Integrand size = 28, antiderivative size = 133 \[ \int (3+3 \sin (e+f x))^3 (c-c \sin (e+f x))^{7/2} \, dx=\frac {2304 c^7 \cos ^7(e+f x)}{1001 f (c-c \sin (e+f x))^{7/2}}+\frac {576 c^6 \cos ^7(e+f x)}{143 f (c-c \sin (e+f x))^{5/2}}+\frac {648 c^5 \cos ^7(e+f x)}{143 f (c-c \sin (e+f x))^{3/2}}+\frac {54 c^4 \cos ^7(e+f x)}{13 f \sqrt {c-c \sin (e+f x)}} \]
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Time = 0.24 (sec) , antiderivative size = 145, normalized size of antiderivative = 1.09, number of steps used = 5, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.107, Rules used = {2815, 2753, 2752} \[ \int (3+3 \sin (e+f x))^3 (c-c \sin (e+f x))^{7/2} \, dx=\frac {256 a^3 c^7 \cos ^7(e+f x)}{3003 f (c-c \sin (e+f x))^{7/2}}+\frac {64 a^3 c^6 \cos ^7(e+f x)}{429 f (c-c \sin (e+f x))^{5/2}}+\frac {24 a^3 c^5 \cos ^7(e+f x)}{143 f (c-c \sin (e+f x))^{3/2}}+\frac {2 a^3 c^4 \cos ^7(e+f x)}{13 f \sqrt {c-c \sin (e+f x)}} \]
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Rule 2752
Rule 2753
Rule 2815
Rubi steps \begin{align*} \text {integral}& = \left (a^3 c^3\right ) \int \cos ^6(e+f x) \sqrt {c-c \sin (e+f x)} \, dx \\ & = \frac {2 a^3 c^4 \cos ^7(e+f x)}{13 f \sqrt {c-c \sin (e+f x)}}+\frac {1}{13} \left (12 a^3 c^4\right ) \int \frac {\cos ^6(e+f x)}{\sqrt {c-c \sin (e+f x)}} \, dx \\ & = \frac {24 a^3 c^5 \cos ^7(e+f x)}{143 f (c-c \sin (e+f x))^{3/2}}+\frac {2 a^3 c^4 \cos ^7(e+f x)}{13 f \sqrt {c-c \sin (e+f x)}}+\frac {1}{143} \left (96 a^3 c^5\right ) \int \frac {\cos ^6(e+f x)}{(c-c \sin (e+f x))^{3/2}} \, dx \\ & = \frac {64 a^3 c^6 \cos ^7(e+f x)}{429 f (c-c \sin (e+f x))^{5/2}}+\frac {24 a^3 c^5 \cos ^7(e+f x)}{143 f (c-c \sin (e+f x))^{3/2}}+\frac {2 a^3 c^4 \cos ^7(e+f x)}{13 f \sqrt {c-c \sin (e+f x)}}+\frac {1}{429} \left (128 a^3 c^6\right ) \int \frac {\cos ^6(e+f x)}{(c-c \sin (e+f x))^{5/2}} \, dx \\ & = \frac {256 a^3 c^7 \cos ^7(e+f x)}{3003 f (c-c \sin (e+f x))^{7/2}}+\frac {64 a^3 c^6 \cos ^7(e+f x)}{429 f (c-c \sin (e+f x))^{5/2}}+\frac {24 a^3 c^5 \cos ^7(e+f x)}{143 f (c-c \sin (e+f x))^{3/2}}+\frac {2 a^3 c^4 \cos ^7(e+f x)}{13 f \sqrt {c-c \sin (e+f x)}} \\ \end{align*}
Time = 2.47 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.55 \[ \int (3+3 \sin (e+f x))^3 (c-c \sin (e+f x))^{7/2} \, dx=-\frac {18 c^3 \cos ^7(e+f x) \sqrt {c-c \sin (e+f x)} \left (-835+1421 \sin (e+f x)-945 \sin ^2(e+f x)+231 \sin ^3(e+f x)\right )}{1001 f (-1+\sin (e+f x))^4} \]
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Time = 38.37 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.61
method | result | size |
default | \(\frac {2 \left (\sin \left (f x +e \right )-1\right ) c^{4} \left (\sin \left (f x +e \right )+1\right )^{4} a^{3} \left (231 \left (\sin ^{3}\left (f x +e \right )\right )-945 \left (\sin ^{2}\left (f x +e \right )\right )+1421 \sin \left (f x +e \right )-835\right )}{3003 \cos \left (f x +e \right ) \sqrt {c -c \sin \left (f x +e \right )}\, f}\) | \(81\) |
parts | \(\frac {2 a^{3} \left (\sin \left (f x +e \right )-1\right ) c^{4} \left (\sin \left (f x +e \right )+1\right ) \left (5 \left (\sin ^{3}\left (f x +e \right )\right )-27 \left (\sin ^{2}\left (f x +e \right )\right )+71 \sin \left (f x +e \right )-177\right )}{35 \cos \left (f x +e \right ) \sqrt {c -c \sin \left (f x +e \right )}\, f}+\frac {2 a^{3} \left (\sin \left (f x +e \right )-1\right ) c^{4} \left (\sin \left (f x +e \right )+1\right ) \left (165 \left (\sin ^{6}\left (f x +e \right )\right )-765 \left (\sin ^{5}\left (f x +e \right )\right )+1565 \left (\sin ^{4}\left (f x +e \right )\right )-2095 \left (\sin ^{3}\left (f x +e \right )\right )+2514 \left (\sin ^{2}\left (f x +e \right )\right )-3352 \sin \left (f x +e \right )+6704\right )}{2145 \cos \left (f x +e \right ) \sqrt {c -c \sin \left (f x +e \right )}\, f}+\frac {2 a^{3} \left (\sin \left (f x +e \right )-1\right ) c^{4} \left (\sin \left (f x +e \right )+1\right ) \left (5 \left (\sin ^{4}\left (f x +e \right )\right )-25 \left (\sin ^{3}\left (f x +e \right )\right )+57 \left (\sin ^{2}\left (f x +e \right )\right )-91 \sin \left (f x +e \right )+182\right )}{15 \cos \left (f x +e \right ) \sqrt {c -c \sin \left (f x +e \right )}\, f}+\frac {2 a^{3} \left (\sin \left (f x +e \right )-1\right ) c^{4} \left (\sin \left (f x +e \right )+1\right ) \left (315 \left (\sin ^{5}\left (f x +e \right )\right )-1505 \left (\sin ^{4}\left (f x +e \right )\right )+3205 \left (\sin ^{3}\left (f x +e \right )\right )-4539 \left (\sin ^{2}\left (f x +e \right )\right )+6052 \sin \left (f x +e \right )-12104\right )}{1155 \cos \left (f x +e \right ) \sqrt {c -c \sin \left (f x +e \right )}\, f}\) | \(374\) |
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Leaf count of result is larger than twice the leaf count of optimal. 265 vs. \(2 (129) = 258\).
Time = 0.26 (sec) , antiderivative size = 265, normalized size of antiderivative = 1.99 \[ \int (3+3 \sin (e+f x))^3 (c-c \sin (e+f x))^{7/2} \, dx=\frac {2 \, {\left (231 \, a^{3} c^{3} \cos \left (f x + e\right )^{7} - 21 \, a^{3} c^{3} \cos \left (f x + e\right )^{6} + 28 \, a^{3} c^{3} \cos \left (f x + e\right )^{5} - 40 \, a^{3} c^{3} \cos \left (f x + e\right )^{4} + 64 \, a^{3} c^{3} \cos \left (f x + e\right )^{3} - 128 \, a^{3} c^{3} \cos \left (f x + e\right )^{2} + 512 \, a^{3} c^{3} \cos \left (f x + e\right ) + 1024 \, a^{3} c^{3} + {\left (231 \, a^{3} c^{3} \cos \left (f x + e\right )^{6} + 252 \, a^{3} c^{3} \cos \left (f x + e\right )^{5} + 280 \, a^{3} c^{3} \cos \left (f x + e\right )^{4} + 320 \, a^{3} c^{3} \cos \left (f x + e\right )^{3} + 384 \, a^{3} c^{3} \cos \left (f x + e\right )^{2} + 512 \, a^{3} c^{3} \cos \left (f x + e\right ) + 1024 \, a^{3} c^{3}\right )} \sin \left (f x + e\right )\right )} \sqrt {-c \sin \left (f x + e\right ) + c}}{3003 \, {\left (f \cos \left (f x + e\right ) - f \sin \left (f x + e\right ) + f\right )}} \]
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Timed out. \[ \int (3+3 \sin (e+f x))^3 (c-c \sin (e+f x))^{7/2} \, dx=\text {Timed out} \]
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\[ \int (3+3 \sin (e+f x))^3 (c-c \sin (e+f x))^{7/2} \, dx=\int { {\left (a \sin \left (f x + e\right ) + a\right )}^{3} {\left (-c \sin \left (f x + e\right ) + c\right )}^{\frac {7}{2}} \,d x } \]
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Time = 0.39 (sec) , antiderivative size = 243, normalized size of antiderivative = 1.83 \[ \int (3+3 \sin (e+f x))^3 (c-c \sin (e+f x))^{7/2} \, dx=-\frac {\sqrt {2} {\left (60060 \, a^{3} c^{3} \cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) + 15015 \, a^{3} c^{3} \cos \left (-\frac {3}{4} \, \pi + \frac {3}{2} \, f x + \frac {3}{2} \, e\right ) \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) - 9009 \, a^{3} c^{3} \cos \left (-\frac {5}{4} \, \pi + \frac {5}{2} \, f x + \frac {5}{2} \, e\right ) \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) - 2574 \, a^{3} c^{3} \cos \left (-\frac {7}{4} \, \pi + \frac {7}{2} \, f x + \frac {7}{2} \, e\right ) \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) + 2002 \, a^{3} c^{3} \cos \left (-\frac {9}{4} \, \pi + \frac {9}{2} \, f x + \frac {9}{2} \, e\right ) \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) + 273 \, a^{3} c^{3} \cos \left (-\frac {11}{4} \, \pi + \frac {11}{2} \, f x + \frac {11}{2} \, e\right ) \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) - 231 \, a^{3} c^{3} \cos \left (-\frac {13}{4} \, \pi + \frac {13}{2} \, f x + \frac {13}{2} \, e\right ) \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )\right )} \sqrt {c}}{96096 \, f} \]
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Timed out. \[ \int (3+3 \sin (e+f x))^3 (c-c \sin (e+f x))^{7/2} \, dx=\int {\left (a+a\,\sin \left (e+f\,x\right )\right )}^3\,{\left (c-c\,\sin \left (e+f\,x\right )\right )}^{7/2} \,d x \]
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