+2 m) (7+2 m) (9+2 m)}+\frac {\left (63 A+189 C+32 A m+44 C m+4 A m^2+4 C m^2\right ) \left (\left (-\frac {1}{4}+\frac {i}{4}\right ) \cos \left (\frac {5}{2} (e+f x)\right )-\left (\frac {1}{4}+\frac {i}{4}\right ) \sin \left (\frac {5}{2} (e+f x)\right )\right )}{(5+2 m) (7+2 m) (9+2 m)}+\frac {\left (63 A+189 C+32 A m+44 C m+4 A m^2+4 C m^2\right ) \left (\left (-\frac {1}{4}-\frac {i}{4}\right ) \cos \left (\frac {5}{2} (e+f x)\right )-\left (\frac {1}{4}-\frac {i}{4}\right ) \sin \left (\frac {5}{2} (e+f x)\right )\right )}{(5+2 m) (7+2 m) (9+2 m)}+\frac {(15+2 m) \left (\left (-\frac {3}{16}-\frac {3 i}{16}\right ) C \cos \left (\frac {7}{2} (e+f x)\right )+\left (\frac {3}{16}-\frac {3 i}{16}\right ) C \sin \left (\frac {7}{2} (e+f x)\right )\right )}{(7+2 m) (9+2 m)}+\frac {(15+2 m) \left (\left (-\frac {3}{16}+\frac {3 i}{16}\right ) C \cos \left (\frac {7}{2} (e+f x)\right )+\left (\frac {3}{16}+\frac {3 i}{16}\right ) C \sin \left (\frac {7}{2} (e+f x)\right )\right )}{(7+2 m) (9+2 m)}+\frac {\left (\frac {1}{16}+\frac {i}{16}\right ) C \cos \left (\frac {9}{2} (e+f x)\right )+\left (\frac {1}{16}-\frac {i}{16}\right ) C \sin \left (\frac {9}{2} (e+f x)\right )}{9+2 m}+\frac {\left (\frac {1}{16}-\frac {i}{16}\right ) C \cos \left (\frac {9}{2} (e+f x)\right )+\left (\frac {1}{16}+\frac {i}{16}\right ) C \sin \left (\frac {9}{2} (e+f x)\right )}{9+2 m}\right )}{f \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^5} \]
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\[\int \left (a +a \sin \left (f x +e \right )\right )^{m} \left (c -c \sin \left (f x +e \right )\right )^{\frac {5}{2}} \left (A +C \left (\sin ^{2}\left (f x +e \right )\right )\right )d x\]
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Leaf count of result is larger than twice the leaf count of optimal. 777 vs. \(2 (364) = 728\).
Time = 0.32 (sec) , antiderivative size = 777, normalized size of antiderivative = 2.02 \[ \int (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{5/2} \left (A+C \sin ^2(e+f x)\right ) \, dx=\frac {2 \, {\left ({\left (16 \, C c^{2} m^{4} + 128 \, C c^{2} m^{3} + 344 \, C c^{2} m^{2} + 352 \, C c^{2} m + 105 \, C c^{2}\right )} \cos \left (f x + e\right )^{5} + 128 \, {\left (A + C\right )} c^{2} m^{2} - {\left (16 \, C c^{2} m^{4} + 224 \, C c^{2} m^{3} + 776 \, C c^{2} m^{2} + 904 \, C c^{2} m + 285 \, C c^{2}\right )} \cos \left (f x + e\right )^{4} + 512 \, {\left (2 \, A - C\right )} c^{2} m - {\left (16 \, {\left (A + 3 \, C\right )} c^{2} m^{4} + 32 \, {\left (5 \, A + 16 \, C\right )} c^{2} m^{3} + 8 \, {\left (65 \, A + 253 \, C\right )} c^{2} m^{2} + 8 \, {\left (75 \, A + 328 \, C\right )} c^{2} m + 3 \, {\left (63 \, A + 289 \, C\right )} c^{2}\right )} \cos \left (f x + e\right )^{3} + 96 \, {\left (21 \, A + 13 \, C\right )} c^{2} + {\left (16 \, {\left (A + C\right )} c^{2} m^{4} + 224 \, {\left (A + C\right )} c^{2} m^{3} + 8 \, {\left (133 \, A + 85 \, C\right )} c^{2} m^{2} + 1864 \, {\left (A + C\right )} c^{2} m + 3 \, {\left (231 \, A + 263 \, C\right )} c^{2}\right )} \cos \left (f x + e\right )^{2} + 2 \, {\left (16 \, {\left (A + C\right )} c^{2} m^{4} + 192 \, {\left (A + C\right )} c^{2} m^{3} + 856 \, {\left (A + C\right )} c^{2} m^{2} + 16 \, {\left (109 \, A + 85 \, C\right )} c^{2} m + 3 \, {\left (483 \, A + 419 \, C\right )} c^{2}\right )} \cos \left (f x + e\right ) + {\left (128 \, {\left (A + C\right )} c^{2} m^{2} + {\left (16 \, C c^{2} m^{4} + 128 \, C c^{2} m^{3} + 344 \, C c^{2} m^{2} + 352 \, C c^{2} m + 105 \, C c^{2}\right )} \cos \left (f x + e\right )^{4} + 512 \, {\left (2 \, A - C\right )} c^{2} m + 2 \, {\left (16 \, C c^{2} m^{4} + 176 \, C c^{2} m^{3} + 560 \, C c^{2} m^{2} + 628 \, C c^{2} m + 195 \, C c^{2}\right )} \cos \left (f x + e\right )^{3} + 96 \, {\left (21 \, A + 13 \, C\right )} c^{2} - {\left (16 \, {\left (A + C\right )} c^{2} m^{4} + 160 \, {\left (A + C\right )} c^{2} m^{3} + 8 \, {\left (65 \, A + 113 \, C\right )} c^{2} m^{2} + 24 \, {\left (25 \, A + 57 \, C\right )} c^{2} m + 9 \, {\left (21 \, A + 53 \, C\right )} c^{2}\right )} \cos \left (f x + e\right )^{2} - 2 \, {\left (16 \, {\left (A + C\right )} c^{2} m^{4} + 192 \, {\left (A + C\right )} c^{2} m^{3} + 792 \, {\left (A + C\right )} c^{2} m^{2} + 16 \, {\left (77 \, A + 101 \, C\right )} c^{2} m + 3 \, {\left (147 \, A + 211 \, C\right )} c^{2}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )\right )} \sqrt {-c \sin \left (f x + e\right ) + c} {\left (a \sin \left (f x + e\right ) + a\right )}^{m}}{32 \, f m^{5} + 400 \, f m^{4} + 1840 \, f m^{3} + 3800 \, f m^{2} + 3378 \, f m + {\left (32 \, f m^{5} + 400 \, f m^{4} + 1840 \, f m^{3} + 3800 \, f m^{2} + 3378 \, f m + 945 \, f\right )} \cos \left (f x + e\right ) - {\left (32 \, f m^{5} + 400 \, f m^{4} + 1840 \, f m^{3} + 3800 \, f m^{2} + 3378 \, f m + 945 \, f\right )} \sin \left (f x + e\right ) + 945 \, f} \]
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Timed out. \[ \int (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{5/2} \left (A+C \sin ^2(e+f x)\right ) \, dx=\text {Timed out} \]
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Leaf count of result is larger than twice the leaf count of optimal. 892 vs. \(2 (364) = 728\).
Time = 0.39 (sec) , antiderivative size = 892, normalized size of antiderivative = 2.32 \[ \int (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{5/2} \left (A+C \sin ^2(e+f x)\right ) \, dx=\text {Too large to display} \]
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\[ \int (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{5/2} \left (A+C \sin ^2(e+f x)\right ) \, dx=\int { {\left (C \sin \left (f x + e\right )^{2} + A\right )} {\left (-c \sin \left (f x + e\right ) + c\right )}^{\frac {5}{2}} {\left (a \sin \left (f x + e\right ) + a\right )}^{m} \,d x } \]
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Time = 22.87 (sec) , antiderivative size = 1110, normalized size of antiderivative = 2.89 \[ \int (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{5/2} \left (A+C \sin ^2(e+f x)\right ) \, dx=\text {Too large to display} \]
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