Integrand size = 26, antiderivative size = 137 \[ \int (a+a \sin (e+f x)) (c-c \sin (e+f x))^{7/2} \, dx=\frac {256 a c^5 \cos ^3(e+f x)}{315 f (c-c \sin (e+f x))^{3/2}}+\frac {64 a c^4 \cos ^3(e+f x)}{105 f \sqrt {c-c \sin (e+f x)}}+\frac {8 a c^3 \cos ^3(e+f x) \sqrt {c-c \sin (e+f x)}}{21 f}+\frac {2 a c^2 \cos ^3(e+f x) (c-c \sin (e+f x))^{3/2}}{9 f} \]
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Time = 0.23 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {2815, 2753, 2752} \[ \int (a+a \sin (e+f x)) (c-c \sin (e+f x))^{7/2} \, dx=\frac {256 a c^5 \cos ^3(e+f x)}{315 f (c-c \sin (e+f x))^{3/2}}+\frac {64 a c^4 \cos ^3(e+f x)}{105 f \sqrt {c-c \sin (e+f x)}}+\frac {8 a c^3 \cos ^3(e+f x) \sqrt {c-c \sin (e+f x)}}{21 f}+\frac {2 a c^2 \cos ^3(e+f x) (c-c \sin (e+f x))^{3/2}}{9 f} \]
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Rule 2752
Rule 2753
Rule 2815
Rubi steps \begin{align*} \text {integral}& = (a c) \int \cos ^2(e+f x) (c-c \sin (e+f x))^{5/2} \, dx \\ & = \frac {2 a c^2 \cos ^3(e+f x) (c-c \sin (e+f x))^{3/2}}{9 f}+\frac {1}{3} \left (4 a c^2\right ) \int \cos ^2(e+f x) (c-c \sin (e+f x))^{3/2} \, dx \\ & = \frac {8 a c^3 \cos ^3(e+f x) \sqrt {c-c \sin (e+f x)}}{21 f}+\frac {2 a c^2 \cos ^3(e+f x) (c-c \sin (e+f x))^{3/2}}{9 f}+\frac {1}{21} \left (32 a c^3\right ) \int \cos ^2(e+f x) \sqrt {c-c \sin (e+f x)} \, dx \\ & = \frac {64 a c^4 \cos ^3(e+f x)}{105 f \sqrt {c-c \sin (e+f x)}}+\frac {8 a c^3 \cos ^3(e+f x) \sqrt {c-c \sin (e+f x)}}{21 f}+\frac {2 a c^2 \cos ^3(e+f x) (c-c \sin (e+f x))^{3/2}}{9 f}+\frac {1}{105} \left (128 a c^4\right ) \int \frac {\cos ^2(e+f x)}{\sqrt {c-c \sin (e+f x)}} \, dx \\ & = \frac {256 a c^5 \cos ^3(e+f x)}{315 f (c-c \sin (e+f x))^{3/2}}+\frac {64 a c^4 \cos ^3(e+f x)}{105 f \sqrt {c-c \sin (e+f x)}}+\frac {8 a c^3 \cos ^3(e+f x) \sqrt {c-c \sin (e+f x)}}{21 f}+\frac {2 a c^2 \cos ^3(e+f x) (c-c \sin (e+f x))^{3/2}}{9 f} \\ \end{align*}
Time = 1.39 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.76 \[ \int (a+a \sin (e+f x)) (c-c \sin (e+f x))^{7/2} \, dx=\frac {a c^3 \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^3 \sqrt {c-c \sin (e+f x)} (1606-330 \cos (2 (e+f x))-1389 \sin (e+f x)+35 \sin (3 (e+f x)))}{630 f \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )} \]
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Time = 2.51 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.58
method | result | size |
default | \(\frac {2 \left (\sin \left (f x +e \right )-1\right ) c^{4} \left (\sin \left (f x +e \right )+1\right )^{2} a \left (35 \left (\sin ^{3}\left (f x +e \right )\right )-165 \left (\sin ^{2}\left (f x +e \right )\right )+321 \sin \left (f x +e \right )-319\right )}{315 \cos \left (f x +e \right ) \sqrt {c -c \sin \left (f x +e \right )}\, f}\) | \(79\) |
parts | \(\frac {2 a \left (\sin \left (f x +e \right )-1\right ) c^{4} \left (\sin \left (f x +e \right )+1\right ) \left (5 \left (\sin ^{3}\left (f x +e \right )\right )-27 \left (\sin ^{2}\left (f x +e \right )\right )+71 \sin \left (f x +e \right )-177\right )}{35 \cos \left (f x +e \right ) \sqrt {c -c \sin \left (f x +e \right )}\, f}+\frac {2 a \left (\sin \left (f x +e \right )-1\right ) c^{4} \left (\sin \left (f x +e \right )+1\right ) \left (5 \left (\sin ^{4}\left (f x +e \right )\right )-25 \left (\sin ^{3}\left (f x +e \right )\right )+57 \left (\sin ^{2}\left (f x +e \right )\right )-91 \sin \left (f x +e \right )+182\right )}{45 \cos \left (f x +e \right ) \sqrt {c -c \sin \left (f x +e \right )}\, f}\) | \(164\) |
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Time = 0.28 (sec) , antiderivative size = 179, normalized size of antiderivative = 1.31 \[ \int (a+a \sin (e+f x)) (c-c \sin (e+f x))^{7/2} \, dx=-\frac {2 \, {\left (35 \, a c^{3} \cos \left (f x + e\right )^{5} - 95 \, a c^{3} \cos \left (f x + e\right )^{4} - 226 \, a c^{3} \cos \left (f x + e\right )^{3} + 32 \, a c^{3} \cos \left (f x + e\right )^{2} - 128 \, a c^{3} \cos \left (f x + e\right ) - 256 \, a c^{3} + {\left (35 \, a c^{3} \cos \left (f x + e\right )^{4} + 130 \, a c^{3} \cos \left (f x + e\right )^{3} - 96 \, a c^{3} \cos \left (f x + e\right )^{2} - 128 \, a c^{3} \cos \left (f x + e\right ) - 256 \, a c^{3}\right )} \sin \left (f x + e\right )\right )} \sqrt {-c \sin \left (f x + e\right ) + c}}{315 \, {\left (f \cos \left (f x + e\right ) - f \sin \left (f x + e\right ) + f\right )}} \]
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Timed out. \[ \int (a+a \sin (e+f x)) (c-c \sin (e+f x))^{7/2} \, dx=\text {Timed out} \]
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\[ \int (a+a \sin (e+f x)) (c-c \sin (e+f x))^{7/2} \, dx=\int { {\left (a \sin \left (f x + e\right ) + a\right )} {\left (-c \sin \left (f x + e\right ) + c\right )}^{\frac {7}{2}} \,d x } \]
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Time = 0.38 (sec) , antiderivative size = 136, normalized size of antiderivative = 0.99 \[ \int (a+a \sin (e+f x)) (c-c \sin (e+f x))^{7/2} \, dx=-\frac {\sqrt {2} {\left (4410 \, a c^{3} \cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) - 504 \, a c^{3} \cos \left (-\frac {5}{4} \, \pi + \frac {5}{2} \, f x + \frac {5}{2} \, e\right ) \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) + 225 \, a c^{3} \cos \left (-\frac {7}{4} \, \pi + \frac {7}{2} \, f x + \frac {7}{2} \, e\right ) \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) - 35 \, a c^{3} \cos \left (-\frac {9}{4} \, \pi + \frac {9}{2} \, f x + \frac {9}{2} \, e\right ) \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )\right )} \sqrt {c}}{2520 \, f} \]
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Timed out. \[ \int (a+a \sin (e+f x)) (c-c \sin (e+f x))^{7/2} \, dx=\int \left (a+a\,\sin \left (e+f\,x\right )\right )\,{\left (c-c\,\sin \left (e+f\,x\right )\right )}^{7/2} \,d x \]
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