Integrand size = 38, antiderivative size = 275 \[ \int (a+a \sin (e+f x))^m (A+B \sin (e+f x)) (c-c \sin (e+f x))^{5/2} \, dx=-\frac {64 c^3 (B (5-2 m)-A (7+2 m)) \cos (e+f x) (a+a \sin (e+f x))^m}{f (5+2 m) (7+2 m) \left (3+8 m+4 m^2\right ) \sqrt {c-c \sin (e+f x)}}-\frac {16 c^2 (B (5-2 m)-A (7+2 m)) \cos (e+f x) (a+a \sin (e+f x))^m \sqrt {c-c \sin (e+f x)}}{f (7+2 m) \left (15+16 m+4 m^2\right )}-\frac {2 c (B (5-2 m)-A (7+2 m)) \cos (e+f x) (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{3/2}}{f (5+2 m) (7+2 m)}-\frac {2 B \cos (e+f x) (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{5/2}}{f (7+2 m)} \]
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Time = 0.35 (sec) , antiderivative size = 275, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.079, Rules used = {3052, 2819, 2817} \[ \int (a+a \sin (e+f x))^m (A+B \sin (e+f x)) (c-c \sin (e+f x))^{5/2} \, dx=-\frac {64 c^3 (B (5-2 m)-A (2 m+7)) \cos (e+f x) (a \sin (e+f x)+a)^m}{f (2 m+5) (2 m+7) \left (4 m^2+8 m+3\right ) \sqrt {c-c \sin (e+f x)}}-\frac {16 c^2 (B (5-2 m)-A (2 m+7)) \cos (e+f x) \sqrt {c-c \sin (e+f x)} (a \sin (e+f x)+a)^m}{f (2 m+7) \left (4 m^2+16 m+15\right )}-\frac {2 c (B (5-2 m)-A (2 m+7)) \cos (e+f x) (c-c \sin (e+f x))^{3/2} (a \sin (e+f x)+a)^m}{f (2 m+5) (2 m+7)}-\frac {2 B \cos (e+f x) (c-c \sin (e+f x))^{5/2} (a \sin (e+f x)+a)^m}{f (2 m+7)} \]
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Rule 2817
Rule 2819
Rule 3052
Rubi steps \begin{align*} \text {integral}& = -\frac {2 B \cos (e+f x) (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{5/2}}{f (7+2 m)}+\frac {\left (B c \left (-\frac {5}{2}+m\right )+A c \left (\frac {7}{2}+m\right )\right ) \int (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{5/2} \, dx}{c \left (\frac {7}{2}+m\right )} \\ & = -\frac {2 c (B (5-2 m)-A (7+2 m)) \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)}-\frac {2 B \cos (e+f x) (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{5/2}}{f (7+2 m)}-\frac {(8 c (B (5-2 m)-A (7+2 m))) \int (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{3/2} \, dx}{(5+2 m) (7+2 m)} \\ & = -\frac {16 c^2 (B (5-2 m)-A (7+2 m)) \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)}-\frac {2 c (B (5-2 m)-A (7+2 m)) \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)}-\frac {2 B \cos (e+f x) (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{5/2}}{f (7+2 m)}-\frac {\left (32 c^2 (B (5-2 m)-A (7+2 m))\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)} \\ & = -\frac {64 c^3 (B (5-2 m)-A (7+2 m)) \cos (e+f x) (a+a \sin (e+f x))^m}{f (1+2 m) (3+2 m) (5+2 m) (7+2 m) \sqrt {c-c \sin (e+f x)}}-\frac {16 c^2 (B (5-2 m)-A (7+2 m)) \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)}-\frac {2 c (B (5-2 m)-A (7+2 m)) \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)}-\frac {2 B \cos (e+f x) (a+a \sin (e+f x))^m (c-c \sin (e+f x))^{5/2}}{f (7+2 m)} \\ \end{align*}
Time = 11.98 (sec) , antiderivative size = 271, normalized size of antiderivative = 0.99 \[ \int (a+a \sin (e+f x))^m (A+B \sin (e+f x)) (c-c \sin (e+f x))^{5/2} \, dx=-\frac {c^2 \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right ) (a (1+\sin (e+f x)))^m \sqrt {c-c \sin (e+f x)} \left (-1246 A+1040 B-1140 A m+664 B m-392 A m^2+256 B m^2-48 A m^3+32 B m^3+2 \left (3+8 m+4 m^2\right ) (-4 B (5+m)+A (7+2 m)) \cos (2 (e+f x))+(1+2 m) \left (8 A (7+2 m)^2-B \left (505+208 m+28 m^2\right )\right ) \sin (e+f x)+15 B \sin (3 (e+f x))+46 B m \sin (3 (e+f x))+36 B m^2 \sin (3 (e+f x))+8 B m^3 \sin (3 (e+f x))\right )}{2 f (1+2 m) (3+2 m) (5+2 m) (7+2 m) \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )} \]
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\[\int \left (a +a \sin \left (f x +e \right )\right )^{m} \left (A +B \sin \left (f x +e \right )\right ) \left (c -c \sin \left (f x +e \right )\right )^{\frac {5}{2}}d x\]
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Leaf count of result is larger than twice the leaf count of optimal. 562 vs. \(2 (262) = 524\).
Time = 0.31 (sec) , antiderivative size = 562, normalized size of antiderivative = 2.04 \[ \int (a+a \sin (e+f x))^m (A+B \sin (e+f x)) (c-c \sin (e+f x))^{5/2} \, dx=\frac {2 \, {\left ({\left (8 \, B c^{2} m^{3} + 36 \, B c^{2} m^{2} + 46 \, B c^{2} m + 15 \, B c^{2}\right )} \cos \left (f x + e\right )^{4} + 64 \, {\left (A + B\right )} c^{2} m - {\left (8 \, {\left (A - 2 \, B\right )} c^{2} m^{3} + 4 \, {\left (11 \, A - 28 \, B\right )} c^{2} m^{2} + 2 \, {\left (31 \, A - 86 \, B\right )} c^{2} m + 3 \, {\left (7 \, A - 20 \, B\right )} c^{2}\right )} \cos \left (f x + e\right )^{3} + 32 \, {\left (7 \, A - 5 \, B\right )} c^{2} + {\left (8 \, {\left (A - B\right )} c^{2} m^{3} + 4 \, {\left (19 \, A - 11 \, B\right )} c^{2} m^{2} + 190 \, {\left (A - B\right )} c^{2} m + {\left (77 \, A - 85 \, B\right )} c^{2}\right )} \cos \left (f x + e\right )^{2} + 2 \, {\left (8 \, {\left (A - B\right )} c^{2} m^{3} + 60 \, {\left (A - B\right )} c^{2} m^{2} + 2 \, {\left (79 \, A - 63 \, B\right )} c^{2} m + {\left (161 \, A - 145 \, B\right )} c^{2}\right )} \cos \left (f x + e\right ) + {\left (64 \, {\left (A + B\right )} c^{2} m - {\left (8 \, B c^{2} m^{3} + 36 \, B c^{2} m^{2} + 46 \, B c^{2} m + 15 \, B c^{2}\right )} \cos \left (f x + e\right )^{3} + 32 \, {\left (7 \, A - 5 \, B\right )} c^{2} - {\left (8 \, {\left (A - B\right )} c^{2} m^{3} + 4 \, {\left (11 \, A - 19 \, B\right )} c^{2} m^{2} + 2 \, {\left (31 \, A - 63 \, B\right )} c^{2} m + 3 \, {\left (7 \, A - 15 \, B\right )} c^{2}\right )} \cos \left (f x + e\right )^{2} - 2 \, {\left (8 \, {\left (A - B\right )} c^{2} m^{3} + 60 \, {\left (A - B\right )} c^{2} m^{2} + 2 \, {\left (63 \, A - 79 \, B\right )} c^{2} m + {\left (49 \, A - 65 \, B\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}}{16 \, f m^{4} + 128 \, f m^{3} + 344 \, f m^{2} + 352 \, f m + {\left (16 \, f m^{4} + 128 \, f m^{3} + 344 \, f m^{2} + 352 \, f m + 105 \, f\right )} \cos \left (f x + e\right ) - {\left (16 \, f m^{4} + 128 \, f m^{3} + 344 \, f m^{2} + 352 \, f m + 105 \, f\right )} \sin \left (f x + e\right ) + 105 \, f} \]
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Timed out. \[ \int (a+a \sin (e+f x))^m (A+B \sin (e+f x)) (c-c \sin (e+f x))^{5/2} \, dx=\text {Timed out} \]
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Leaf count of result is larger than twice the leaf count of optimal. 725 vs. \(2 (262) = 524\).
Time = 0.34 (sec) , antiderivative size = 725, normalized size of antiderivative = 2.64 \[ \int (a+a \sin (e+f x))^m (A+B \sin (e+f x)) (c-c \sin (e+f x))^{5/2} \, dx=\text {Too large to display} \]
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\[ \int (a+a \sin (e+f x))^m (A+B \sin (e+f x)) (c-c \sin (e+f x))^{5/2} \, dx=\int { {\left (B \sin \left (f x + e\right ) + A\right )} {\left (-c \sin \left (f x + e\right ) + c\right )}^{\frac {5}{2}} {\left (a \sin \left (f x + e\right ) + a\right )}^{m} \,d x } \]
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Time = 21.31 (sec) , antiderivative size = 749, normalized size of antiderivative = 2.72 \[ \int (a+a \sin (e+f x))^m (A+B \sin (e+f x)) (c-c \sin (e+f x))^{5/2} \, dx=-\frac {\sqrt {c-c\,\sin \left (e+f\,x\right )}\,\left (\frac {B\,c^2\,{\left (a+a\,\sin \left (e+f\,x\right )\right )}^m\,\left (m^3\,8{}\mathrm {i}+m^2\,36{}\mathrm {i}+m\,46{}\mathrm {i}+15{}\mathrm {i}\right )}{4\,f\,\left (16\,m^4+128\,m^3+344\,m^2+352\,m+105\right )}-\frac {c^2\,{\mathrm {e}}^{e\,3{}\mathrm {i}+f\,x\,3{}\mathrm {i}}\,{\left (a+a\,\sin \left (e+f\,x\right )\right )}^m\,\left (2100\,A-1575\,B+1272\,A\,m-110\,B\,m+304\,A\,m^2+32\,A\,m^3-68\,B\,m^2-8\,B\,m^3\right )}{4\,f\,\left (16\,m^4+128\,m^3+344\,m^2+352\,m+105\right )}+\frac {c^2\,{\mathrm {e}}^{e\,4{}\mathrm {i}+f\,x\,4{}\mathrm {i}}\,{\left (a+a\,\sin \left (e+f\,x\right )\right )}^m\,\left (A\,2100{}\mathrm {i}-B\,1575{}\mathrm {i}+A\,m\,1272{}\mathrm {i}-B\,m\,110{}\mathrm {i}+A\,m^2\,304{}\mathrm {i}+A\,m^3\,32{}\mathrm {i}-B\,m^2\,68{}\mathrm {i}-B\,m^3\,8{}\mathrm {i}\right )}{4\,f\,\left (16\,m^4+128\,m^3+344\,m^2+352\,m+105\right )}-\frac {c^2\,{\mathrm {e}}^{e\,5{}\mathrm {i}+f\,x\,5{}\mathrm {i}}\,\left (2\,m+1\right )\,{\left (a+a\,\sin \left (e+f\,x\right )\right )}^m\,\left (350\,A-385\,B+184\,A\,m-104\,B\,m+24\,A\,m^2-12\,B\,m^2\right )}{4\,f\,\left (16\,m^4+128\,m^3+344\,m^2+352\,m+105\right )}+\frac {c^2\,{\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}\,\left (2\,m+1\right )\,{\left (a+a\,\sin \left (e+f\,x\right )\right )}^m\,\left (A\,350{}\mathrm {i}-B\,385{}\mathrm {i}+A\,m\,184{}\mathrm {i}-B\,m\,104{}\mathrm {i}+A\,m^2\,24{}\mathrm {i}-B\,m^2\,12{}\mathrm {i}\right )}{4\,f\,\left (16\,m^4+128\,m^3+344\,m^2+352\,m+105\right )}-\frac {B\,c^2\,{\mathrm {e}}^{e\,7{}\mathrm {i}+f\,x\,7{}\mathrm {i}}\,{\left (a+a\,\sin \left (e+f\,x\right )\right )}^m\,\left (8\,m^3+36\,m^2+46\,m+15\right )}{4\,f\,\left (16\,m^4+128\,m^3+344\,m^2+352\,m+105\right )}+\frac {c^2\,{\mathrm {e}}^{e\,1{}\mathrm {i}+f\,x\,1{}\mathrm {i}}\,{\left (a+a\,\sin \left (e+f\,x\right )\right )}^m\,\left (4\,m^2+8\,m+3\right )\,\left (14\,A-35\,B+4\,A\,m-6\,B\,m\right )}{4\,f\,\left (16\,m^4+128\,m^3+344\,m^2+352\,m+105\right )}-\frac {c^2\,{\mathrm {e}}^{e\,6{}\mathrm {i}+f\,x\,6{}\mathrm {i}}\,{\left (a+a\,\sin \left (e+f\,x\right )\right )}^m\,\left (4\,m^2+8\,m+3\right )\,\left (A\,14{}\mathrm {i}-B\,35{}\mathrm {i}+A\,m\,4{}\mathrm {i}-B\,m\,6{}\mathrm {i}\right )}{4\,f\,\left (16\,m^4+128\,m^3+344\,m^2+352\,m+105\right )}\right )}{{\mathrm {e}}^{e\,4{}\mathrm {i}+f\,x\,4{}\mathrm {i}}-\frac {{\mathrm {e}}^{e\,3{}\mathrm {i}+f\,x\,3{}\mathrm {i}}\,\left (m^4\,16{}\mathrm {i}+m^3\,128{}\mathrm {i}+m^2\,344{}\mathrm {i}+m\,352{}\mathrm {i}+105{}\mathrm {i}\right )}{16\,m^4+128\,m^3+344\,m^2+352\,m+105}} \]
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