Integrand size = 28, antiderivative size = 145 \[ \int (a+a \sin (e+f x))^2 (c-c \sin (e+f x))^{7/2} \, dx=\frac {256 a^2 c^6 \cos ^5(e+f x)}{1155 f (c-c \sin (e+f x))^{5/2}}+\frac {64 a^2 c^5 \cos ^5(e+f x)}{231 f (c-c \sin (e+f x))^{3/2}}+\frac {8 a^2 c^4 \cos ^5(e+f x)}{33 f \sqrt {c-c \sin (e+f x)}}+\frac {2 a^2 c^3 \cos ^5(e+f x) \sqrt {c-c \sin (e+f x)}}{11 f} \]
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Time = 0.24 (sec) , antiderivative size = 145, normalized size of antiderivative = 1.00, 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} \[ \int (a+a \sin (e+f x))^2 (c-c \sin (e+f x))^{7/2} \, dx=\frac {256 a^2 c^6 \cos ^5(e+f x)}{1155 f (c-c \sin (e+f x))^{5/2}}+\frac {64 a^2 c^5 \cos ^5(e+f x)}{231 f (c-c \sin (e+f x))^{3/2}}+\frac {8 a^2 c^4 \cos ^5(e+f x)}{33 f \sqrt {c-c \sin (e+f x)}}+\frac {2 a^2 c^3 \cos ^5(e+f x) \sqrt {c-c \sin (e+f x)}}{11 f} \]
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Rule 2752
Rule 2753
Rule 2815
Rubi steps \begin{align*} \text {integral}& = \left (a^2 c^2\right ) \int \cos ^4(e+f x) (c-c \sin (e+f x))^{3/2} \, dx \\ & = \frac {2 a^2 c^3 \cos ^5(e+f x) \sqrt {c-c \sin (e+f x)}}{11 f}+\frac {1}{11} \left (12 a^2 c^3\right ) \int \cos ^4(e+f x) \sqrt {c-c \sin (e+f x)} \, dx \\ & = \frac {8 a^2 c^4 \cos ^5(e+f x)}{33 f \sqrt {c-c \sin (e+f x)}}+\frac {2 a^2 c^3 \cos ^5(e+f x) \sqrt {c-c \sin (e+f x)}}{11 f}+\frac {1}{33} \left (32 a^2 c^4\right ) \int \frac {\cos ^4(e+f x)}{\sqrt {c-c \sin (e+f x)}} \, dx \\ & = \frac {64 a^2 c^5 \cos ^5(e+f x)}{231 f (c-c \sin (e+f x))^{3/2}}+\frac {8 a^2 c^4 \cos ^5(e+f x)}{33 f \sqrt {c-c \sin (e+f x)}}+\frac {2 a^2 c^3 \cos ^5(e+f x) \sqrt {c-c \sin (e+f x)}}{11 f}+\frac {1}{231} \left (128 a^2 c^5\right ) \int \frac {\cos ^4(e+f x)}{(c-c \sin (e+f x))^{3/2}} \, dx \\ & = \frac {256 a^2 c^6 \cos ^5(e+f x)}{1155 f (c-c \sin (e+f x))^{5/2}}+\frac {64 a^2 c^5 \cos ^5(e+f x)}{231 f (c-c \sin (e+f x))^{3/2}}+\frac {8 a^2 c^4 \cos ^5(e+f x)}{33 f \sqrt {c-c \sin (e+f x)}}+\frac {2 a^2 c^3 \cos ^5(e+f x) \sqrt {c-c \sin (e+f x)}}{11 f} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(1105\) vs. \(2(145)=290\).
Time = 10.84 (sec) , antiderivative size = 1105, normalized size of antiderivative = 7.62 \[ \int (a+a \sin (e+f x))^2 (c-c \sin (e+f x))^{7/2} \, dx=\frac {7 \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 {\cos \left (\frac {3}{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 {11 \cos \left (\frac {5}{2} (e+f x)\right ) (a+a \sin (e+f x))^2 (c-c \sin (e+f x))^{7/2}}{80 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 {\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 {\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 {\cos \left (\frac {11}{2} (e+f x)\right ) (a+a \sin (e+f x))^2 (c-c \sin (e+f x))^{7/2}}{176 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 \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 {(a+a \sin (e+f x))^2 (c-c \sin (e+f x))^{7/2} \sin \left (\frac {3}{2} (e+f x)\right )}{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 {11 (a+a \sin (e+f x))^2 (c-c \sin (e+f x))^{7/2} \sin \left (\frac {5}{2} (e+f x)\right )}{80 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 \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 \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 {(a+a \sin (e+f x))^2 (c-c \sin (e+f x))^{7/2} \sin \left (\frac {11}{2} (e+f x)\right )}{176 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 = 7.74 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.56
method | result | size |
default | \(\frac {2 \left (\sin \left (f x +e \right )-1\right ) c^{4} \left (\sin \left (f x +e \right )+1\right )^{3} a^{2} \left (105 \left (\sin ^{3}\left (f x +e \right )\right )-455 \left (\sin ^{2}\left (f x +e \right )\right )+755 \sin \left (f x +e \right )-533\right )}{1155 \cos \left (f x +e \right ) \sqrt {c -c \sin \left (f x +e \right )}\, f}\) | \(81\) |
parts | \(\frac {2 a^{2} \left (\sin \left (f x +e \right )-1\right ) c^{4} \left (\sin \left (f x +e \right )+1\right ) \left (5 \left (\sin ^{3}\left (f x +e \right )\right )-27 \left (\sin ^{2}\left (f x +e \right )\right )+71 \sin \left (f x +e \right )-177\right )}{35 \cos \left (f x +e \right ) \sqrt {c -c \sin \left (f x +e \right )}\, f}+\frac {2 a^{2} \left (\sin \left (f x +e \right )-1\right ) c^{4} \left (\sin \left (f x +e \right )+1\right ) \left (315 \left (\sin ^{5}\left (f x +e \right )\right )-1505 \left (\sin ^{4}\left (f x +e \right )\right )+3205 \left (\sin ^{3}\left (f x +e \right )\right )-4539 \left (\sin ^{2}\left (f x +e \right )\right )+6052 \sin \left (f x +e \right )-12104\right )}{3465 \cos \left (f x +e \right ) \sqrt {c -c \sin \left (f x +e \right )}\, f}+\frac {4 a^{2} \left (\sin \left (f x +e \right )-1\right ) c^{4} \left (\sin \left (f x +e \right )+1\right ) \left (5 \left (\sin ^{4}\left (f x +e \right )\right )-25 \left (\sin ^{3}\left (f x +e \right )\right )+57 \left (\sin ^{2}\left (f x +e \right )\right )-91 \sin \left (f x +e \right )+182\right )}{45 \cos \left (f x +e \right ) \sqrt {c -c \sin \left (f x +e \right )}\, f}\) | \(266\) |
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Time = 0.36 (sec) , antiderivative size = 234, normalized size of antiderivative = 1.61 \[ \int (a+a \sin (e+f x))^2 (c-c \sin (e+f x))^{7/2} \, dx=\frac {2 \, {\left (105 \, a^{2} c^{3} \cos \left (f x + e\right )^{6} + 245 \, a^{2} c^{3} \cos \left (f x + e\right )^{5} - 20 \, a^{2} c^{3} \cos \left (f x + e\right )^{4} + 32 \, a^{2} c^{3} \cos \left (f x + e\right )^{3} - 64 \, a^{2} c^{3} \cos \left (f x + e\right )^{2} + 256 \, a^{2} c^{3} \cos \left (f x + e\right ) + 512 \, a^{2} c^{3} - {\left (105 \, a^{2} c^{3} \cos \left (f x + e\right )^{5} - 140 \, a^{2} c^{3} \cos \left (f x + e\right )^{4} - 160 \, a^{2} c^{3} \cos \left (f x + e\right )^{3} - 192 \, a^{2} c^{3} \cos \left (f x + e\right )^{2} - 256 \, a^{2} c^{3} \cos \left (f x + e\right ) - 512 \, a^{2} c^{3}\right )} \sin \left (f x + e\right )\right )} \sqrt {-c \sin \left (f x + e\right ) + c}}{1155 \, {\left (f \cos \left (f x + e\right ) - f \sin \left (f x + e\right ) + f\right )}} \]
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Timed out. \[ \int (a+a \sin (e+f x))^2 (c-c \sin (e+f x))^{7/2} \, dx=\text {Timed out} \]
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\[ \int (a+a \sin (e+f x))^2 (c-c \sin (e+f x))^{7/2} \, dx=\int { {\left (a \sin \left (f x + e\right ) + a\right )}^{2} {\left (-c \sin \left (f x + e\right ) + c\right )}^{\frac {7}{2}} \,d x } \]
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Time = 0.42 (sec) , antiderivative size = 210, normalized size of antiderivative = 1.45 \[ \int (a+a \sin (e+f x))^2 (c-c \sin (e+f x))^{7/2} \, dx=-\frac {\sqrt {2} {\left (16170 \, a^{2} c^{3} \cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) + 2310 \, a^{2} c^{3} \cos \left (-\frac {3}{4} \, \pi + \frac {3}{2} \, f x + \frac {3}{2} \, e\right ) \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) - 2541 \, a^{2} c^{3} \cos \left (-\frac {5}{4} \, \pi + \frac {5}{2} \, f x + \frac {5}{2} \, e\right ) \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) + 165 \, a^{2} c^{3} \cos \left (-\frac {7}{4} \, \pi + \frac {7}{2} \, f x + \frac {7}{2} \, e\right ) \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) + 385 \, a^{2} c^{3} \cos \left (-\frac {9}{4} \, \pi + \frac {9}{2} \, f x + \frac {9}{2} \, e\right ) \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) - 105 \, a^{2} c^{3} \cos \left (-\frac {11}{4} \, \pi + \frac {11}{2} \, f x + \frac {11}{2} \, e\right ) \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )\right )} \sqrt {c}}{18480 \, f} \]
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Timed out. \[ \int (a+a \sin (e+f x))^2 (c-c \sin (e+f x))^{7/2} \, dx=\int {\left (a+a\,\sin \left (e+f\,x\right )\right )}^2\,{\left (c-c\,\sin \left (e+f\,x\right )\right )}^{7/2} \,d x \]
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