Integrand size = 26, antiderivative size = 103 \[ \int (a+a \sin (e+f x)) (c-c \sin (e+f x))^{5/2} \, dx=\frac {64 a c^4 \cos ^3(e+f x)}{105 f (c-c \sin (e+f x))^{3/2}}+\frac {16 a c^3 \cos ^3(e+f x)}{35 f \sqrt {c-c \sin (e+f x)}}+\frac {2 a c^2 \cos ^3(e+f x) \sqrt {c-c \sin (e+f x)}}{7 f} \]
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Time = 0.17 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {2815, 2753, 2752} \[ \int (a+a \sin (e+f x)) (c-c \sin (e+f x))^{5/2} \, dx=\frac {64 a c^4 \cos ^3(e+f x)}{105 f (c-c \sin (e+f x))^{3/2}}+\frac {16 a c^3 \cos ^3(e+f x)}{35 f \sqrt {c-c \sin (e+f x)}}+\frac {2 a c^2 \cos ^3(e+f x) \sqrt {c-c \sin (e+f x)}}{7 f} \]
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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))^{3/2} \, dx \\ & = \frac {2 a c^2 \cos ^3(e+f x) \sqrt {c-c \sin (e+f x)}}{7 f}+\frac {1}{7} \left (8 a c^2\right ) \int \cos ^2(e+f x) \sqrt {c-c \sin (e+f x)} \, dx \\ & = \frac {16 a c^3 \cos ^3(e+f x)}{35 f \sqrt {c-c \sin (e+f x)}}+\frac {2 a c^2 \cos ^3(e+f x) \sqrt {c-c \sin (e+f x)}}{7 f}+\frac {1}{35} \left (32 a c^3\right ) \int \frac {\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 (c-c \sin (e+f x))^{3/2}}+\frac {16 a c^3 \cos ^3(e+f x)}{35 f \sqrt {c-c \sin (e+f x)}}+\frac {2 a c^2 \cos ^3(e+f x) \sqrt {c-c \sin (e+f x)}}{7 f} \\ \end{align*}
Time = 0.91 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.91 \[ \int (a+a \sin (e+f x)) (c-c \sin (e+f x))^{5/2} \, dx=-\frac {a c^2 \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^3 (-157+15 \cos (2 (e+f x))+108 \sin (e+f x)) \sqrt {c-c \sin (e+f x)}}{105 f \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )} \]
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Time = 2.06 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.67
method | result | size |
default | \(-\frac {2 \left (\sin \left (f x +e \right )-1\right ) c^{3} \left (\sin \left (f x +e \right )+1\right )^{2} a \left (15 \left (\sin ^{2}\left (f x +e \right )\right )-54 \sin \left (f x +e \right )+71\right )}{105 \cos \left (f x +e \right ) \sqrt {c -c \sin \left (f x +e \right )}\, f}\) | \(69\) |
parts | \(-\frac {2 a \left (\sin \left (f x +e \right )-1\right ) c^{3} \left (\sin \left (f x +e \right )+1\right ) \left (3 \left (\sin ^{2}\left (f x +e \right )\right )-14 \sin \left (f x +e \right )+43\right )}{15 \cos \left (f x +e \right ) \sqrt {c -c \sin \left (f x +e \right )}\, f}-\frac {2 a \left (\sin \left (f x +e \right )-1\right ) c^{3} \left (\sin \left (f x +e \right )+1\right ) \left (3 \left (\sin ^{3}\left (f x +e \right )\right )-12 \left (\sin ^{2}\left (f x +e \right )\right )+23 \sin \left (f x +e \right )-46\right )}{21 \cos \left (f x +e \right ) \sqrt {c -c \sin \left (f x +e \right )}\, f}\) | \(144\) |
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Time = 0.25 (sec) , antiderivative size = 152, normalized size of antiderivative = 1.48 \[ \int (a+a \sin (e+f x)) (c-c \sin (e+f x))^{5/2} \, dx=\frac {2 \, {\left (15 \, a c^{2} \cos \left (f x + e\right )^{4} + 39 \, a c^{2} \cos \left (f x + e\right )^{3} - 8 \, a c^{2} \cos \left (f x + e\right )^{2} + 32 \, a c^{2} \cos \left (f x + e\right ) + 64 \, a c^{2} - {\left (15 \, a c^{2} \cos \left (f x + e\right )^{3} - 24 \, a c^{2} \cos \left (f x + e\right )^{2} - 32 \, a c^{2} \cos \left (f x + e\right ) - 64 \, a c^{2}\right )} \sin \left (f x + e\right )\right )} \sqrt {-c \sin \left (f x + e\right ) + c}}{105 \, {\left (f \cos \left (f x + e\right ) - f \sin \left (f x + e\right ) + f\right )}} \]
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\[ \int (a+a \sin (e+f x)) (c-c \sin (e+f x))^{5/2} \, dx=a \left (\int c^{2} \sqrt {- c \sin {\left (e + f x \right )} + c}\, dx + \int \left (- c^{2} \sqrt {- c \sin {\left (e + f x \right )} + c} \sin {\left (e + f x \right )}\right )\, dx + \int \left (- c^{2} \sqrt {- c \sin {\left (e + f x \right )} + c} \sin ^{2}{\left (e + f x \right )}\right )\, dx + \int c^{2} \sqrt {- c \sin {\left (e + f x \right )} + c} \sin ^{3}{\left (e + f x \right )}\, dx\right ) \]
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\[ \int (a+a \sin (e+f x)) (c-c \sin (e+f x))^{5/2} \, dx=\int { {\left (a \sin \left (f x + e\right ) + a\right )} {\left (-c \sin \left (f x + e\right ) + c\right )}^{\frac {5}{2}} \,d x } \]
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Time = 0.38 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.32 \[ \int (a+a \sin (e+f x)) (c-c \sin (e+f x))^{5/2} \, dx=-\frac {\sqrt {2} {\left (525 \, a c^{2} \cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) + 35 \, a c^{2} \cos \left (-\frac {3}{4} \, \pi + \frac {3}{2} \, f x + \frac {3}{2} \, e\right ) \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) - 63 \, a c^{2} \cos \left (-\frac {5}{4} \, \pi + \frac {5}{2} \, f x + \frac {5}{2} \, e\right ) \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) + 15 \, a c^{2} \cos \left (-\frac {7}{4} \, \pi + \frac {7}{2} \, f x + \frac {7}{2} \, e\right ) \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )\right )} \sqrt {c}}{420 \, f} \]
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Timed out. \[ \int (a+a \sin (e+f x)) (c-c \sin (e+f x))^{5/2} \, dx=\int \left (a+a\,\sin \left (e+f\,x\right )\right )\,{\left (c-c\,\sin \left (e+f\,x\right )\right )}^{5/2} \,d x \]
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