Integrand size = 26, antiderivative size = 69 \[ \int (a+a \sin (e+f x)) (c-c \sin (e+f x))^{3/2} \, dx=\frac {8 a c^3 \cos ^3(e+f x)}{15 f (c-c \sin (e+f x))^{3/2}}+\frac {2 a c^2 \cos ^3(e+f x)}{5 f \sqrt {c-c \sin (e+f x)}} \]
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Time = 0.11 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {2815, 2753, 2752} \[ \int (a+a \sin (e+f x)) (c-c \sin (e+f x))^{3/2} \, dx=\frac {8 a c^3 \cos ^3(e+f x)}{15 f (c-c \sin (e+f x))^{3/2}}+\frac {2 a c^2 \cos ^3(e+f x)}{5 f \sqrt {c-c \sin (e+f x)}} \]
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Rule 2752
Rule 2753
Rule 2815
Rubi steps \begin{align*} \text {integral}& = (a c) \int \cos ^2(e+f x) \sqrt {c-c \sin (e+f x)} \, dx \\ & = \frac {2 a c^2 \cos ^3(e+f x)}{5 f \sqrt {c-c \sin (e+f x)}}+\frac {1}{5} \left (4 a c^2\right ) \int \frac {\cos ^2(e+f x)}{\sqrt {c-c \sin (e+f x)}} \, dx \\ & = \frac {8 a c^3 \cos ^3(e+f x)}{15 f (c-c \sin (e+f x))^{3/2}}+\frac {2 a c^2 \cos ^3(e+f x)}{5 f \sqrt {c-c \sin (e+f x)}} \\ \end{align*}
Time = 0.72 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.19 \[ \int (a+a \sin (e+f x)) (c-c \sin (e+f x))^{3/2} \, dx=-\frac {2 a c \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^3 (-7+3 \sin (e+f x)) \sqrt {c-c \sin (e+f x)}}{15 f \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )} \]
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Time = 1.20 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.86
method | result | size |
default | \(\frac {2 \left (\sin \left (f x +e \right )-1\right ) c^{2} \left (\sin \left (f x +e \right )+1\right )^{2} a \left (3 \sin \left (f x +e \right )-7\right )}{15 \cos \left (f x +e \right ) \sqrt {c -c \sin \left (f x +e \right )}\, f}\) | \(59\) |
parts | \(\frac {2 a \left (\sin \left (f x +e \right )-1\right ) c^{2} \left (\sin \left (f x +e \right )+1\right ) \left (\sin \left (f x +e \right )-5\right )}{3 \cos \left (f x +e \right ) \sqrt {c -c \sin \left (f x +e \right )}\, f}+\frac {2 a \left (\sin \left (f x +e \right )-1\right ) c^{2} \left (\sin \left (f x +e \right )+1\right ) \left (\sin ^{2}\left (f x +e \right )-3 \sin \left (f x +e \right )+6\right )}{5 \cos \left (f x +e \right ) \sqrt {c -c \sin \left (f x +e \right )}\, f}\) | \(120\) |
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Time = 0.27 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.58 \[ \int (a+a \sin (e+f x)) (c-c \sin (e+f x))^{3/2} \, dx=\frac {2 \, {\left (3 \, a c \cos \left (f x + e\right )^{3} - a c \cos \left (f x + e\right )^{2} + 4 \, a c \cos \left (f x + e\right ) + 8 \, a c + {\left (3 \, a c \cos \left (f x + e\right )^{2} + 4 \, a c \cos \left (f x + e\right ) + 8 \, a c\right )} \sin \left (f x + e\right )\right )} \sqrt {-c \sin \left (f x + e\right ) + c}}{15 \, {\left (f \cos \left (f x + e\right ) - f \sin \left (f x + e\right ) + f\right )}} \]
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\[ \int (a+a \sin (e+f x)) (c-c \sin (e+f x))^{3/2} \, dx=a \left (\int c \sqrt {- c \sin {\left (e + f x \right )} + c}\, dx + \int \left (- c \sqrt {- c \sin {\left (e + f x \right )} + c} \sin ^{2}{\left (e + f x \right )}\right )\, dx\right ) \]
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\[ \int (a+a \sin (e+f x)) (c-c \sin (e+f x))^{3/2} \, dx=\int { {\left (a \sin \left (f x + e\right ) + a\right )} {\left (-c \sin \left (f x + e\right ) + c\right )}^{\frac {3}{2}} \,d x } \]
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Time = 0.34 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.43 \[ \int (a+a \sin (e+f x)) (c-c \sin (e+f x))^{3/2} \, dx=-\frac {\sqrt {2} {\left (30 \, a c \cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) + 5 \, a c \cos \left (-\frac {3}{4} \, \pi + \frac {3}{2} \, f x + \frac {3}{2} \, e\right ) \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) - 3 \, a c \cos \left (-\frac {5}{4} \, \pi + \frac {5}{2} \, f x + \frac {5}{2} \, e\right ) \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )\right )} \sqrt {c}}{30 \, f} \]
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Timed out. \[ \int (a+a \sin (e+f x)) (c-c \sin (e+f x))^{3/2} \, dx=\int \left (a+a\,\sin \left (e+f\,x\right )\right )\,{\left (c-c\,\sin \left (e+f\,x\right )\right )}^{3/2} \,d x \]
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